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miarc.c

/* $XFree86: xc/programs/Xserver/mi/miarc.c,v 3.13 2002/01/12 22:20:33 dawes Exp $ */
/***********************************************************

Copyright 1987, 1998  The Open Group

Permission to use, copy, modify, distribute, and sell this software and its
documentation for any purpose is hereby granted without fee, provided that
the above copyright notice appear in all copies and that both that
copyright notice and this permission notice appear in supporting
documentation.

The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
OPEN GROUP BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN
AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Except as contained in this notice, the name of The Open Group shall not be
used in advertising or otherwise to promote the sale, use or other dealings
in this Software without prior written authorization from The Open Group.


Copyright 1987 by Digital Equipment Corporation, Maynard, Massachusetts.

                        All Rights Reserved

Permission to use, copy, modify, and distribute this software and its 
documentation for any purpose and without fee is hereby granted, 
provided that the above copyright notice appear in all copies and that
both that copyright notice and this permission notice appear in 
supporting documentation, and that the name of Digital not be
used in advertising or publicity pertaining to distribution of the
software without specific, written prior permission.  

DIGITAL DISCLAIMS ALL WARRANTIES WITH REGARD TO THIS SOFTWARE, INCLUDING
ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS, IN NO EVENT SHALL
DIGITAL BE LIABLE FOR ANY SPECIAL, INDIRECT OR CONSEQUENTIAL DAMAGES OR
ANY DAMAGES WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS,
WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION,
ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS
SOFTWARE.

******************************************************************/
/* $Xorg: miarc.c,v 1.4 2001/02/09 02:05:20 xorgcvs Exp $ */
/* Author: Keith Packard and Bob Scheifler */
/* Warning: this code is toxic, do not dally very long here. */

#if defined(_XOPEN_SOURCE) || defined(__QNXNTO__)
#include <math.h>
#else
#define _XOPEN_SOURCE   /* to get prototype for hypot on some systems */
#include <math.h>
#undef _XOPEN_SOURCE
#endif
#include "X.h"
#include "Xprotostr.h"
#include "misc.h"
#include "gcstruct.h"
#include "scrnintstr.h"
#include "pixmapstr.h"
#include "windowstr.h"
#include "mifpoly.h"
#include "mi.h"
#include "mifillarc.h"
#include "Xfuncproto.h"

static double miDsin(double a);
static double miDcos(double a);
static double miDasin(double v);
static double miDatan2(double dy, double dx);
double      cbrt(double);

#ifdef ICEILTEMPDECL
ICEILTEMPDECL
#endif

/*
 * some interesting sematic interpretation of the protocol:
 *
 * Self intersecting arcs (i.e. those spanning 360 degrees) 
 *  never join with other arcs, and are drawn without caps
 *  (unless on/off dashed, in which case each dash segment
 *  is capped, except when the last segment meets the
 *  first segment, when no caps are drawn)
 *
 * double dash arcs are drawn in two parts, first the
 *  odd dashes (drawn in background) then the even dashes
 *  (drawn in foreground).  This means that overlapping
 *  sections of foreground/background are drawn twice,
 *  first in background then in foreground.  The double-draw
 *  occurs even when the function uses the destination values
 *  (e.g. xor mode).  This is the same way the wide-line
 *  code works and should be "fixed".
 *
 */

#undef max
#undef min

#if defined (__GNUC__) && !defined (__STRICT_ANSI__)
#define USE_INLINE
#endif

#ifdef USE_INLINE
inline static const int max (const int x, const int y)
{
      return x>y? x:y;
}

inline static const int min (const int x, const int y)
{
      return x<y? x:y;
}

#else

static int
max (int x, int y)
{
      return x>y? x:y;
}

static int
min (int x, int y)
{
      return x<y? x:y;
}

#endif

struct bound {
      double      min, max;
};

struct ibound {
      int   min, max;
};

#define boundedLe(value, bounds)\
      ((bounds).min <= (value) && (value) <= (bounds).max)

struct line {
      double      m, b;
      int   valid;
};

#define intersectLine(y,line) (line.m * (y) + line.b)

/*
 * these are all y value bounds
 */

struct arc_bound {
      struct bound      ellipse;
      struct bound      inner;
      struct bound      outer;
      struct bound      right;
      struct bound      left;
      struct ibound     inneri;
      struct ibound     outeri;
};

struct accelerators {
      double            tail_y;
      double            h2;
      double            w2;
      double            h4;
      double            w4;
      double            h2mw2;
      double            h2l;
      double            w2l;
      double            fromIntX;
      double            fromIntY;
      struct line left, right;
      int         yorgu;
      int         yorgl;
      int         xorg;
};

struct arc_def {
      double      w, h, l;
      double      a0, a1;
};

# define todeg(xAngle)  (((double) (xAngle)) / 64.0)

# define RIGHT_END      0
# define LEFT_END 1

typedef struct _miArcJoin {
      int   arcIndex0, arcIndex1;
      int   phase0, phase1;
      int   end0, end1;
} miArcJoinRec, *miArcJoinPtr;

typedef struct _miArcCap {
      int         arcIndex;
      int         end;        
} miArcCapRec, *miArcCapPtr;

typedef struct _miArcFace {
      SppPointRec clock;
      SppPointRec center;
      SppPointRec counterClock;
} miArcFaceRec, *miArcFacePtr;

typedef struct _miArcData {
      xArc        arc;
      int         render;           /* non-zero means render after drawing */
      int         join;       /* related join */
      int         cap;        /* related cap */
      int         selfJoin;   /* final dash meets first dash */
      miArcFaceRec      bounds[2];
      double            x0, y0, x1, y1;
} miArcDataRec, *miArcDataPtr;

/*
 * This is an entire sequence of arcs, computed and categorized according
 * to operation.  miDashArcs generates either one or two of these.
 */

typedef struct _miPolyArc {
      int         narcs;
      miArcDataPtr      arcs;
      int         ncaps;
      miArcCapPtr caps;
      int         njoins;
      miArcJoinPtr      joins;
} miPolyArcRec, *miPolyArcPtr;

#define GCValsFunction        0
#define GCValsForeground      1
#define GCValsBackground      2
#define GCValsLineWidth       3
#define GCValsCapStyle        4
#define GCValsJoinStyle       5
#define GCValsMask            (GCFunction | GCForeground | GCBackground | \
                         GCLineWidth | GCCapStyle | GCJoinStyle)
static CARD32 gcvals[6];

static void fillSpans(DrawablePtr pDrawable, GCPtr pGC);
static void newFinalSpan(int y, register int xmin, register int xmax);
static void drawArc(xArc *tarc, int l, int a0, int a1, miArcFacePtr right,
                miArcFacePtr left);
static void drawZeroArc(DrawablePtr pDraw, GCPtr pGC, xArc *tarc, int lw,
                  miArcFacePtr left, miArcFacePtr right);
static void miArcJoin(DrawablePtr pDraw, GCPtr pGC, miArcFacePtr pLeft,
                  miArcFacePtr pRight, int xOrgLeft, int yOrgLeft,
                  double xFtransLeft, double yFtransLeft,
                  int xOrgRight, int yOrgRight,
                  double xFtransRight, double yFtransRight);
static void miArcCap(DrawablePtr pDraw, GCPtr pGC, miArcFacePtr pFace,
                 int end, int xOrg, int yOrg, double xFtrans,
                 double yFtrans);
static void miRoundCap(DrawablePtr pDraw, GCPtr pGC, SppPointRec pCenter,
                   SppPointRec pEnd, SppPointRec pCorner,
                   SppPointRec pOtherCorner, int fLineEnd,
                   int xOrg, int yOrg, double xFtrans, double yFtrans);
static void miFreeArcs(miPolyArcPtr arcs, GCPtr pGC);
static miPolyArcPtr miComputeArcs(xArc *parcs, int narcs, GCPtr pGC);
static int miGetArcPts(SppArcPtr parc, int cpt, SppPointPtr *ppPts);

# define CUBED_ROOT_2   1.2599210498948732038115849718451499938964
# define CUBED_ROOT_4   1.5874010519681993173435330390930175781250

/*
 * draw one segment of the arc using the arc spans generation routines
 */

static void
miArcSegment(
    DrawablePtr   pDraw,
    GCPtr         pGC,
    xArc          tarc,
    miArcFacePtr  right,
    miArcFacePtr  left)
{
    int l = pGC->lineWidth;
    int a0, a1, startAngle, endAngle;
    miArcFacePtr  temp;

    if (!l)
      l = 1;

    if (tarc.width == 0 || tarc.height == 0) {
      drawZeroArc (pDraw, pGC, &tarc, l, left, right);
      return;
    }

    if (pGC->miTranslate) {
      tarc.x += pDraw->x;
      tarc.y += pDraw->y;
    }

    a0 = tarc.angle1;
    a1 = tarc.angle2;
    if (a1 > FULLCIRCLE)
      a1 = FULLCIRCLE;
    else if (a1 < -FULLCIRCLE)
      a1 = -FULLCIRCLE;
    if (a1 < 0) {
      startAngle = a0 + a1;
      endAngle = a0;
      temp = right;
      right = left;
      left = temp;
    } else {
      startAngle = a0;
      endAngle = a0 + a1;
    }
    /*
     * bounds check the two angles
     */
    if (startAngle < 0)
      startAngle = FULLCIRCLE - (-startAngle) % FULLCIRCLE;
    if (startAngle >= FULLCIRCLE)
      startAngle = startAngle % FULLCIRCLE;
    if (endAngle < 0)
      endAngle = FULLCIRCLE - (-endAngle) % FULLCIRCLE;
    if (endAngle > FULLCIRCLE)
      endAngle = (endAngle-1) % FULLCIRCLE + 1;
    if ((startAngle == endAngle) && a1) {
      startAngle = 0;
      endAngle = FULLCIRCLE;
    }

    drawArc (&tarc, l, startAngle, endAngle, right, left);
}

/*

Three equations combine to describe the boundaries of the arc

x^2/w^2 + y^2/h^2 = 1               ellipse itself
(X-x)^2 + (Y-y)^2 = r^2             circle at (x, y) on the ellipse
(Y-y) = (X-x)*w^2*y/(h^2*x)         normal at (x, y) on the ellipse

These lead to a quartic relating Y and y

y^4 - (2Y)y^3 + (Y^2 + (h^4 - w^2*r^2)/(w^2 - h^2))y^2
    - (2Y*h^4/(w^2 - h^2))y + (Y^2*h^4)/(w^2 - h^2) = 0

The reducible cubic obtained from this quartic is

z^3 - (3N)z^2 - 2V = 0

where

N = (Y^2 + (h^4 - w^2*r^2/(w^2 - h^2)))/6
V = w^2*r^2*Y^2*h^4/(4 *(w^2 - h^2)^2)

Let

t = z - N
p = -N^2
q = -N^3 - V

Then we get

t^3 + 3pt + 2q = 0

The discriminant of this cubic is

D = q^2 + p^3

When D > 0, a real root is obtained as

z = N + cbrt(-q+sqrt(D)) + cbrt(-q-sqrt(D))

When D < 0, a real root is obtained as

z = N - 2m*cos(acos(-q/m^3)/3)

where

m = sqrt(|p|) * sign(q)

Given a real root Z of the cubic, the roots of the quartic are the roots
of the two quadratics

y^2 + ((b+A)/2)y + (Z + (bZ - d)/A) = 0

where 

A = +/- sqrt(8Z + b^2 - 4c)
b, c, d are the cubic, quadratic, and linear coefficients of the quartic

Some experimentation is then required to determine which solutions
correspond to the inner and outer boundaries.

*/

typedef struct {
    short lx, lw, rx, rw;
} miArcSpan;

typedef struct {
    miArcSpan *spans;
    int count1, count2, k;
    char top, bot, hole;
} miArcSpanData;

typedef struct {
    unsigned long lrustamp;
    unsigned short lw;
    unsigned short width, height;
    miArcSpanData *spdata;
} arcCacheRec;

#define CACHESIZE 25

static void drawQuadrant(struct arc_def *def, struct accelerators *acc,
                   int a0, int a1, int mask, miArcFacePtr right,
                   miArcFacePtr left, miArcSpanData *spdata);

static arcCacheRec arcCache[CACHESIZE];
static unsigned long lrustamp;
static arcCacheRec *lastCacheHit = &arcCache[0];
static RESTYPE cacheType;

/*
 * External so it can be called when low on memory.
 * Call with a zero ID in that case.
 */
/*ARGSUSED*/
int
miFreeArcCache (data, id)
    pointer     data;
    XID               id;
{
    int k;
    arcCacheRec *cent;

    if (id)
      cacheType = 0;

    for (k = CACHESIZE, cent = &arcCache[0]; --k >= 0; cent++)
    {
      if (cent->spdata)
      {
          cent->lrustamp = 0;
          cent->lw = 0;
          xfree(cent->spdata);
          cent->spdata = NULL;
      }
    }
    lrustamp = 0;
    return Success;
}

static void
miComputeCircleSpans(
    int lw,
    xArc *parc,
    miArcSpanData *spdata)
{
    register miArcSpan *span;
    int doinner;
    register int x, y, e;
    int xk, yk, xm, ym, dx, dy;
    register int slw, inslw;
    int inx = 0, iny, ine = 0;
    int inxk = 0, inyk = 0, inxm = 0, inym = 0;

    doinner = -lw;
    slw = parc->width - doinner;
    y = parc->height >> 1;
    dy = parc->height & 1;
    dx = 1 - dy;
    MIWIDEARCSETUP(x, y, dy, slw, e, xk, xm, yk, ym);
    inslw = parc->width + doinner;
    if (inslw > 0)
    {
      spdata->hole = spdata->top;
      MIWIDEARCSETUP(inx, iny, dy, inslw, ine, inxk, inxm, inyk, inym);
    }
    else
    {
      spdata->hole = FALSE;
      doinner = -y;
    }
    spdata->count1 = -doinner - spdata->top;
    spdata->count2 = y + doinner;
    span = spdata->spans;
    while (y)
    {
      MIFILLARCSTEP(slw);
      span->lx = dy - x;
      if (++doinner <= 0)
      {
          span->lw = slw;
          span->rx = 0;
          span->rw = span->lx + slw;
      }
      else
      {
          MIFILLINARCSTEP(inslw);
          span->lw = x - inx;
          span->rx = dy - inx + inslw;
          span->rw = inx - x + slw - inslw;
      }
      span++;
    }
    if (spdata->bot)
    {
      if (spdata->count2)
          spdata->count2--;
      else
      {
          if (lw > (int)parc->height)
            span[-1].rx = span[-1].rw = -((lw - (int)parc->height) >> 1);
          else
            span[-1].rw = 0;
          spdata->count1--;
      }
    }
}

static void
miComputeEllipseSpans(
    int lw,
    xArc *parc,
    miArcSpanData *spdata)
{
    register miArcSpan *span;
    double w, h, r, xorg;
    double Hs, Hf, WH, K, Vk, Nk, Fk, Vr, N, Nc, Z, rs;
    double A, T, b, d, x, y, t, inx, outx = 0.0, hepp, hepm;
    int flip, solution;

    w = (double)parc->width / 2.0;
    h = (double)parc->height / 2.0;
    r = lw / 2.0;
    rs = r * r;
    Hs = h * h;
    WH = w * w - Hs;
    Nk = w * r;
    Vk = (Nk * Hs) / (WH + WH);
    Hf = Hs * Hs;
    Nk = (Hf - Nk * Nk) / WH;
    Fk = Hf / WH;
    hepp = h + EPSILON;
    hepm = h - EPSILON;
    K = h + ((lw - 1) >> 1);
    span = spdata->spans;
    if (parc->width & 1)
      xorg = .5;
    else
      xorg = 0.0;
    if (spdata->top)
    {
      span->lx = 0;
      span->lw = 1;
      span++;
    }
    spdata->count1 = 0;
    spdata->count2 = 0;
    spdata->hole = (spdata->top &&
             (int)parc->height * lw <= (int)(parc->width * parc->width) &&
                lw < (int)parc->height);
    for (; K > 0.0; K -= 1.0)
    {
      N = (K * K + Nk) / 6.0;
      Nc = N * N * N;
      Vr = Vk * K;
      t = Nc + Vr * Vr;
      d = Nc + t;
      if (d < 0.0) {
          d = Nc;
          b = N;
          if ( (b < 0.0) == (t < 0.0) )
          {
            b = -b;
            d = -d;
          }
          Z = N - 2.0 * b * cos(acos(-t / d) / 3.0);
          if ( (Z < 0.0) == (Vr < 0.0) )
            flip = 2;
          else
            flip = 1;
      }
      else
      {
          d = Vr * sqrt(d);
          Z = N + cbrt(t + d) + cbrt(t - d);
          flip = 0;
      }
      A = sqrt((Z + Z) - Nk);
      T = (Fk - Z) * K / A;
      inx = 0.0;
      solution = FALSE;
      b = -A + K;
      d = b * b - 4 * (Z + T);
      if (d >= 0)
      {
          d = sqrt(d);
          y = (b + d) / 2;
          if ((y >= 0.0) && (y < hepp))
          {
            solution = TRUE;
            if (y > hepm)
                y = h;
            t = y / h;
            x = w * sqrt(1 - (t * t));
            t = K - y;
            if (rs - (t * t) >= 0)
               t = sqrt(rs - (t * t));
            else
               t = 0;
            if (flip == 2)
                inx = x - t;
            else
                outx = x + t;
          }
      }
      b = A + K;
      d = b * b - 4 * (Z - T);
      /* Because of the large magnitudes involved, we lose enough precision
       * that sometimes we end up with a negative value near the axis, when
       * it should be positive.  This is a workaround.
       */
      if (d < 0 && !solution)
          d = 0.0;
      if (d >= 0) {
          d = sqrt(d);
          y = (b + d) / 2;
          if (y < hepp)
          {
            if (y > hepm)
                y = h;
            t = y / h;
            x = w * sqrt(1 - (t * t));
            t = K - y;
            if (rs - (t * t) >= 0)
               inx = x - sqrt(rs - (t * t));
            else
               inx = x;
          }
          y = (b - d) / 2;
          if (y >= 0.0)
          {
            if (y > hepm)
                y = h;
            t = y / h;
            x = w * sqrt(1 - (t * t));
            t = K - y;
            if (rs - (t * t) >= 0)
               t = sqrt(rs - (t * t));
            else 
               t = 0;
            if (flip == 1)
                inx = x - t;
            else
                outx = x + t;
          }
      }
      span->lx = ICEIL(xorg - outx);
      if (inx <= 0.0)
      {
          spdata->count1++;
          span->lw = ICEIL(xorg + outx) - span->lx;
          span->rx = ICEIL(xorg + inx);
          span->rw = -ICEIL(xorg - inx);
      }
      else
      {
          spdata->count2++;
          span->lw = ICEIL(xorg - inx) - span->lx;
          span->rx = ICEIL(xorg + inx);
          span->rw = ICEIL(xorg + outx) - span->rx;
      }
      span++;
    }
    if (spdata->bot)
    {
      outx = w + r;
      if (r >= h && r <= w)
          inx = 0.0;
      else if (Nk < 0.0 && -Nk < Hs)
      {
          inx = w * sqrt(1 + Nk / Hs) - sqrt(rs + Nk);
          if (inx > w - r)
            inx = w - r;
      }
      else
          inx = w - r;
      span->lx = ICEIL(xorg - outx);
      if (inx <= 0.0)
      {
          span->lw = ICEIL(xorg + outx) - span->lx;
          span->rx = ICEIL(xorg + inx);
          span->rw = -ICEIL(xorg - inx);
      }
      else
      {
          span->lw = ICEIL(xorg - inx) - span->lx;
          span->rx = ICEIL(xorg + inx);
          span->rw = ICEIL(xorg + outx) - span->rx;
      }
    }
    if (spdata->hole)
    {
      span = &spdata->spans[spdata->count1];
      span->lw = -span->lx;
      span->rx = 1;
      span->rw = span->lw;
      spdata->count1--;
      spdata->count2++;
    }
}

static double
tailX(
    double K,
    struct arc_def *def,
    struct arc_bound *bounds,
    struct accelerators *acc)
{
    double w, h, r;
    double Hs, Hf, WH, Vk, Nk, Fk, Vr, N, Nc, Z, rs;
    double A, T, b, d, x, y, t, hepp, hepm;
    int flip, solution;
    double xs[2];
    double *xp;

    w = def->w;
    h = def->h;
    r = def->l;
    rs = r * r;
    Hs = acc->h2;
    WH = -acc->h2mw2;
    Nk = def->w * r;
    Vk = (Nk * Hs) / (WH + WH);
    Hf = acc->h4;
    Nk = (Hf - Nk * Nk) / WH;
    if (K == 0.0) {
      if (Nk < 0.0 && -Nk < Hs) {
          xs[0] = w * sqrt(1 + Nk / Hs) - sqrt(rs + Nk);
          xs[1] = w - r;
          if (acc->left.valid && boundedLe(K, bounds->left) &&
            !boundedLe(K, bounds->outer) && xs[0] >= 0.0 && xs[1] >= 0.0)
            return xs[1];
          if (acc->right.valid && boundedLe(K, bounds->right) &&
            !boundedLe(K, bounds->inner) && xs[0] <= 0.0 && xs[1] <= 0.0)
            return xs[1];
          return xs[0];
      }
      return w - r;
    }
    Fk = Hf / WH;
    hepp = h + EPSILON;
    hepm = h - EPSILON;
    N = (K * K + Nk) / 6.0;
    Nc = N * N * N;
    Vr = Vk * K;
    xp = xs;
    xs[0] = 0.0;
    t = Nc + Vr * Vr;
    d = Nc + t;
    if (d < 0.0) {
      d = Nc;
      b = N;
      if ( (b < 0.0) == (t < 0.0) )
      {
          b = -b;
          d = -d;
      }
      Z = N - 2.0 * b * cos(acos(-t / d) / 3.0);
      if ( (Z < 0.0) == (Vr < 0.0) )
          flip = 2;
      else
          flip = 1;
    }
    else
    {
      d = Vr * sqrt(d);
      Z = N + cbrt(t + d) + cbrt(t - d);
      flip = 0;
    }
    A = sqrt((Z + Z) - Nk);
    T = (Fk - Z) * K / A;
    solution = FALSE;
    b = -A + K;
    d = b * b - 4 * (Z + T);
    if (d >= 0 && flip == 2)
    {
      d = sqrt(d);
      y = (b + d) / 2;
      if ((y >= 0.0) && (y < hepp))
      {
          solution = TRUE;
          if (y > hepm)
            y = h;
          t = y / h;
          x = w * sqrt(1 - (t * t));
          t = K - y;
          if (rs - (t * t) >= 0)
             t = sqrt(rs - (t * t));
          else
             t = 0;
          *xp++ = x - t;
      }
    }
    b = A + K;
    d = b * b - 4 * (Z - T);
    /* Because of the large magnitudes involved, we lose enough precision
     * that sometimes we end up with a negative value near the axis, when
     * it should be positive.  This is a workaround.
     */
    if (d < 0 && !solution)
      d = 0.0;
    if (d >= 0) {
      d = sqrt(d);
      y = (b + d) / 2;
      if (y < hepp)
      {
          if (y > hepm)
            y = h;
          t = y / h;
          x = w * sqrt(1 - (t * t));
          t = K - y;
          if (rs - (t * t) >= 0)
             *xp++ = x - sqrt(rs - (t * t));
          else
             *xp++ = x;
      }
      y = (b - d) / 2;
      if (y >= 0.0 && flip == 1)
      {
          if (y > hepm)
            y = h;
          t = y / h;
          x = w * sqrt(1 - (t * t));
          t = K - y;
          if (rs - (t * t) >= 0)
             t = sqrt(rs - (t * t));
          else
             t = 0;
          *xp++ = x - t;
      }
    }
    if (xp > &xs[1]) {
      if (acc->left.valid && boundedLe(K, bounds->left) &&
          !boundedLe(K, bounds->outer) && xs[0] >= 0.0 && xs[1] >= 0.0)
          return xs[1];
      if (acc->right.valid && boundedLe(K, bounds->right) &&
          !boundedLe(K, bounds->inner) && xs[0] <= 0.0 && xs[1] <= 0.0)
          return xs[1];
    }
    return xs[0];
}

static miArcSpanData *
miComputeWideEllipse(
    int              lw,
    register xArc *parc,
    Bool      *mustFree)
{
    register miArcSpanData *spdata;
    register arcCacheRec *cent, *lruent;
    register int k;
    arcCacheRec fakeent;

    if (!lw)
      lw = 1;
    if (parc->height <= 1500)
    {
      *mustFree = FALSE;
      cent = lastCacheHit;
      if (cent->lw == lw &&
          cent->width == parc->width && cent->height == parc->height)
      {
          cent->lrustamp = ++lrustamp;
          return cent->spdata;
      }
      lruent = &arcCache[0];
      for (k = CACHESIZE, cent = lruent; --k >= 0; cent++)
      {
          if (cent->lw == lw &&
            cent->width == parc->width && cent->height == parc->height)
          {
            cent->lrustamp = ++lrustamp;
            lastCacheHit = cent;
            return cent->spdata;
          }
          if (cent->lrustamp < lruent->lrustamp)
            lruent = cent;
      }
      if (!cacheType)
      {
          cacheType = CreateNewResourceType(miFreeArcCache);
          (void) AddResource(FakeClientID(0), cacheType, NULL);
      }
    } else {
      lruent = &fakeent;
      lruent->spdata = NULL;
      *mustFree = TRUE;
    }
    k = (parc->height >> 1) + ((lw - 1) >> 1);
    spdata = lruent->spdata;
    if (!spdata || spdata->k != k)
    {
      if (spdata)
          xfree(spdata);
      spdata = (miArcSpanData *)xalloc(sizeof(miArcSpanData) +
                               sizeof(miArcSpan) * (k + 2));
      lruent->spdata = spdata;
      if (!spdata)
      {
          lruent->lrustamp = 0;
          lruent->lw = 0;
          return spdata;
      }
      spdata->spans = (miArcSpan *)(spdata + 1);
      spdata->k = k;
    }
    spdata->top = !(lw & 1) && !(parc->width & 1);
    spdata->bot = !(parc->height & 1);
    lruent->lrustamp = ++lrustamp;
    lruent->lw = lw;
    lruent->width = parc->width;
    lruent->height = parc->height;
    if (lruent != &fakeent)
      lastCacheHit = lruent;
    if (parc->width == parc->height)
      miComputeCircleSpans(lw, parc, spdata);
    else
      miComputeEllipseSpans(lw, parc, spdata);
    return spdata;
}

static void
miFillWideEllipse(
    DrawablePtr   pDraw,
    GCPtr   pGC,
    xArc    *parc)
{
    DDXPointPtr points;
    register DDXPointPtr pts;
    int *widths;
    register int *wids;
    miArcSpanData *spdata;
    Bool mustFree;
    register miArcSpan *span;
    register int xorg, yorgu, yorgl;
    register int n;

    yorgu = parc->height + pGC->lineWidth;
    n = (sizeof(int) * 2) * yorgu;
    widths = (int *)ALLOCATE_LOCAL(n + (sizeof(DDXPointRec) * 2) * yorgu);
    if (!widths)
      return;
    points = (DDXPointPtr)((char *)widths + n);
    spdata = miComputeWideEllipse((int)pGC->lineWidth, parc, &mustFree);
    if (!spdata)
    {
      DEALLOCATE_LOCAL(widths);
      return;
    }
    pts = points;
    wids = widths;
    span = spdata->spans;
    xorg = parc->x + (parc->width >> 1);
    yorgu = parc->y + (parc->height >> 1);
    yorgl = yorgu + (parc->height & 1);
    if (pGC->miTranslate)
    {
      xorg += pDraw->x;
      yorgu += pDraw->y;
      yorgl += pDraw->y;
    }
    yorgu -= spdata->k;
    yorgl += spdata->k;
    if (spdata->top)
    {
      pts->x = xorg;
      pts->y = yorgu - 1;
      pts++;
      *wids++ = 1;
      span++;
    }
    for (n = spdata->count1; --n >= 0; )
    {
      pts[0].x = xorg + span->lx;
      pts[0].y = yorgu;
      wids[0] = span->lw;
      pts[1].x = pts[0].x;
      pts[1].y = yorgl;
      wids[1] = wids[0];
      yorgu++;
      yorgl--;
      pts += 2;
      wids += 2;
      span++;
    }
    if (spdata->hole)
    {
      pts[0].x = xorg;
      pts[0].y = yorgl;
      wids[0] = 1;
      pts++;
      wids++;
    }
    for (n = spdata->count2; --n >= 0; )
    {
      pts[0].x = xorg + span->lx;
      pts[0].y = yorgu;
      wids[0] = span->lw;
      pts[1].x = xorg + span->rx;
      pts[1].y = pts[0].y;
      wids[1] = span->rw;
      pts[2].x = pts[0].x;
      pts[2].y = yorgl;
      wids[2] = wids[0];
      pts[3].x = pts[1].x;
      pts[3].y = pts[2].y;
      wids[3] = wids[1];
      yorgu++;
      yorgl--;
      pts += 4;
      wids += 4;
      span++;
    }
    if (spdata->bot)
    {
      if (span->rw <= 0)
      {
          pts[0].x = xorg + span->lx;
          pts[0].y = yorgu;
          wids[0] = span->lw;
          pts++;
          wids++;
      }     
      else
      {
          pts[0].x = xorg + span->lx;
          pts[0].y = yorgu;
          wids[0] = span->lw;
          pts[1].x = xorg + span->rx;
          pts[1].y = pts[0].y;
          wids[1] = span->rw;
          pts += 2;
          wids += 2;
      }
    }
    if (mustFree)
      xfree(spdata);
    (*pGC->ops->FillSpans)(pDraw, pGC, pts - points, points, widths, FALSE);

    DEALLOCATE_LOCAL(widths);
}

/*
 * miPolyArc strategy:
 *
 * If arc is zero width and solid, we don't have to worry about the rasterop
 * or join styles.  For wide solid circles, we use a fast integer algorithm.
 * For wide solid ellipses, we use special case floating point code.
 * Otherwise, we set up pDrawTo and pGCTo according to the rasterop, then
 * draw using pGCTo and pDrawTo.  If the raster-op was "tricky," that is,
 * if it involves the destination, then we use PushPixels to move the bits
 * from the scratch drawable to pDraw. (See the wide line code for a
 * fuller explanation of this.)
 */

void
miPolyArc(pDraw, pGC, narcs, parcs)
    DrawablePtr   pDraw;
    GCPtr   pGC;
    int           narcs;
    xArc    *parcs;
{
    register int        i;
    xArc                *parc;
    int                       xMin, xMax, yMin, yMax;
    int                       pixmapWidth = 0, pixmapHeight = 0;
    int                       xOrg = 0, yOrg = 0;
    int                       width;
    Bool                fTricky;
    DrawablePtr               pDrawTo;
    CARD32              fg, bg;
    GCPtr               pGCTo;
    miPolyArcPtr        polyArcs;
    int                       cap[2], join[2];
    int                       iphase;
    int                       halfWidth;

    width = pGC->lineWidth;
    if(width == 0 && pGC->lineStyle == LineSolid)
    {
      for(i = narcs, parc = parcs; --i >= 0; parc++)
          miArcSegment( pDraw, pGC, *parc,
          (miArcFacePtr) 0, (miArcFacePtr) 0 );
      fillSpans (pDraw, pGC);
    }
    else 
    {
      if ((pGC->lineStyle == LineSolid) && narcs)
      {
          while (parcs->width && parcs->height &&
               (parcs->angle2 >= FULLCIRCLE ||
                parcs->angle2 <= -FULLCIRCLE))
          {
            miFillWideEllipse(pDraw, pGC, parcs);
            if (!--narcs)
                return;
            parcs++;
          }
      }

      /* Set up pDrawTo and pGCTo based on the rasterop */
      switch(pGC->alu)
      {
        case GXclear:         /* 0 */
        case GXcopy:          /* src */
        case GXcopyInverted:  /* NOT src */
        case GXset:           /* 1 */
          fTricky = FALSE;
          pDrawTo = pDraw;
          pGCTo = pGC;
          break;
        default:
          fTricky = TRUE;

          /* find bounding box around arcs */
          xMin = yMin = MAXSHORT;
          xMax = yMax = MINSHORT;

          for(i = narcs, parc = parcs; --i >= 0; parc++)
          {
            xMin = min (xMin, parc->x);
            yMin = min (yMin, parc->y);
            xMax = max (xMax, (parc->x + (int) parc->width));
            yMax = max (yMax, (parc->y + (int) parc->height));
          }

          /* expand box to deal with line widths */
          halfWidth = (width + 1)/2;
          xMin -= halfWidth;
          yMin -= halfWidth;
          xMax += halfWidth;
          yMax += halfWidth;

          /* compute pixmap size; limit it to size of drawable */
          xOrg = max(xMin, 0);
          yOrg = max(yMin, 0);
          pixmapWidth = min(xMax, pDraw->width) - xOrg;
          pixmapHeight = min(yMax, pDraw->height) - yOrg;

          /* if nothing left, return */
          if ( (pixmapWidth <= 0) || (pixmapHeight <= 0) ) return;

          for(i = narcs, parc = parcs; --i >= 0; parc++)
          {
            parc->x -= xOrg;
            parc->y -= yOrg;
          }
          if (pGC->miTranslate)
          {
            xOrg += pDraw->x;
            yOrg += pDraw->y;
          }

          /* set up scratch GC */

          pGCTo = GetScratchGC(1, pDraw->pScreen);
          if (!pGCTo)
            return;
          gcvals[GCValsFunction] = GXcopy;
          gcvals[GCValsForeground] = 1;
          gcvals[GCValsBackground] = 0;
          gcvals[GCValsLineWidth] = pGC->lineWidth;
          gcvals[GCValsCapStyle] = pGC->capStyle;
          gcvals[GCValsJoinStyle] = pGC->joinStyle;
          dixChangeGC(NullClient, pGCTo, GCValsMask, gcvals, NULL);
    
          /* allocate a 1 bit deep pixmap of the appropriate size, and
           * validate it */
          pDrawTo = (DrawablePtr)(*pDraw->pScreen->CreatePixmap)
                        (pDraw->pScreen, pixmapWidth, pixmapHeight, 1);
          if (!pDrawTo)
          {
            FreeScratchGC(pGCTo);
            return;
          }
          ValidateGC(pDrawTo, pGCTo);
          miClearDrawable(pDrawTo, pGCTo);
      }

      fg = pGC->fgPixel;
      bg = pGC->bgPixel;
      if ((pGC->fillStyle == FillTiled) ||
          (pGC->fillStyle == FillOpaqueStippled))
          bg = fg; /* the protocol sez these don't cause color changes */

      polyArcs = miComputeArcs (parcs, narcs, pGC);

      if (!polyArcs)
      {
          if (fTricky) {
            (*pDraw->pScreen->DestroyPixmap) ((PixmapPtr)pDrawTo);
            FreeScratchGC (pGCTo);
          }
          return;
      }

      cap[0] = cap[1] = 0;
      join[0] = join[1] = 0;
      for (iphase = ((pGC->lineStyle == LineDoubleDash) ? 1 : 0);
           iphase >= 0;
           iphase--)
      {
          if (iphase == 1) {
            dixChangeGC (NullClient, pGC, GCForeground, &bg, NULL);
            ValidateGC (pDraw, pGC);
          } else if (pGC->lineStyle == LineDoubleDash) {
            dixChangeGC (NullClient, pGC, GCForeground, &fg, NULL);
            ValidateGC (pDraw, pGC);
          }
          for (i = 0; i < polyArcs[iphase].narcs; i++) {
            miArcDataPtr      arcData;

            arcData = &polyArcs[iphase].arcs[i];
            miArcSegment(pDrawTo, pGCTo, arcData->arc,
                       &arcData->bounds[RIGHT_END],
                       &arcData->bounds[LEFT_END]);
            if (polyArcs[iphase].arcs[i].render) {
                fillSpans (pDrawTo, pGCTo);
                /*
                 * don't cap self-joining arcs
                 */
                if (polyArcs[iphase].arcs[i].selfJoin &&
                    cap[iphase] < polyArcs[iphase].arcs[i].cap)
                  cap[iphase]++;
                while (cap[iphase] < polyArcs[iphase].arcs[i].cap) {
                  int   arcIndex, end;
                  miArcDataPtr      arcData0;

                  arcIndex = polyArcs[iphase].caps[cap[iphase]].arcIndex;
                  end = polyArcs[iphase].caps[cap[iphase]].end;
                  arcData0 = &polyArcs[iphase].arcs[arcIndex];
                  miArcCap (pDrawTo, pGCTo,
                          &arcData0->bounds[end], end,
                          arcData0->arc.x, arcData0->arc.y,
                          (double) arcData0->arc.width / 2.0,
                          (double) arcData0->arc.height / 2.0);
                  ++cap[iphase];
                }
                while (join[iphase] < polyArcs[iphase].arcs[i].join) {
                  int   arcIndex0, arcIndex1, end0, end1;
                  int   phase0, phase1;
                  miArcDataPtr      arcData0, arcData1;
                  miArcJoinPtr      joinp;

                  joinp = &polyArcs[iphase].joins[join[iphase]];
                  arcIndex0 = joinp->arcIndex0;
                  end0 = joinp->end0;
                  arcIndex1 = joinp->arcIndex1;
                  end1 = joinp->end1;
                  phase0 = joinp->phase0;
                  phase1 = joinp->phase1;
                  arcData0 = &polyArcs[phase0].arcs[arcIndex0];
                  arcData1 = &polyArcs[phase1].arcs[arcIndex1];
                  miArcJoin (pDrawTo, pGCTo,
                          &arcData0->bounds[end0],
                          &arcData1->bounds[end1],
                          arcData0->arc.x, arcData0->arc.y,
                          (double) arcData0->arc.width / 2.0,
                          (double) arcData0->arc.height / 2.0,
                          arcData1->arc.x, arcData1->arc.y,
                          (double) arcData1->arc.width / 2.0,
                          (double) arcData1->arc.height / 2.0);
                  ++join[iphase];
                }
                if (fTricky) {
                  if (pGC->serialNumber != pDraw->serialNumber)
                      ValidateGC (pDraw, pGC);
                  (*pGC->ops->PushPixels) (pGC, (PixmapPtr)pDrawTo,
                         pDraw, pixmapWidth, pixmapHeight, xOrg, yOrg);
                  miClearDrawable ((DrawablePtr) pDrawTo, pGCTo);
                }
            }
          }
      }
      miFreeArcs(polyArcs, pGC);

      if(fTricky)
      {
          (*pGCTo->pScreen->DestroyPixmap)((PixmapPtr)pDrawTo);
          FreeScratchGC(pGCTo);
      }
    }
}

static double
angleBetween (SppPointRec center, SppPointRec point1, SppPointRec point2)
{
      double      a1, a2, a;
      
      /*
       * reflect from X coordinates back to ellipse
       * coordinates -- y increasing upwards
       */
      a1 = miDatan2 (- (point1.y - center.y), point1.x - center.x);
      a2 = miDatan2 (- (point2.y - center.y), point2.x - center.x);
      a = a2 - a1;
      if (a <= -180.0)
            a += 360.0;
      else if (a > 180.0)
            a -= 360.0;
      return a;
}

static void
translateBounds (
      miArcFacePtr      b,
      int         x,
      int         y,
      double            fx,
      double            fy)
{
      fx += x;
      fy += y;
      b->clock.x -= fx;
      b->clock.y -= fy;
      b->center.x -= fx;
      b->center.y -= fy;
      b->counterClock.x -= fx;
      b->counterClock.y -= fy;
}

static void
miArcJoin(DrawablePtr pDraw, GCPtr pGC, miArcFacePtr pLeft,
        miArcFacePtr pRight, int xOrgLeft, int yOrgLeft,
        double xFtransLeft, double yFtransLeft,
        int xOrgRight, int yOrgRight,
        double xFtransRight, double yFtransRight)
{
      SppPointRec center, corner, otherCorner;
      SppPointRec poly[5], e;
      SppPointPtr pArcPts;
      int         cpt;
      SppArcRec   arc;
      miArcFaceRec      Right, Left;
      int         polyLen = 0;
      int         xOrg, yOrg;
      double            xFtrans, yFtrans;
      double            a;
      double            ae, ac2, ec2, bc2, de;
      double            width;
      
      xOrg = (xOrgRight + xOrgLeft) / 2;
      yOrg = (yOrgRight + yOrgLeft) / 2;
      xFtrans = (xFtransLeft + xFtransRight) / 2;
      yFtrans = (yFtransLeft + yFtransRight) / 2;
      Right = *pRight;
      translateBounds (&Right, xOrg - xOrgRight, yOrg - yOrgRight,
                         xFtrans - xFtransRight, yFtrans - yFtransRight);
      Left = *pLeft;
      translateBounds (&Left, xOrg - xOrgLeft, yOrg - yOrgLeft,
                         xFtrans - xFtransLeft, yFtrans - yFtransLeft);
      pRight = &Right;
      pLeft = &Left;

      if (pRight->clock.x == pLeft->counterClock.x &&
          pRight->clock.y == pLeft->counterClock.y)
            return;
      center = pRight->center;
      if (0 <= (a = angleBetween (center, pRight->clock, pLeft->counterClock))
          && a <= 180.0)
      {
            corner = pRight->clock;
            otherCorner = pLeft->counterClock;
      } else {
            a = angleBetween (center, pLeft->clock, pRight->counterClock);
            corner = pLeft->clock;
            otherCorner = pRight->counterClock;
      }
      switch (pGC->joinStyle) {
      case JoinRound:
            width = (pGC->lineWidth ? (double)pGC->lineWidth : (double)1);

            arc.x = center.x - width/2;
            arc.y = center.y - width/2;
            arc.width = width;
            arc.height = width;
            arc.angle1 = -miDatan2 (corner.y - center.y, corner.x - center.x);
            arc.angle2 = a;
            pArcPts = (SppPointPtr) xalloc (3 * sizeof (SppPointRec));
            if (!pArcPts)
                return;
            pArcPts[0].x = otherCorner.x;
            pArcPts[0].y = otherCorner.y;
            pArcPts[1].x = center.x;
            pArcPts[1].y = center.y;
            pArcPts[2].x = corner.x;
            pArcPts[2].y = corner.y;
            if( (cpt = miGetArcPts(&arc, 3, &pArcPts)) )
            {
                  /* by drawing with miFillSppPoly and setting the endpoints of the arc
                   * to be the corners, we assure that the cap will meet up with the
                   * rest of the line */
                  miFillSppPoly(pDraw, pGC, cpt, pArcPts, xOrg, yOrg, xFtrans, yFtrans);
            }
            xfree(pArcPts);
            return;
      case JoinMiter:
            /*
             * don't miter arcs with less than 11 degrees between them
             */
            if (a < 169.0) {
                  poly[0] = corner;
                  poly[1] = center;
                  poly[2] = otherCorner;
                  bc2 = (corner.x - otherCorner.x) * (corner.x - otherCorner.x) +
                        (corner.y - otherCorner.y) * (corner.y - otherCorner.y);
                  ec2 = bc2 / 4;
                  ac2 = (corner.x - center.x) * (corner.x - center.x) +
                        (corner.y - center.y) * (corner.y - center.y);
                  ae = sqrt (ac2 - ec2);
                  de = ec2 / ae;
                  e.x = (corner.x + otherCorner.x) / 2;
                  e.y = (corner.y + otherCorner.y) / 2;
                  poly[3].x = e.x + de * (e.x - center.x) / ae;
                  poly[3].y = e.y + de * (e.y - center.y) / ae;
                  poly[4] = corner;
                  polyLen = 5;
                  break;
            }
      case JoinBevel:
            poly[0] = corner;
            poly[1] = center;
            poly[2] = otherCorner;
            poly[3] = corner;
            polyLen = 4;
            break;
      }
      miFillSppPoly (pDraw, pGC, polyLen, poly, xOrg, yOrg, xFtrans, yFtrans);
}

/*ARGSUSED*/
static void
miArcCap (
      DrawablePtr pDraw,
      GCPtr       pGC,
      miArcFacePtr      pFace,
      int         end,
      int         xOrg,
      int         yOrg,
      double            xFtrans,
      double            yFtrans)
{
      SppPointRec corner, otherCorner, center, endPoint, poly[5];

      corner = pFace->clock;
      otherCorner = pFace->counterClock;
      center = pFace->center;
      switch (pGC->capStyle) {
      case CapProjecting:
            poly[0].x = otherCorner.x;
            poly[0].y = otherCorner.y;
            poly[1].x = corner.x;
            poly[1].y = corner.y;
            poly[2].x = corner.x -
                        (center.y - corner.y);
            poly[2].y = corner.y +
                        (center.x - corner.x);
            poly[3].x = otherCorner.x -
                        (otherCorner.y - center.y);
            poly[3].y = otherCorner.y +
                        (otherCorner.x - center.x);
            poly[4].x = otherCorner.x;
            poly[4].y = otherCorner.y;
            miFillSppPoly (pDraw, pGC, 5, poly, xOrg, yOrg, xFtrans, yFtrans);
            break;
      case CapRound:
            /*
             * miRoundCap just needs these to be unequal.
             */
            endPoint = center;
            endPoint.x = endPoint.x + 100;
            miRoundCap (pDraw, pGC, center, endPoint, corner, otherCorner, 0,
                      -xOrg, -yOrg, xFtrans, yFtrans);
            break;
      }
}

/* MIROUNDCAP -- a private helper function
 * Put Rounded cap on end. pCenter is the center of this end of the line
 * pEnd is the center of the other end of the line. pCorner is one of the
 * two corners at this end of the line.  
 * NOTE:  pOtherCorner must be counter-clockwise from pCorner.
 */
/*ARGSUSED*/
static void
miRoundCap(
    DrawablePtr   pDraw,
    GCPtr   pGC,
    SppPointRec   pCenter,
    SppPointRec   pEnd,
    SppPointRec   pCorner,
    SppPointRec   pOtherCorner,
    int           fLineEnd,
    int           xOrg,
    int           yOrg,
    double  xFtrans,
    double  yFtrans)
{
    int           cpt;
    double  width;
    SppArcRec     arc;
    SppPointPtr   pArcPts;

    width = (pGC->lineWidth ? (double)pGC->lineWidth : (double)1);

    arc.x = pCenter.x - width/2;
    arc.y = pCenter.y - width/2;
    arc.width = width;
    arc.height = width;
    arc.angle1 = -miDatan2 (pCorner.y - pCenter.y, pCorner.x - pCenter.x);
    if(PTISEQUAL(pCenter, pEnd))
      arc.angle2 = - 180.0;
    else {
      arc.angle2 = -miDatan2 (pOtherCorner.y - pCenter.y, pOtherCorner.x - pCenter.x) - arc.angle1;
      if (arc.angle2 < 0)
          arc.angle2 += 360.0;
    }
    pArcPts = (SppPointPtr) NULL;
    if( (cpt = miGetArcPts(&arc, 0, &pArcPts)) )
    {
      /* by drawing with miFillSppPoly and setting the endpoints of the arc
       * to be the corners, we assure that the cap will meet up with the
       * rest of the line */
      miFillSppPoly(pDraw, pGC, cpt, pArcPts, -xOrg, -yOrg, xFtrans, yFtrans);
    }
    xfree(pArcPts);
}

/*
 * To avoid inaccuracy at the cardinal points, use trig functions
 * which are exact for those angles
 */

#ifndef M_PI
#define M_PI      3.14159265358979323846
#endif
#ifndef M_PI_2
#define M_PI_2    1.57079632679489661923
#endif

# define Dsin(d)  ((d) == 0.0 ? 0.0 : ((d) == 90.0 ? 1.0 : sin(d*M_PI/180.0)))
# define Dcos(d)  ((d) == 0.0 ? 1.0 : ((d) == 90.0 ? 0.0 : cos(d*M_PI/180.0)))
# define mod(a,b) ((a) >= 0 ? (a) % (b) : (b) - (-a) % (b))

static double
miDcos (double a)
{
      int   i;

      if (floor (a/90) == a/90) {
            i = (int) (a/90.0);
            switch (mod (i, 4)) {
            case 0: return 1;
            case 1: return 0;
            case 2: return -1;
            case 3: return 0;
            }
      }
      return cos (a * M_PI / 180.0);
}

static double
miDsin (double a)
{
      int   i;

      if (floor (a/90) == a/90) {
            i = (int) (a/90.0);
            switch (mod (i, 4)) {
            case 0: return 0;
            case 1: return 1;
            case 2: return 0;
            case 3: return -1;
            }
      }
      return sin (a * M_PI / 180.0);
}

static double
miDasin (double v)
{
    if (v == 0)
      return 0.0;
    if (v == 1.0)
      return 90.0;
    if (v == -1.0)
      return -90.0;
    return asin(v) * (180.0 / M_PI);
}

static double
miDatan2 (double dy, double dx)
{
    if (dy == 0) {
      if (dx >= 0)
          return 0.0;
      return 180.0;
    } else if (dx == 0) {
      if (dy > 0)
          return 90.0;
      return -90.0;
    } else if (Fabs (dy) == Fabs (dx)) {
      if (dy > 0) {
          if (dx > 0)
            return 45.0;
          return 135.0;
      } else {
          if (dx > 0)
            return 315.0;
          return 225.0;
      }
    } else {
      return atan2 (dy, dx) * (180.0 / M_PI);
    }
}

/* MIGETARCPTS -- Converts an arc into a set of line segments -- a helper
 * routine for filled arc and line (round cap) code.
 * Returns the number of points in the arc.  Note that it takes a pointer
 * to a pointer to where it should put the points and an index (cpt).
 * This procedure allocates the space necessary to fit the arc points.
 * Sometimes it's convenient for those points to be at the end of an existing
 * array. (For example, if we want to leave a spare point to make sectors
 * instead of segments.)  So we pass in the xalloc()ed chunk that contains the
 * array and an index saying where we should start stashing the points.
 * If there isn't an array already, we just pass in a null pointer and 
 * count on xrealloc() to handle the null pointer correctly.
 */
static int
miGetArcPts(
    SppArcPtr     parc, /* points to an arc */
    int           cpt,  /* number of points already in arc list */
    SppPointPtr   *ppPts) /* pointer to pointer to arc-list -- modified */
{
    double  st,   /* Start Theta, start angle */
                et,     /* End Theta, offset from start theta */
            dt,   /* Delta Theta, angle to sweep ellipse */
            cdt,  /* Cos Delta Theta, actually 2 cos(dt) */
            x0, y0,     /* the recurrence formula needs two points to start */
            x1, y1,
            x2, y2, /* this will be the new point generated */
            xc, yc; /* the center point */
    int           count, i;
    SppPointPtr   poly;
    DDXPointRec last;         /* last point on integer boundaries */

    /* The spec says that positive angles indicate counterclockwise motion.
     * Given our coordinate system (with 0,0 in the upper left corner), 
     * the screen appears flipped in Y.  The easiest fix is to negate the
     * angles given */
    
    st = - parc->angle1;

    et = - parc->angle2;

    /* Try to get a delta theta that is within 1/2 pixel.  Then adjust it
     * so that it divides evenly into the total.
     * I'm just using cdt 'cause I'm lazy.
     */
    cdt = parc->width;
    if (parc->height > cdt)
      cdt = parc->height;
    cdt /= 2.0;
    if(cdt <= 0)
      return 0;
    if (cdt < 1.0)
      cdt = 1.0;
    dt = miDasin ( 1.0 / cdt ); /* minimum step necessary */
    count = et/dt;
    count = abs(count) + 1;
    dt = et/count;      
    count++;

    cdt = 2 * miDcos(dt);
    if (!(poly = (SppPointPtr) xrealloc((pointer)*ppPts,
                              (cpt + count) * sizeof(SppPointRec))))
      return(0);
    *ppPts = poly;

    xc = parc->width/2.0;           /* store half width and half height */
    yc = parc->height/2.0;
    
    x0 = xc * miDcos(st);
    y0 = yc * miDsin(st);
    x1 = xc * miDcos(st + dt);
    y1 = yc * miDsin(st + dt);
    xc += parc->x;            /* by adding initial point, these become */
    yc += parc->y;            /* the center point */

    poly[cpt].x = (xc + x0);
    poly[cpt].y = (yc + y0);
    last.x = ROUNDTOINT( poly[cpt + 1].x = (xc + x1) );
    last.y = ROUNDTOINT( poly[cpt + 1].y = (yc + y1) );

    for(i = 2; i < count; i++)
    {
      x2 = cdt * x1 - x0;
      y2 = cdt * y1 - y0;

      poly[cpt + i].x = (xc + x2);
      poly[cpt + i].y = (yc + y2);

      x0 = x1; y0 = y1;
      x1 = x2; y1 = y2;
    }
    /* adjust the last point */
    if (abs(parc->angle2) >= 360.0)
      poly[cpt +i -1] = poly[0];
    else {
      poly[cpt +i -1].x = (miDcos(st + et) * parc->width/2.0 + xc);
      poly[cpt +i -1].y = (miDsin(st + et) * parc->height/2.0 + yc);
    }

    return(count);
}

struct arcData {
      double      x0, y0, x1, y1;
      int   selfJoin;
};

# define ADD_REALLOC_STEP     20

static void
addCap (
      miArcCapPtr *capsp,
      int         *ncapsp,
      int         *sizep,
      int         end,
      int         arcIndex)
{
      int newsize;
      miArcCapPtr cap;

      if (*ncapsp == *sizep)
      {
          newsize = *sizep + ADD_REALLOC_STEP;
          cap = (miArcCapPtr) xrealloc (*capsp,
                                newsize * sizeof (**capsp));
          if (!cap)
            return;
          *sizep = newsize;
          *capsp = cap;
      }
      cap = &(*capsp)[*ncapsp];
      cap->end = end;
      cap->arcIndex = arcIndex;
      ++*ncapsp;
}

static void
addJoin (
      miArcJoinPtr      *joinsp,
      int         *njoinsp,
      int         *sizep,
      int         end0,
      int         index0,
      int         phase0,
      int         end1,
      int         index1,
      int         phase1)
{
      int newsize;
      miArcJoinPtr      join;

      if (*njoinsp == *sizep)
      {
          newsize = *sizep + ADD_REALLOC_STEP;
          join = (miArcJoinPtr) xrealloc (*joinsp,
                                  newsize * sizeof (**joinsp));
          if (!join)
            return;
          *sizep = newsize;
          *joinsp = join;
      }
      join = &(*joinsp)[*njoinsp];
      join->end0 = end0;
      join->arcIndex0 = index0;
      join->phase0 = phase0;
      join->end1 = end1;
      join->arcIndex1 = index1;
      join->phase1 = phase1;
      ++*njoinsp;
}

static miArcDataPtr
addArc (
      miArcDataPtr      *arcsp,
      int         *narcsp,
      int         *sizep,
      xArc        *xarc)
{
      int newsize;
      miArcDataPtr      arc;

      if (*narcsp == *sizep)
      {
          newsize = *sizep + ADD_REALLOC_STEP;
          arc = (miArcDataPtr) xrealloc (*arcsp,
                                 newsize * sizeof (**arcsp));
          if (!arc)
            return (miArcDataPtr)NULL;
          *sizep = newsize;
          *arcsp = arc;
      }
      arc = &(*arcsp)[*narcsp];
      arc->arc = *xarc;
      ++*narcsp;
      return arc;
}

static void
miFreeArcs(
    miPolyArcPtr arcs,
    GCPtr pGC)
{
      int iphase;

      for (iphase = ((pGC->lineStyle == LineDoubleDash) ? 1 : 0);
           iphase >= 0;
           iphase--)
      {
          if (arcs[iphase].narcs > 0)
            xfree(arcs[iphase].arcs);
          if (arcs[iphase].njoins > 0)
            xfree(arcs[iphase].joins);
          if (arcs[iphase].ncaps > 0)
            xfree(arcs[iphase].caps);
      }
      xfree(arcs);
}

/*
 * map angles to radial distance.  This only deals with the first quadrant
 */

/*
 * a polygonal approximation to the arc for computing arc lengths
 */

# define DASH_MAP_SIZE  91

# define dashIndexToAngle(di) ((((double) (di)) * 90.0) / ((double) DASH_MAP_SIZE - 1))
# define xAngleToDashIndex(xa)      ((((long) (xa)) * (DASH_MAP_SIZE - 1)) / (90 * 64))
# define dashIndexToXAngle(di)      ((((long) (di)) * (90 * 64)) / (DASH_MAP_SIZE - 1))
# define dashXAngleStep (((double) (90 * 64)) / ((double) (DASH_MAP_SIZE - 1)))

typedef struct {
      double      map[DASH_MAP_SIZE];
} dashMap;

static int computeAngleFromPath(int startAngle, int endAngle, dashMap *map,
                        int *lenp, int backwards);

static void
computeDashMap (
      xArc  *arcp,
      dashMap     *map)
{
      int   di;
      double      a, x, y, prevx = 0.0, prevy = 0.0, dist;

      for (di = 0; di < DASH_MAP_SIZE; di++) {
            a = dashIndexToAngle (di);
            x = ((double) arcp->width / 2.0) * miDcos (a);
            y = ((double) arcp->height / 2.0) * miDsin (a);
            if (di == 0) {
                  map->map[di] = 0.0;
            } else {
                  dist = hypot (x - prevx, y - prevy);
                  map->map[di] = map->map[di - 1] + dist;
            }
            prevx = x;
            prevy = y;
      }
}

typedef enum {HORIZONTAL, VERTICAL, OTHER} arcTypes;

/* this routine is a bit gory */

static miPolyArcPtr
miComputeArcs (
      xArc  *parcs,
      int   narcs,
      GCPtr pGC)
{
      int         isDashed, isDoubleDash;
      int         dashOffset;
      miPolyArcPtr      arcs;
      int         start, i, j, k = 0, nexti, nextk = 0;
      int         joinSize[2];
      int         capSize[2];
      int         arcSize[2];
      int         angle2;
      double            a0, a1;
      struct arcData    *data;
      miArcDataPtr      arc;
      xArc        xarc;
      int         iphase, prevphase = 0, joinphase;
      int         arcsJoin;
      int         selfJoin;

      int         iDash = 0, dashRemaining;
      int         iDashStart = 0, dashRemainingStart = 0, iphaseStart;
      int         startAngle, spanAngle, endAngle, backwards = 0;
      int         prevDashAngle, dashAngle;
      dashMap           map;

      isDashed = !(pGC->lineStyle == LineSolid);
      isDoubleDash = (pGC->lineStyle == LineDoubleDash);
      dashOffset = pGC->dashOffset;

      data = (struct arcData *) ALLOCATE_LOCAL (narcs * sizeof (struct arcData));
      if (!data)
          return (miPolyArcPtr)NULL;
      arcs = (miPolyArcPtr) xalloc (sizeof (*arcs) * (isDoubleDash ? 2 : 1));
      if (!arcs)
      {
          DEALLOCATE_LOCAL(data);
          return (miPolyArcPtr)NULL;
      }
      for (i = 0; i < narcs; i++) {
            a0 = todeg (parcs[i].angle1);
            angle2 = parcs[i].angle2;
            if (angle2 > FULLCIRCLE)
                  angle2 = FULLCIRCLE;
            else if (angle2 < -FULLCIRCLE)
                  angle2 = -FULLCIRCLE;
            data[i].selfJoin = angle2 == FULLCIRCLE || angle2 == -FULLCIRCLE;
            a1 = todeg (parcs[i].angle1 + angle2);
            data[i].x0 = parcs[i].x + (double) parcs[i].width / 2 * (1 + miDcos (a0));
            data[i].y0 = parcs[i].y + (double) parcs[i].height / 2 * (1 - miDsin (a0));
            data[i].x1 = parcs[i].x + (double) parcs[i].width / 2 * (1 + miDcos (a1));
            data[i].y1 = parcs[i].y + (double) parcs[i].height / 2 * (1 - miDsin (a1));
      }

      for (iphase = 0; iphase < (isDoubleDash ? 2 : 1); iphase++) {
            arcs[iphase].njoins = 0;
            arcs[iphase].joins = 0;
            joinSize[iphase] = 0;
            
            arcs[iphase].ncaps = 0;
            arcs[iphase].caps = 0;
            capSize[iphase] = 0;
            
            arcs[iphase].narcs = 0;
            arcs[iphase].arcs = 0;
            arcSize[iphase] = 0;
      }

      iphase = 0;
      if (isDashed) {
            iDash = 0;
            dashRemaining = pGC->dash[0];
            while (dashOffset > 0) {
                  if (dashOffset >= dashRemaining) {
                        dashOffset -= dashRemaining;
                        iphase = iphase ? 0 : 1;
                        iDash++;
                        if (iDash == pGC->numInDashList)
                            iDash = 0;
                        dashRemaining = pGC->dash[iDash];
                  } else {
                        dashRemaining -= dashOffset;
                        dashOffset = 0;
                  }
            }
            iDashStart = iDash;
            dashRemainingStart = dashRemaining;
      }
      iphaseStart = iphase;

      for (i = narcs - 1; i >= 0; i--) {
            j = i + 1;
            if (j == narcs)
                  j = 0;
            if (data[i].selfJoin || i == j ||
                 (UNEQUAL (data[i].x1, data[j].x0) ||
                  UNEQUAL (data[i].y1, data[j].y0)))
            {
                  if (iphase == 0 || isDoubleDash)
                        addCap (&arcs[iphase].caps, &arcs[iphase].ncaps,
                              &capSize[iphase], RIGHT_END, 0);
                  break;
            }
      }
      start = i + 1;
      if (start == narcs)
            start = 0;
      i = start;
      for (;;) {
            j = i + 1;
            if (j == narcs)
                  j = 0;
            nexti = i+1;
            if (nexti == narcs)
                  nexti = 0;
            if (isDashed) {
                  /*
                  ** deal with dashed arcs.  Use special rules for certain 0 area arcs.
                  ** Presumably, the other 0 area arcs still aren't done right.
                  */
                  arcTypes    arcType = OTHER;
                  CARD16            thisLength;

                  if (parcs[i].height == 0
                      && (parcs[i].angle1 % FULLCIRCLE) == 0x2d00
                      && parcs[i].angle2 == 0x2d00) 
                        arcType = HORIZONTAL;
                  else if (parcs[i].width == 0
                      && (parcs[i].angle1 % FULLCIRCLE) == 0x1680
                      && parcs[i].angle2 == 0x2d00)
                        arcType = VERTICAL;
                  if (arcType == OTHER) {
                        /*
                         * precompute an approximation map
                         */
                        computeDashMap (&parcs[i], &map);
                        /*
                         * compute each individual dash segment using the path
                         * length function
                         */
                        startAngle = parcs[i].angle1;
                        spanAngle = parcs[i].angle2;
                        if (spanAngle > FULLCIRCLE)
                              spanAngle = FULLCIRCLE;
                        else if (spanAngle < -FULLCIRCLE)
                              spanAngle = -FULLCIRCLE;
                        if (startAngle < 0)
                              startAngle = FULLCIRCLE - (-startAngle) % FULLCIRCLE;
                        if (startAngle >= FULLCIRCLE)
                              startAngle = startAngle % FULLCIRCLE;
                        endAngle = startAngle + spanAngle;
                        backwards = spanAngle < 0;
                  } else {
                        xarc = parcs[i];
                        if (arcType == VERTICAL) {
                              xarc.angle1 = 0x1680;
                              startAngle = parcs[i].y;
                              endAngle = startAngle + parcs[i].height;
                        } else {
                              xarc.angle1 = 0x2d00;
                              startAngle = parcs[i].x;
                              endAngle = startAngle + parcs[i].width;
                        }
                  }
                  dashAngle = startAngle;
                  selfJoin = data[i].selfJoin &&
                            (iphase == 0 || isDoubleDash);
                  /*
                   * add dashed arcs to each bucket
                   */
                  arc = 0;
                  while (dashAngle != endAngle) {
                        prevDashAngle = dashAngle;
                        if (arcType == OTHER) {
                              dashAngle = computeAngleFromPath (prevDashAngle, endAngle,
                                                &map, &dashRemaining, backwards);
                              /* avoid troubles with huge arcs and small dashes */
                              if (dashAngle == prevDashAngle) {
                                    if (backwards)
                                          dashAngle--;
                                    else
                                          dashAngle++;
                              }
                        } else {
                              thisLength = (dashAngle + dashRemaining <= endAngle) ? 
                                  dashRemaining : endAngle - dashAngle;
                              if (arcType == VERTICAL) {
                                    xarc.y = dashAngle;
                                    xarc.height = thisLength;
                              } else {
                                    xarc.x = dashAngle;
                                    xarc.width = thisLength;
                              }
                              dashAngle += thisLength;
                              dashRemaining -= thisLength;
                        }
                        if (iphase == 0 || isDoubleDash) {
                              if (arcType == OTHER) {
                                    xarc = parcs[i];
                                    spanAngle = prevDashAngle;
                                    if (spanAngle < 0)
                                        spanAngle = FULLCIRCLE - (-spanAngle) % FULLCIRCLE;
                                    if (spanAngle >= FULLCIRCLE)
                                        spanAngle = spanAngle % FULLCIRCLE;
                                    xarc.angle1 = spanAngle;
                                    spanAngle = dashAngle - prevDashAngle;
                                    if (backwards) {
                                          if (dashAngle > prevDashAngle)
                                                spanAngle = - FULLCIRCLE + spanAngle;
                                    } else {
                                          if (dashAngle < prevDashAngle)
                                                spanAngle = FULLCIRCLE + spanAngle;
                                    }
                                    if (spanAngle > FULLCIRCLE)
                                        spanAngle = FULLCIRCLE;
                                    if (spanAngle < -FULLCIRCLE)
                                        spanAngle = -FULLCIRCLE;
                                    xarc.angle2 = spanAngle;
                              }
                              arc = addArc (&arcs[iphase].arcs, &arcs[iphase].narcs,
                                          &arcSize[iphase], &xarc);
                              if (!arc)
                                  goto arcfail;
                              /*
                               * cap each end of an on/off dash
                               */
                              if (!isDoubleDash) {
                                    if (prevDashAngle != startAngle) {
                                          addCap (&arcs[iphase].caps,
                                                &arcs[iphase].ncaps,
                                                &capSize[iphase], RIGHT_END,
                                                arc - arcs[iphase].arcs);
                                          
                                    }
                                    if (dashAngle != endAngle) {
                                          addCap (&arcs[iphase].caps,
                                                &arcs[iphase].ncaps,
                                                &capSize[iphase], LEFT_END,
                                                arc - arcs[iphase].arcs);
                                    }
                              }
                              arc->cap = arcs[iphase].ncaps;
                              arc->join = arcs[iphase].njoins;
                              arc->render = 0;
                              arc->selfJoin = 0;
                              if (dashAngle == endAngle)
                                    arc->selfJoin = selfJoin;
                        }
                        prevphase = iphase;
                        if (dashRemaining <= 0) {
                              ++iDash;
                              if (iDash == pGC->numInDashList)
                                    iDash = 0;
                              iphase = iphase ? 0:1;
                              dashRemaining = pGC->dash[iDash];
                        }
                  }
                  /*
                   * make sure a place exists for the position data when
                   * drawing a zero-length arc
                   */
                  if (startAngle == endAngle) {
                        prevphase = iphase;
                        if (!isDoubleDash && iphase == 1)
                              prevphase = 0;
                        arc = addArc (&arcs[prevphase].arcs, &arcs[prevphase].narcs,
                                    &arcSize[prevphase], &parcs[i]);
                        if (!arc)
                            goto arcfail;
                        arc->join = arcs[prevphase].njoins;
                        arc->cap = arcs[prevphase].ncaps;
                        arc->selfJoin = data[i].selfJoin;
                  }
            } else {
                  arc = addArc (&arcs[iphase].arcs, &arcs[iphase].narcs,
                              &arcSize[iphase], &parcs[i]);
                  if (!arc)
                      goto arcfail;
                  arc->join = arcs[iphase].njoins;
                  arc->cap = arcs[iphase].ncaps;
                  arc->selfJoin = data[i].selfJoin;
                  prevphase = iphase;
            }
            if (prevphase == 0 || isDoubleDash)
                  k = arcs[prevphase].narcs - 1;
            if (iphase == 0 || isDoubleDash)
                  nextk = arcs[iphase].narcs;
            if (nexti == start) {
                  nextk = 0;
                  if (isDashed) {
                        iDash = iDashStart;
                        iphase = iphaseStart;
                        dashRemaining = dashRemainingStart;
                  }
            }
            arcsJoin = narcs > 1 && i != j &&
                      ISEQUAL (data[i].x1, data[j].x0) &&
                      ISEQUAL (data[i].y1, data[j].y0) &&
                      !data[i].selfJoin && !data[j].selfJoin;
            if (arc)
            {
                  if (arcsJoin)
                        arc->render = 0;
                  else
                        arc->render = 1;
            }
            if (arcsJoin &&
                (prevphase == 0 || isDoubleDash) &&
                (iphase == 0 || isDoubleDash))
            {
                  joinphase = iphase;
                  if (isDoubleDash) {
                        if (nexti == start)
                              joinphase = iphaseStart;
                        /*
                         * if the join is right at the dash,
                         * draw the join in foreground
                         * This is because the foreground
                         * arcs are computed second, the results
                         * of which are needed to draw the join
                         */
                        if (joinphase != prevphase)
                              joinphase = 0;
                  }
                  if (joinphase == 0 || isDoubleDash) {
                        addJoin (&arcs[joinphase].joins,
                               &arcs[joinphase].njoins,
                               &joinSize[joinphase],
                               LEFT_END, k, prevphase,
                               RIGHT_END, nextk, iphase);
                        arc->join = arcs[prevphase].njoins;
                  }
            } else {
                  /*
                   * cap the left end of this arc
                   * unless it joins itself
                   */
                  if ((prevphase == 0 || isDoubleDash) &&
                      !arc->selfJoin)
                  {
                        addCap (&arcs[prevphase].caps, &arcs[prevphase].ncaps,
                              &capSize[prevphase], LEFT_END, k);
                        arc->cap = arcs[prevphase].ncaps;
                  }
                  if (isDashed && !arcsJoin) {
                        iDash = iDashStart;
                        iphase = iphaseStart;
                        dashRemaining = dashRemainingStart;
                  }
                  nextk = arcs[iphase].narcs;
                  if (nexti == start) {
                        nextk = 0;
                        iDash = iDashStart;
                        iphase = iphaseStart;
                        dashRemaining = dashRemainingStart;
                  }
                  /*
                   * cap the right end of the next arc.  If the
                   * next arc is actually the first arc, only
                   * cap it if it joins with this arc.  This
                   * case will occur when the final dash segment
                   * of an on/off dash is off.  Of course, this
                   * cap will be drawn at a strange time, but that
                   * hardly matters...
                   */
                  if ((iphase == 0 || isDoubleDash) &&
                      (nexti != start || (arcsJoin && isDashed)))
                        addCap (&arcs[iphase].caps, &arcs[iphase].ncaps,
                              &capSize[iphase], RIGHT_END, nextk);
            }
            i = nexti;
            if (i == start)
                  break;
      }
      /*
       * make sure the last section is rendered
       */
      for (iphase = 0; iphase < (isDoubleDash ? 2 : 1); iphase++)
            if (arcs[iphase].narcs > 0) {
                  arcs[iphase].arcs[arcs[iphase].narcs-1].render = 1;
                  arcs[iphase].arcs[arcs[iphase].narcs-1].join =
                           arcs[iphase].njoins;
                  arcs[iphase].arcs[arcs[iphase].narcs-1].cap =
                           arcs[iphase].ncaps;
            }
      DEALLOCATE_LOCAL(data);
      return arcs;
arcfail:
      miFreeArcs(arcs, pGC);
      DEALLOCATE_LOCAL(data);
      return (miPolyArcPtr)NULL;
}

static double
angleToLength (
      int   angle,
      dashMap     *map)
{
      double      len, excesslen, sidelen = map->map[DASH_MAP_SIZE - 1], totallen;
      int   di;
      int   excess;
      Bool  oddSide = FALSE;

      totallen = 0;
      if (angle >= 0) {
            while (angle >= 90 * 64) {
                  angle -= 90 * 64;
                  totallen += sidelen;
                  oddSide = !oddSide;
            }
      } else {
            while (angle < 0) {
                  angle += 90 * 64;
                  totallen -= sidelen;
                  oddSide = !oddSide;
            }
      }
      if (oddSide)
            angle = 90 * 64 - angle;
            
      di = xAngleToDashIndex (angle);
      excess = angle - dashIndexToXAngle (di);

      len = map->map[di];
      /*
       * linearly interpolate between this point and the next
       */
      if (excess > 0) {
            excesslen = (map->map[di + 1] - map->map[di]) *
                        ((double) excess) / dashXAngleStep;
            len += excesslen;
      }
      if (oddSide)
            totallen += (sidelen - len);
      else
            totallen += len;
      return totallen;
}

/*
 * len is along the arc, but may be more than one rotation
 */

static int
lengthToAngle (
      double      len,
      dashMap     *map)
{
      double      sidelen = map->map[DASH_MAP_SIZE - 1];
      int   angle, angleexcess;
      Bool  oddSide = FALSE;
      int   a0, a1, a;

      angle = 0;
      /*
       * step around the ellipse, subtracting sidelens and
       * adding 90 degrees.  oddSide will tell if the
       * map should be interpolated in reverse
       */
      if (len >= 0) {
            if (sidelen == 0)
                  return 2 * FULLCIRCLE;  /* infinity */
            while (len >= sidelen) {
                  angle += 90 * 64;
                  len -= sidelen;
                  oddSide = !oddSide;
            }
      } else {
            if (sidelen == 0)
                  return -2 * FULLCIRCLE; /* infinity */
            while (len < 0) {
                  angle -= 90 * 64;
                  len += sidelen;
                  oddSide = !oddSide;
            }
      }
      if (oddSide)
            len = sidelen - len;
      a0 = 0;
      a1 = DASH_MAP_SIZE - 1;
      /*
       * binary search for the closest pre-computed length
       */
      while (a1 - a0 > 1) {
            a = (a0 + a1) / 2;
            if (len > map->map[a])
                  a0 = a;
            else
                  a1 = a;
      }
      angleexcess = dashIndexToXAngle (a0);
      /*
       * linearly interpolate to the next point
       */
      angleexcess += (len - map->map[a0]) /
                  (map->map[a0+1] - map->map[a0]) * dashXAngleStep;
      if (oddSide)
            angle += (90 * 64) - angleexcess;
      else
            angle += angleexcess;
      return angle;
}

/*
 * compute the angle of an ellipse which cooresponds to
 * the given path length.  Note that the correct solution
 * to this problem is an eliptic integral, we'll punt and
 * approximate (it's only for dashes anyway).  This
 * approximation uses a polygon.
 *
 * The remaining portion of len is stored in *lenp -
 * this will be negative if the arc extends beyond
 * len and positive if len extends beyond the arc.
 */

static int
computeAngleFromPath (
      int   startAngle,
      int   endAngle,   /* normalized absolute angles in *64 degrees */
      dashMap     *map,
      int   *lenp,
      int   backwards)
{
      int   a0, a1, a;
      double      len0;
      int   len;

      a0 = startAngle;
      a1 = endAngle;
      len = *lenp;
      if (backwards) {
            /*
             * flip the problem around to always be
             * forwards
             */
            a0 = FULLCIRCLE - a0;
            a1 = FULLCIRCLE - a1;
      }
      if (a1 < a0)
            a1 += FULLCIRCLE;
      len0 = angleToLength (a0, map);
      a = lengthToAngle (len0 + len, map);
      if (a > a1) {
            a = a1;
            len -= angleToLength (a1, map) - len0;
      } else
            len = 0;
      if (backwards)
            a = FULLCIRCLE - a;
      *lenp = len;
      return a;
}

/*
 * scan convert wide arcs.
 */

/*
 * draw zero width/height arcs
 */

static void
drawZeroArc (
    DrawablePtr   pDraw,
    GCPtr         pGC,
    xArc          *tarc,
    int             lw,
    miArcFacePtr  left,
    miArcFacePtr  right)
{
      double      x0 = 0.0, y0 = 0.0, x1 = 0.0, y1 = 0.0, w, h, x, y;
      double      xmax, ymax, xmin, ymin;
      int   a0, a1;
      double      a, startAngle, endAngle;
      double      l, lx, ly;

      l = lw / 2.0;
      a0 = tarc->angle1;
      a1 = tarc->angle2;
      if (a1 > FULLCIRCLE)
            a1 = FULLCIRCLE;
      else if (a1 < -FULLCIRCLE)
            a1 = -FULLCIRCLE;
      w = (double)tarc->width / 2.0;
      h = (double)tarc->height / 2.0;
      /*
       * play in X coordinates right away
       */
      startAngle = - ((double) a0 / 64.0);
      endAngle = - ((double) (a0 + a1) / 64.0);
      
      xmax = -w;
      xmin = w;
      ymax = -h;
      ymin = h;
      a = startAngle;
      for (;;)
      {
            x = w * miDcos(a);
            y = h * miDsin(a);
            if (a == startAngle)
            {
                  x0 = x;
                  y0 = y;
            }
            if (a == endAngle)
            {
                  x1 = x;
                  y1 = y;
            }
            if (x > xmax)
                  xmax = x;
            if (x < xmin)
                  xmin = x;
            if (y > ymax)
                  ymax = y;
            if (y < ymin)
                  ymin = y;
            if (a == endAngle)
                  break;
            if (a1 < 0) /* clockwise */
            {
                  if (floor (a / 90.0) == floor (endAngle / 90.0))
                        a = endAngle;
                  else
                        a = 90 * (floor (a/90.0) + 1);
            }
            else
            {
                  if (ceil (a / 90.0) == ceil (endAngle / 90.0))
                        a = endAngle;
                  else
                        a = 90 * (ceil (a/90.0) - 1);
            }
      }
      lx = ly = l;
      if ((x1 - x0) + (y1 - y0) < 0)
          lx = ly = -l;
      if (h)
      {
          ly = 0.0;
          lx = -lx;
      }
      else
          lx = 0.0;
      if (right)
      {
          right->center.x = x0;
          right->center.y = y0;
          right->clock.x = x0 - lx;
          right->clock.y = y0 - ly;
          right->counterClock.x = x0 + lx;
          right->counterClock.y = y0 + ly;
      }
      if (left)
      {
          left->center.x = x1;
          left->center.y = y1;
          left->clock.x = x1 + lx;
          left->clock.y = y1 + ly;
          left->counterClock.x = x1 - lx;
          left->counterClock.y = y1 - ly;
      }
      
      x0 = xmin;
      x1 = xmax;
      y0 = ymin;
      y1 = ymax;
      if (ymin != y1) {
            xmin = -l;
            xmax = l;
      } else {
            ymin = -l;
            ymax = l;
      }
      if (xmax != xmin && ymax != ymin) {
            int   minx, maxx, miny, maxy;
            xRectangle  rect;

            minx = ICEIL (xmin + w) + tarc->x;
            maxx = ICEIL (xmax + w) + tarc->x;
            miny = ICEIL (ymin + h) + tarc->y;
            maxy = ICEIL (ymax + h) + tarc->y;
            rect.x = minx;
            rect.y = miny;
            rect.width = maxx - minx;
            rect.height = maxy - miny;
            (*pGC->ops->PolyFillRect) (pDraw, pGC, 1, &rect);
      }
}

/*
 * this computes the ellipse y value associated with the
 * bottom of the tail.
 */

static void
tailEllipseY (
      struct arc_def          *def,
      struct accelerators     *acc)
{
      double            t;

      acc->tail_y = 0.0;
      if (def->w == def->h)
          return;
      t = def->l * def->w;
      if (def->w > def->h) {
          if (t < acc->h2)
            return;
      } else {
          if (t > acc->h2)
            return;
      }
      t = 2.0 * def->h * t;
      t = (CUBED_ROOT_4 * acc->h2 - cbrt(t * t)) / acc->h2mw2;
      if (t > 0.0)
          acc->tail_y = def->h / CUBED_ROOT_2 * sqrt(t);
}

/*
 * inverse functions -- compute edge coordinates
 * from the ellipse
 */

static double
outerXfromXY (
      double                  x,
      double                  y,
      struct arc_def          *def,
      struct accelerators     *acc)
{
      return x + (x * acc->h2l) / sqrt (x*x * acc->h4 + y*y * acc->w4);
}

static double
outerYfromXY (
      double            x,
      double            y,
      struct arc_def          *def,
      struct accelerators     *acc)
{
      return y + (y * acc->w2l) / sqrt (x*x * acc->h4 + y*y * acc->w4);
}

static double
innerXfromXY (
      double                  x,
      double                  y,
      struct arc_def          *def,
      struct accelerators     *acc)
{
      return x - (x * acc->h2l) / sqrt (x*x * acc->h4 + y*y * acc->w4);
}

static double
innerYfromXY (
      double                  x,
      double                  y,
      struct arc_def          *def,
      struct accelerators     *acc)
{
      return y - (y * acc->w2l) / sqrt (x*x * acc->h4 + y*y * acc->w4);
}

static double
innerYfromY (
      double      y,
      struct arc_def          *def,
      struct accelerators     *acc)
{
      double      x;

      x = (def->w / def->h) * sqrt (acc->h2 - y*y);

      return y - (y * acc->w2l) / sqrt (x*x * acc->h4 + y*y * acc->w4);
}

static void
computeLine (
      double            x1,
      double            y1,
      double            x2,
      double            y2,
      struct line *line)
{
      if (y1 == y2)
            line->valid = 0;
      else {
            line->m = (x1 - x2) / (y1 - y2);
            line->b = x1  - y1 * line->m;
            line->valid = 1;
      }
}

/*
 * compute various accelerators for an ellipse.  These
 * are simply values that are used repeatedly in
 * the computations
 */

static void
computeAcc (
      xArc              *tarc,
      int               lw,
      struct arc_def          *def,
      struct accelerators     *acc)
{
      def->w = ((double) tarc->width) / 2.0;
      def->h = ((double) tarc->height) / 2.0;
      def->l = ((double) lw) / 2.0;
      acc->h2 = def->h * def->h;
      acc->w2 = def->w * def->w;
      acc->h4 = acc->h2 * acc->h2;
      acc->w4 = acc->w2 * acc->w2;
      acc->h2l = acc->h2 * def->l;
      acc->w2l = acc->w2 * def->l;
      acc->h2mw2 = acc->h2 - acc->w2;
      acc->fromIntX = (tarc->width & 1) ? 0.5 : 0.0;
      acc->fromIntY = (tarc->height & 1) ? 0.5 : 0.0;
      acc->xorg = tarc->x + (tarc->width >> 1);
      acc->yorgu = tarc->y + (tarc->height >> 1);
      acc->yorgl = acc->yorgu + (tarc->height & 1);
      tailEllipseY (def, acc);
}
            
/*
 * compute y value bounds of various portions of the arc,
 * the outer edge, the ellipse and the inner edge.
 */

static void
computeBound (
      struct arc_def          *def,
      struct arc_bound  *bound,
      struct accelerators     *acc,
      miArcFacePtr            right,
      miArcFacePtr            left)
{
      double            t;
      double            innerTaily;
      double            tail_y;
      struct bound      innerx, outerx;
      struct bound      ellipsex;

      bound->ellipse.min = Dsin (def->a0) * def->h;
      bound->ellipse.max = Dsin (def->a1) * def->h;
      if (def->a0 == 45 && def->w == def->h)
            ellipsex.min = bound->ellipse.min;
      else
            ellipsex.min = Dcos (def->a0) * def->w;
      if (def->a1 == 45 && def->w == def->h)
            ellipsex.max = bound->ellipse.max;
      else
            ellipsex.max = Dcos (def->a1) * def->w;
      bound->outer.min = outerYfromXY (ellipsex.min, bound->ellipse.min, def, acc);
      bound->outer.max = outerYfromXY (ellipsex.max, bound->ellipse.max, def, acc);
      bound->inner.min = innerYfromXY (ellipsex.min, bound->ellipse.min, def, acc);
      bound->inner.max = innerYfromXY (ellipsex.max, bound->ellipse.max, def, acc);

      outerx.min = outerXfromXY (ellipsex.min, bound->ellipse.min, def, acc);
      outerx.max = outerXfromXY (ellipsex.max, bound->ellipse.max, def, acc);
      innerx.min = innerXfromXY (ellipsex.min, bound->ellipse.min, def, acc);
      innerx.max = innerXfromXY (ellipsex.max, bound->ellipse.max, def, acc);
      
      /*
       * save the line end points for the
       * cap code to use.  Careful here, these are
       * in cartesean coordinates (y increasing upwards)
       * while the cap code uses inverted coordinates
       * (y increasing downwards)
       */

      if (right) {
            right->counterClock.y = bound->outer.min;
            right->counterClock.x = outerx.min;
            right->center.y = bound->ellipse.min;
            right->center.x = ellipsex.min;
            right->clock.y = bound->inner.min;
            right->clock.x = innerx.min;
      }

      if (left) {
            left->clock.y = bound->outer.max;
            left->clock.x = outerx.max;
            left->center.y = bound->ellipse.max;
            left->center.x = ellipsex.max;
            left->counterClock.y = bound->inner.max;
            left->counterClock.x = innerx.max;
      }

      bound->left.min = bound->inner.max;
      bound->left.max = bound->outer.max;
      bound->right.min = bound->inner.min;
      bound->right.max = bound->outer.min;

      computeLine (innerx.min, bound->inner.min, outerx.min, bound->outer.min,
                  &acc->right);
      computeLine (innerx.max, bound->inner.max, outerx.max, bound->outer.max,
                 &acc->left);

      if (bound->inner.min > bound->inner.max) {
            t = bound->inner.min;
            bound->inner.min = bound->inner.max;
            bound->inner.max = t;
      }
      tail_y = acc->tail_y;
      if (tail_y > bound->ellipse.max)
            tail_y = bound->ellipse.max;
      else if (tail_y < bound->ellipse.min)
            tail_y = bound->ellipse.min;
      innerTaily = innerYfromY (tail_y, def, acc);
      if (bound->inner.min > innerTaily)
            bound->inner.min = innerTaily;
      if (bound->inner.max < innerTaily)
            bound->inner.max = innerTaily;
      bound->inneri.min = ICEIL(bound->inner.min - acc->fromIntY);
      bound->inneri.max = floor(bound->inner.max - acc->fromIntY);
      bound->outeri.min = ICEIL(bound->outer.min - acc->fromIntY);
      bound->outeri.max = floor(bound->outer.max - acc->fromIntY);
}

/*
 * this section computes the x value of the span at y 
 * intersected with the specified face of the ellipse.
 *
 * this is the min/max X value over the set of normal
 * lines to the entire ellipse,  the equation of the
 * normal lines is:
 *
 *     ellipse_x h^2                   h^2
 * x = ------------ y + ellipse_x (1 - --- )
 *     ellipse_y w^2                   w^2
 *
 * compute the derivative with-respect-to ellipse_y and solve
 * for zero:
 *    
 *       (w^2 - h^2) ellipse_y^3 + h^4 y
 * 0 = - ----------------------------------
 *       h w ellipse_y^2 sqrt (h^2 - ellipse_y^2)
 *
 *             (   h^4 y     )
 * ellipse_y = ( ----------  ) ^ (1/3)
 *             ( (h^2 - w^2) )
 *
 * The other two solutions to the equation are imaginary.
 *
 * This gives the position on the ellipse which generates
 * the normal with the largest/smallest x intersection point.
 *
 * Now compute the second derivative to check whether
 * the intersection is a minimum or maximum:
 *
 *    h (y0^3 (w^2 - h^2) + h^2 y (3y0^2 - 2h^2))
 * -  -------------------------------------------
 *          w y0^3 (sqrt (h^2 - y^2)) ^ 3
 *
 * as we only care about the sign,
 *
 * - (y0^3 (w^2 - h^2) + h^2 y (3y0^2 - 2h^2))
 *
 * or (to use accelerators),
 *
 * y0^3 (h^2 - w^2) - h^2 y (3y0^2 - 2h^2) 
 *
 */

/*
 * computes the position on the ellipse whose normal line
 * intersects the given scan line maximally
 */

static double
hookEllipseY (
      double                  scan_y,
      struct arc_bound  *bound,
      struct accelerators     *acc,
      int               left)
{
      double      ret;

      if (acc->h2mw2 == 0) {
            if ( (scan_y > 0 && !left) || (scan_y < 0 && left) )
                  return bound->ellipse.min;
            return bound->ellipse.max;
      }
      ret = (acc->h4 * scan_y) / (acc->h2mw2);
      if (ret >= 0)
            return cbrt (ret);
      else
            return -cbrt (-ret);
}

/*
 * computes the X value of the intersection of the
 * given scan line with the right side of the lower hook
 */

static double
hookX (
      double                  scan_y,
      struct arc_def          *def,
      struct arc_bound  *bound,
      struct accelerators     *acc,
      int               left)
{
      double      ellipse_y, x;
      double      maxMin;

      if (def->w != def->h) {
            ellipse_y = hookEllipseY (scan_y, bound, acc, left);
            if (boundedLe (ellipse_y, bound->ellipse)) {
                  /*
                   * compute the value of the second
                   * derivative
                   */
                  maxMin = ellipse_y*ellipse_y*ellipse_y * acc->h2mw2 -
                   acc->h2 * scan_y * (3 * ellipse_y*ellipse_y - 2*acc->h2);
                  if ((left && maxMin > 0) || (!left && maxMin < 0)) {
                        if (ellipse_y == 0)
                              return def->w + left ? -def->l : def->l;
                        x = (acc->h2 * scan_y - ellipse_y * acc->h2mw2) *
                              sqrt (acc->h2 - ellipse_y * ellipse_y) /
                              (def->h * def->w * ellipse_y);
                        return x;
                  }
            }
      }
      if (left) {
            if (acc->left.valid && boundedLe (scan_y, bound->left)) {
                  x = intersectLine (scan_y, acc->left);
            } else {
                  if (acc->right.valid)
                        x = intersectLine (scan_y, acc->right);
                  else
                        x = def->w - def->l;
            }
      } else {
            if (acc->right.valid && boundedLe (scan_y, bound->right)) {
                  x = intersectLine (scan_y, acc->right);
            } else {
                  if (acc->left.valid)
                        x = intersectLine (scan_y, acc->left);
                  else
                        x = def->w - def->l;
            }
      }
      return x;
}

/*
 * generate the set of spans with
 * the given y coordinate
 */

static void
arcSpan (
      int               y,
      int               lx,
      int               lw,
      int               rx,
      int               rw,
      struct arc_def          *def,
      struct arc_bound  *bounds,
      struct accelerators     *acc,
      int               mask)
{
      int linx, loutx, rinx, routx;
      double x, altx;

      if (boundedLe (y, bounds->inneri)) {
          linx = -(lx + lw);
          rinx = rx;
      } else {
          /*
           * intersection with left face
           */
          x = hookX (y + acc->fromIntY, def, bounds, acc, 1);
          if (acc->right.valid &&
            boundedLe (y + acc->fromIntY, bounds->right))
          {
            altx = intersectLine (y + acc->fromIntY, acc->right);
            if (altx < x)
                x = altx;
          }
          linx = -ICEIL(acc->fromIntX - x);
          rinx = ICEIL(acc->fromIntX + x);
      }
      if (boundedLe (y, bounds->outeri)) {
          loutx = -lx;
          routx = rx + rw;
      } else {
          /*
           * intersection with right face
           */
          x = hookX (y + acc->fromIntY, def, bounds, acc, 0);
          if (acc->left.valid &&
            boundedLe (y + acc->fromIntY, bounds->left))
          {
            altx = x;
            x = intersectLine (y + acc->fromIntY, acc->left);
            if (x < altx)
                x = altx;
          }
          loutx = -ICEIL(acc->fromIntX - x);
          routx = ICEIL(acc->fromIntX + x);
      }
      if (routx > rinx) {
          if (mask & 1)
            newFinalSpan (acc->yorgu - y,
                        acc->xorg + rinx, acc->xorg + routx);
          if (mask & 8)
            newFinalSpan (acc->yorgl + y,
                        acc->xorg + rinx, acc->xorg + routx);
      }
      if (loutx > linx) {
          if (mask & 2)
            newFinalSpan (acc->yorgu - y,
                        acc->xorg - loutx, acc->xorg - linx);
          if (mask & 4)
            newFinalSpan (acc->yorgl + y,
                        acc->xorg - loutx, acc->xorg - linx);
      }
}

static void
arcSpan0 (
      int               lx,
      int               lw,
      int               rx,
      int               rw,
      struct arc_def          *def,
      struct arc_bound  *bounds,
      struct accelerators     *acc,
      int               mask)
{
    double x;

    if (boundedLe (0, bounds->inneri) &&
      acc->left.valid && boundedLe (0, bounds->left) &&
      acc->left.b > 0)
    {
      x = def->w - def->l;
      if (acc->left.b < x)
          x = acc->left.b;
      lw = ICEIL(acc->fromIntX - x) - lx;
      rw += rx;
      rx = ICEIL(acc->fromIntX + x);
      rw -= rx;
    }
    arcSpan (0, lx, lw, rx, rw, def, bounds, acc, mask);
}

static void
tailSpan (
      int               y,
      int               lw,
      int               rw,
      struct arc_def          *def,
      struct arc_bound  *bounds,
      struct accelerators     *acc,
      int               mask)
{
    double yy, xalt, x, lx, rx;
    int n;

    if (boundedLe(y, bounds->outeri))
      arcSpan (y, 0, lw, -rw, rw, def, bounds, acc, mask);
    else if (def->w != def->h) {
      yy = y + acc->fromIntY;
      x = tailX(yy, def, bounds, acc);
      if (yy == 0.0 && x == -rw - acc->fromIntX)
          return;
      if (acc->right.valid && boundedLe (yy, bounds->right)) {
          rx = x;
          lx = -x;
          xalt = intersectLine (yy, acc->right);
          if (xalt >= -rw - acc->fromIntX && xalt <= rx)
            rx = xalt;
          n = ICEIL(acc->fromIntX + lx);
          if (lw > n) {
            if (mask & 2)
                newFinalSpan (acc->yorgu - y,
                          acc->xorg + n, acc->xorg + lw);
            if (mask & 4)
                newFinalSpan (acc->yorgl + y,
                          acc->xorg + n, acc->xorg + lw);
          }
          n = ICEIL(acc->fromIntX + rx);
          if (n > -rw) {
            if (mask & 1)
                newFinalSpan (acc->yorgu - y,
                          acc->xorg - rw, acc->xorg + n);
            if (mask & 8)
                newFinalSpan (acc->yorgl + y,
                          acc->xorg - rw, acc->xorg + n);
          }
      }
      arcSpan (y,
             ICEIL(acc->fromIntX - x), 0,
             ICEIL(acc->fromIntX + x), 0,
             def, bounds, acc, mask);
    }
}

/*
 * create whole arcs out of pieces.  This code is
 * very bad.
 */

static struct finalSpan **finalSpans = NULL;
static int        finalMiny = 0, finalMaxy = -1;
static int        finalSize = 0;

static int        nspans = 0; /* total spans, not just y coords */

struct finalSpan {
      struct finalSpan  *next;
      int               min, max;   /* x values */
};

static struct finalSpan    *freeFinalSpans, *tmpFinalSpan;

# define allocFinalSpan()   (freeFinalSpans ?\
                        ((tmpFinalSpan = freeFinalSpans), \
                         (freeFinalSpans = freeFinalSpans->next), \
                         (tmpFinalSpan->next = 0), \
                         tmpFinalSpan) : \
                       realAllocSpan ())

# define SPAN_CHUNK_SIZE    128

struct finalSpanChunk {
      struct finalSpan  data[SPAN_CHUNK_SIZE];
      struct finalSpanChunk   *next;
};

static struct finalSpanChunk  *chunks;

struct finalSpan *
realAllocSpan ()
{
      register struct finalSpanChunk      *newChunk;
      register struct finalSpan     *span;
      register int                  i;

      newChunk = (struct finalSpanChunk *) xalloc (sizeof (struct finalSpanChunk));
      if (!newChunk)
            return (struct finalSpan *) NULL;
      newChunk->next = chunks;
      chunks = newChunk;
      freeFinalSpans = span = newChunk->data + 1;
      for (i = 1; i < SPAN_CHUNK_SIZE-1; i++) {
            span->next = span+1;
            span++;
      }
      span->next = 0;
      span = newChunk->data;
      span->next = 0;
      return span;
}

static void
disposeFinalSpans (void)
{
      struct finalSpanChunk   *chunk, *next;

      for (chunk = chunks; chunk; chunk = next) {
            next = chunk->next;
            xfree (chunk);
      }
      chunks = 0;
      freeFinalSpans = 0;
      xfree(finalSpans);
      finalSpans = 0;
}

static void
fillSpans (
    DrawablePtr   pDrawable,
    GCPtr   pGC)
{
      register struct finalSpan     *span;
      register DDXPointPtr          xSpan;
      register int                  *xWidth;
      register int                  i;
      register struct finalSpan     **f;
      register int                  spany;
      DDXPointPtr             xSpans;
      int                     *xWidths;

      if (nspans == 0)
            return;
      xSpan = xSpans = (DDXPointPtr) ALLOCATE_LOCAL (nspans * sizeof (DDXPointRec));
      xWidth = xWidths = (int *) ALLOCATE_LOCAL (nspans * sizeof (int));
      if (xSpans && xWidths)
      {
          i = 0;
          f = finalSpans;
          for (spany = finalMiny; spany <= finalMaxy; spany++, f++) {
                for (span = *f; span; span=span->next) {
                      if (span->max <= span->min)
                            continue;
                      xSpan->x = span->min;
                      xSpan->y = spany;
                      ++xSpan;
                      *xWidth++ = span->max - span->min;
                      ++i;
                }
          }
          (*pGC->ops->FillSpans) (pDrawable, pGC, i, xSpans, xWidths, TRUE);
      }
      disposeFinalSpans ();
      if (xSpans)
          DEALLOCATE_LOCAL (xSpans);
      if (xWidths)
          DEALLOCATE_LOCAL (xWidths);
      finalMiny = 0;
      finalMaxy = -1;
      finalSize = 0;
      nspans = 0;
}

# define SPAN_REALLOC   100

# define findSpan(y) ((finalMiny <= (y) && (y) <= finalMaxy) ? \
                    &finalSpans[(y) - finalMiny] : \
                    realFindSpan (y))

static struct finalSpan **
realFindSpan (int y)
{
      struct finalSpan  **newSpans;
      int               newSize, newMiny, newMaxy;
      int               change;
      int               i;

      if (y < finalMiny || y > finalMaxy) {
            if (!finalSize) {
                  finalMiny = y;
                  finalMaxy = y - 1;
            }
            if (y < finalMiny)
                  change = finalMiny - y;
            else
                  change = y - finalMaxy;
            if (change >= SPAN_REALLOC)
                  change += SPAN_REALLOC;
            else
                  change = SPAN_REALLOC;
            newSize = finalSize + change;
            newSpans = (struct finalSpan **) xalloc
                              (newSize * sizeof (struct finalSpan *));
            if (!newSpans)
                return (struct finalSpan **)NULL;
            newMiny = finalMiny;
            newMaxy = finalMaxy;
            if (y < finalMiny)
                  newMiny = finalMiny - change;
            else
                  newMaxy = finalMaxy + change;
            if (finalSpans) {
                  memmove(((char *) newSpans) + (finalMiny-newMiny) * sizeof (struct finalSpan *),
                        (char *) finalSpans,
                         finalSize * sizeof (struct finalSpan *));
                  xfree (finalSpans);
            }
            if ((i = finalMiny - newMiny) > 0)
                  bzero ((char *)newSpans, i * sizeof (struct finalSpan *));
            if ((i = newMaxy - finalMaxy) > 0)
                  bzero ((char *)(newSpans + newSize - i),
                         i * sizeof (struct finalSpan *));
            finalSpans = newSpans;
            finalMaxy = newMaxy;
            finalMiny = newMiny;
            finalSize = newSize;
      }
      return &finalSpans[y - finalMiny];
}

static void
newFinalSpan (
    int           y,
    register int  xmin,
    register int  xmax)
{
      register struct finalSpan     *x;
      register struct finalSpan     **f;
      struct finalSpan        *oldx;
      struct finalSpan        *prev;

      f = findSpan (y);
      if (!f)
          return;
      oldx = 0;
      for (;;) {
            prev = 0;
            for (x = *f; x; x=x->next) {
                  if (x == oldx) {
                        prev = x;
                        continue;
                  }
                  if (x->min <= xmax && xmin <= x->max) {
                        if (oldx) {
                              oldx->min = min (x->min, xmin);
                              oldx->max = max (x->max, xmax);
                              if (prev)
                                    prev->next = x->next;
                              else
                                    *f = x->next;
                              --nspans;
                        } else {
                              x->min = min (x->min, xmin);
                              x->max = max (x->max, xmax);
                              oldx = x;
                        }
                        xmin = oldx->min;
                        xmax = oldx->max;
                        break;
                  }
                  prev = x;
            }
            if (!x)
                  break;
      }
      if (!oldx) {
            x = allocFinalSpan ();
            if (x)
            {
                x->min = xmin;
                x->max = xmax;
                x->next = *f;
                *f = x;
                ++nspans;
            }
      }
}

static void
mirrorSppPoint (
      int         quadrant,
      SppPointPtr sppPoint)
{
      switch (quadrant) {
      case 0:
            break;
      case 1:
            sppPoint->x = -sppPoint->x;
            break;
      case 2:
            sppPoint->x = -sppPoint->x;
            sppPoint->y = -sppPoint->y;
            break;
      case 3:
            sppPoint->y = -sppPoint->y;
            break;
      }
      /*
       * and translate to X coordinate system
       */
      sppPoint->y = -sppPoint->y;
}

/*
 * split an arc into pieces which are scan-converted
 * in the first-quadrant and mirrored into position.
 * This is necessary as the scan-conversion code can
 * only deal with arcs completely contained in the
 * first quadrant.
 */

static void
drawArc (
      xArc *tarc,
      int   l,
      int   a0,
      int   a1,
      miArcFacePtr      right,
      miArcFacePtr      left) /* save end line points */
{
      struct arc_def          def;
      struct accelerators     acc;
      int               startq, endq, curq;
      int               rightq, leftq = 0, righta = 0, lefta = 0;
      miArcFacePtr            passRight, passLeft;
      int               q0 = 0, q1 = 0, mask;
      struct band {
            int   a0, a1;
            int   mask;
      }     band[5], sweep[20];
      int               bandno, sweepno;
      int               i, j;
      int               flipRight = 0, flipLeft = 0;              
      int               copyEnd = 0;
      miArcSpanData           *spdata;
      Bool              mustFree;

      spdata = miComputeWideEllipse(l, tarc, &mustFree);
      if (!spdata)
          return;

      if (a1 < a0)
            a1 += 360 * 64;
      startq = a0 / (90 * 64);
      if (a0 == a1)
          endq = startq;
      else
          endq = (a1-1) / (90 * 64);
      bandno = 0;
      curq = startq;
      rightq = -1;
      for (;;) {
            switch (curq) {
            case 0:
                  if (a0 > 90 * 64)
                        q0 = 0;
                  else
                        q0 = a0;
                  if (a1 < 360 * 64)
                        q1 = min (a1, 90 * 64);
                  else
                        q1 = 90 * 64;
                  if (curq == startq && a0 == q0 && rightq < 0) {
                        righta = q0;
                        rightq = curq;
                  }
                  if (curq == endq && a1 == q1) {
                        lefta = q1;
                        leftq = curq;
                  }
                  break;
            case 1:
                  if (a1 < 90 * 64)
                        q0 = 0;
                  else
                        q0 = 180 * 64 - min (a1, 180 * 64);
                  if (a0 > 180 * 64)
                        q1 = 90 * 64;
                  else
                        q1 = 180 * 64 - max (a0, 90 * 64);
                  if (curq == startq && 180 * 64 - a0 == q1) {
                        righta = q1;
                        rightq = curq;
                  }
                  if (curq == endq && 180 * 64 - a1 == q0) {
                        lefta = q0;
                        leftq = curq;
                  }
                  break;
            case 2:
                  if (a0 > 270 * 64)
                        q0 = 0;
                  else
                        q0 = max (a0, 180 * 64) - 180 * 64;
                  if (a1 < 180 * 64)
                        q1 = 90 * 64;
                  else
                        q1 = min (a1, 270 * 64) - 180 * 64;
                  if (curq == startq && a0 - 180*64 == q0) {
                        righta = q0;
                        rightq = curq;
                  }
                  if (curq == endq && a1 - 180 * 64 == q1) {
                        lefta = q1;
                        leftq = curq;
                  }
                  break;
            case 3:
                  if (a1 < 270 * 64)
                        q0 = 0;
                  else
                        q0 = 360 * 64 - min (a1, 360 * 64);
                  q1 = 360 * 64 - max (a0, 270 * 64);
                  if (curq == startq && 360 * 64 - a0 == q1) {
                        righta = q1;
                        rightq = curq;
                  }
                  if (curq == endq && 360 * 64 - a1 == q0) {
                        lefta = q0;
                        leftq = curq;
                  }
                  break;
            }
            band[bandno].a0 = q0;
            band[bandno].a1 = q1;
            band[bandno].mask = 1 << curq;
            bandno++;
            if (curq == endq)
                  break;
            curq++;
            if (curq == 4) {
                  a0 = 0;
                  a1 -= 360 * 64;
                  curq = 0;
                  endq -= 4;
            }
      }
      sweepno = 0;
      for (;;) {
            q0 = 90 * 64;
            mask = 0;
            /*
             * find left-most point
             */
            for (i = 0; i < bandno; i++)
                  if (band[i].a0 <= q0) {
                        q0 = band[i].a0;
                        q1 = band[i].a1;
                        mask = band[i].mask;
                  }
            if (!mask)
                  break;
            /*
             * locate next point of change
             */
            for (i = 0; i < bandno; i++)
                  if (!(mask & band[i].mask)) {
                        if (band[i].a0 == q0) {
                              if (band[i].a1 < q1)
                                    q1 = band[i].a1;
                              mask |= band[i].mask;
                        } else if (band[i].a0 < q1)
                              q1 = band[i].a0;
                  }
            /*
             * create a new sweep
             */
            sweep[sweepno].a0 = q0;
            sweep[sweepno].a1 = q1;
            sweep[sweepno].mask = mask;
            sweepno++;
            /*
             * subtract the sweep from the affected bands
             */
            for (i = 0; i < bandno; i++)
                  if (band[i].a0 == q0) {
                        band[i].a0 = q1;
                        /*
                         * check if this band is empty
                         */
                        if (band[i].a0 == band[i].a1)
                              band[i].a1 = band[i].a0 = 90 * 64 + 1;
                  }
      }
      computeAcc (tarc, l, &def, &acc);
      for (j = 0; j < sweepno; j++) {
            mask = sweep[j].mask;
            passRight = passLeft = 0;
            if (mask & (1 << rightq)) {
                  if (sweep[j].a0 == righta)
                        passRight = right;
                  else if (sweep[j].a1 == righta) {
                        passLeft = right;
                        flipRight = 1;
                  }
            }
            if (mask & (1 << leftq)) {
                  if (sweep[j].a1 == lefta)
                  {
                        if (passLeft)
                              copyEnd = 1;
                        passLeft = left;
                  }
                  else if (sweep[j].a0 == lefta) {
                        if (passRight)
                              copyEnd = 1;
                        passRight = left;
                        flipLeft = 1;
                  }
            }
            drawQuadrant (&def, &acc, sweep[j].a0, sweep[j].a1, mask, 
                        passRight, passLeft, spdata);
      }
      /*
       * when copyEnd is set, both ends of the arc were computed
       * at the same time; drawQuadrant only takes one end though,
       * so the left end will be the only one holding the data.  Copy
       * it from there.
       */
      if (copyEnd)
            *right = *left;
      /*
       * mirror the coordinates generated for the
       * faces of the arc
       */
      if (right) {
            mirrorSppPoint (rightq, &right->clock);
            mirrorSppPoint (rightq, &right->center);
            mirrorSppPoint (rightq, &right->counterClock);
            if (flipRight) {
                  SppPointRec temp;

                  temp = right->clock;
                  right->clock = right->counterClock;
                  right->counterClock = temp;
            }
      }
      if (left) {
            mirrorSppPoint (leftq,  &left->counterClock);
            mirrorSppPoint (leftq,  &left->center);
            mirrorSppPoint (leftq,  &left->clock);
            if (flipLeft) {
                  SppPointRec temp;

                  temp = left->clock;
                  left->clock = left->counterClock;
                  left->counterClock = temp;
            }
      }
      if (mustFree)
          xfree(spdata);
}

static void
drawQuadrant (
      struct arc_def          *def,
      struct accelerators     *acc,
      int               a0,
      int               a1,
      int               mask,
      miArcFacePtr            right,
      miArcFacePtr            left,
      miArcSpanData           *spdata)
{
      struct arc_bound  bound;
      double                  yy, x, xalt;
      int               y, miny, maxy;
      int               n;
      miArcSpan         *span;

      def->a0 = ((double) a0) / 64.0;
      def->a1 = ((double) a1) / 64.0;
      computeBound (def, &bound, acc, right, left);
      yy = bound.inner.min;
      if (bound.outer.min < yy)
          yy = bound.outer.min;
      miny = ICEIL(yy - acc->fromIntY);
      yy = bound.inner.max;
      if (bound.outer.max > yy)
          yy = bound.outer.max;
      maxy = floor(yy - acc->fromIntY);
      y = spdata->k;
      span = spdata->spans;
      if (spdata->top)
      {
          if (a1 == 90 * 64 && (mask & 1))
            newFinalSpan (acc->yorgu - y - 1, acc->xorg, acc->xorg + 1);
          span++;
      }
      for (n = spdata->count1; --n >= 0; )
      {
          if (y < miny)
            return;
          if (y <= maxy) {
            arcSpan (y,
                   span->lx, -span->lx, 0, span->lx + span->lw,
                   def, &bound, acc, mask);
            if (span->rw + span->rx)
                tailSpan (y, -span->rw, -span->rx, def, &bound, acc, mask);
          }
          y--;
          span++;
      }
      if (y < miny)
          return;
      if (spdata->hole)
      {
          if (y <= maxy)
            arcSpan (y, 0, 0, 0, 1, def, &bound, acc, mask & 0xc);
      }
      for (n = spdata->count2; --n >= 0; )
      {
          if (y < miny)
            return;
          if (y <= maxy)
            arcSpan (y, span->lx, span->lw, span->rx, span->rw,
                   def, &bound, acc, mask);
          y--;
          span++;
      }
      if (spdata->bot && miny <= y && y <= maxy)
      {
          n = mask;
          if (y == miny)
            n &= 0xc;
          if (span->rw <= 0) {
            arcSpan0 (span->lx, -span->lx, 0, span->lx + span->lw,
                    def, &bound, acc, n);
            if (span->rw + span->rx)
                tailSpan (y, -span->rw, -span->rx, def, &bound, acc, n);
          }
          else
            arcSpan0 (span->lx, span->lw, span->rx, span->rw,
                    def, &bound, acc, n);
          y--;
      }
      while (y >= miny) {
          yy = y + acc->fromIntY;
          if (def->w == def->h) {
            xalt = def->w - def->l;
            x = -sqrt(xalt * xalt - yy * yy);
          } else {
            x = tailX(yy, def, &bound, acc);
            if (acc->left.valid && boundedLe (yy, bound.left)) {
                xalt = intersectLine (yy, acc->left);
                if (xalt < x)
                  x = xalt;
            }
            if (acc->right.valid && boundedLe (yy, bound.right)) {
                xalt = intersectLine (yy, acc->right);
                if (xalt < x)
                  x = xalt;
            }
          }
          arcSpan (y,
                 ICEIL(acc->fromIntX - x), 0,
                 ICEIL(acc->fromIntX + x), 0,
                 def, &bound, acc, mask);
          y--;
      }
}

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