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tkTrig.c

/* 
 * tkTrig.c --
 *
 *    This file contains a collection of trigonometry utility
 *    routines that are used by Tk and in particular by the
 *    canvas code.  It also has miscellaneous geometry functions
 *    used by canvases.
 *
 * Copyright (c) 1992-1994 The Regents of the University of California.
 * Copyright (c) 1994 Sun Microsystems, Inc.
 *
 * See the file "license.terms" for information on usage and redistribution
 * of this file, and for a DISCLAIMER OF ALL WARRANTIES.
 *
 * RCS: @(#) $Id: tkTrig.c,v 1.2 1998/09/14 18:23:20 stanton Exp $
 */

#include <stdio.h>
#include "tkInt.h"
#include "tkPort.h"
#include "tkCanvas.h"

#undef MIN
#define MIN(a,b) (((a) < (b)) ? (a) : (b))
#undef MAX
#define MAX(a,b) (((a) > (b)) ? (a) : (b))
#ifndef PI
#   define PI 3.14159265358979323846
#endif /* PI */

/*
 *--------------------------------------------------------------
 *
 * TkLineToPoint --
 *
 *    Compute the distance from a point to a finite line segment.
 *
 * Results:
 *    The return value is the distance from the line segment
 *    whose end-points are *end1Ptr and *end2Ptr to the point
 *    given by *pointPtr.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

double
TkLineToPoint(end1Ptr, end2Ptr, pointPtr)
    double end1Ptr[2];        /* Coordinates of first end-point of line. */
    double end2Ptr[2];        /* Coordinates of second end-point of line. */
    double pointPtr[2];       /* Points to coords for point. */
{
    double x, y;

    /*
     * Compute the point on the line that is closest to the
     * point.  This must be done separately for vertical edges,
     * horizontal edges, and other edges.
     */

    if (end1Ptr[0] == end2Ptr[0]) {

      /*
       * Vertical edge.
       */

      x = end1Ptr[0];
      if (end1Ptr[1] >= end2Ptr[1]) {
          y = MIN(end1Ptr[1], pointPtr[1]);
          y = MAX(y, end2Ptr[1]);
      } else {
          y = MIN(end2Ptr[1], pointPtr[1]);
          y = MAX(y, end1Ptr[1]);
      }
    } else if (end1Ptr[1] == end2Ptr[1]) {

      /*
       * Horizontal edge.
       */

      y = end1Ptr[1];
      if (end1Ptr[0] >= end2Ptr[0]) {
          x = MIN(end1Ptr[0], pointPtr[0]);
          x = MAX(x, end2Ptr[0]);
      } else {
          x = MIN(end2Ptr[0], pointPtr[0]);
          x = MAX(x, end1Ptr[0]);
      }
    } else {
      double m1, b1, m2, b2;

      /*
       * The edge is neither horizontal nor vertical.  Convert the
       * edge to a line equation of the form y = m1*x + b1.  Then
       * compute a line perpendicular to this edge but passing
       * through the point, also in the form y = m2*x + b2.
       */

      m1 = (end2Ptr[1] - end1Ptr[1])/(end2Ptr[0] - end1Ptr[0]);
      b1 = end1Ptr[1] - m1*end1Ptr[0];
      m2 = -1.0/m1;
      b2 = pointPtr[1] - m2*pointPtr[0];
      x = (b2 - b1)/(m1 - m2);
      y = m1*x + b1;
      if (end1Ptr[0] > end2Ptr[0]) {
          if (x > end1Ptr[0]) {
            x = end1Ptr[0];
            y = end1Ptr[1];
          } else if (x < end2Ptr[0]) {
            x = end2Ptr[0];
            y = end2Ptr[1];
          }
      } else {
          if (x > end2Ptr[0]) {
            x = end2Ptr[0];
            y = end2Ptr[1];
          } else if (x < end1Ptr[0]) {
            x = end1Ptr[0];
            y = end1Ptr[1];
          }
      }
    }

    /*
     * Compute the distance to the closest point.
     */

    return hypot(pointPtr[0] - x, pointPtr[1] - y);
}

/*
 *--------------------------------------------------------------
 *
 * TkLineToArea --
 *
 *    Determine whether a line lies entirely inside, entirely
 *    outside, or overlapping a given rectangular area.
 *
 * Results:
 *    -1 is returned if the line given by end1Ptr and end2Ptr
 *    is entirely outside the rectangle given by rectPtr.  0 is
 *    returned if the polygon overlaps the rectangle, and 1 is
 *    returned if the polygon is entirely inside the rectangle.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

int
TkLineToArea(end1Ptr, end2Ptr, rectPtr)
    double end1Ptr[2];        /* X and y coordinates for one endpoint
                         * of line. */
    double end2Ptr[2];        /* X and y coordinates for other endpoint
                         * of line. */
    double rectPtr[4];        /* Points to coords for rectangle, in the
                         * order x1, y1, x2, y2.  X1 must be no
                         * larger than x2, and y1 no larger than y2. */
{
    int inside1, inside2;

    /*
     * First check the two points individually to see whether they
     * are inside the rectangle or not.
     */

    inside1 = (end1Ptr[0] >= rectPtr[0]) && (end1Ptr[0] <= rectPtr[2])
          && (end1Ptr[1] >= rectPtr[1]) && (end1Ptr[1] <= rectPtr[3]);
    inside2 = (end2Ptr[0] >= rectPtr[0]) && (end2Ptr[0] <= rectPtr[2])
          && (end2Ptr[1] >= rectPtr[1]) && (end2Ptr[1] <= rectPtr[3]);
    if (inside1 != inside2) {
      return 0;
    }
    if (inside1 & inside2) {
      return 1;
    }

    /*
     * Both points are outside the rectangle, but still need to check
     * for intersections between the line and the rectangle.  Horizontal
     * and vertical lines are particularly easy, so handle them
     * separately.
     */

    if (end1Ptr[0] == end2Ptr[0]) {
      /*
       * Vertical line.
       */
    
      if (((end1Ptr[1] >= rectPtr[1]) ^ (end2Ptr[1] >= rectPtr[1]))
            && (end1Ptr[0] >= rectPtr[0])
            && (end1Ptr[0] <= rectPtr[2])) {
          return 0;
      }
    } else if (end1Ptr[1] == end2Ptr[1]) {
      /*
       * Horizontal line.
       */
    
      if (((end1Ptr[0] >= rectPtr[0]) ^ (end2Ptr[0] >= rectPtr[0]))
            && (end1Ptr[1] >= rectPtr[1])
            && (end1Ptr[1] <= rectPtr[3])) {
          return 0;
      }
    } else {
      double m, x, y, low, high;
    
      /*
       * Diagonal line.  Compute slope of line and use
       * for intersection checks against each of the
       * sides of the rectangle: left, right, bottom, top.
       */
    
      m = (end2Ptr[1] - end1Ptr[1])/(end2Ptr[0] - end1Ptr[0]);
      if (end1Ptr[0] < end2Ptr[0]) {
          low = end1Ptr[0];  high = end2Ptr[0];
      } else {
          low = end2Ptr[0]; high = end1Ptr[0];
      }
    
      /*
       * Left edge.
       */
    
      y = end1Ptr[1] + (rectPtr[0] - end1Ptr[0])*m;
      if ((rectPtr[0] >= low) && (rectPtr[0] <= high)
            && (y >= rectPtr[1]) && (y <= rectPtr[3])) {
          return 0;
      }
    
      /*
       * Right edge.
       */
    
      y += (rectPtr[2] - rectPtr[0])*m;
      if ((y >= rectPtr[1]) && (y <= rectPtr[3])
            && (rectPtr[2] >= low) && (rectPtr[2] <= high)) {
          return 0;
      }
    
      /*
       * Bottom edge.
       */
    
      if (end1Ptr[1] < end2Ptr[1]) {
          low = end1Ptr[1];  high = end2Ptr[1];
      } else {
          low = end2Ptr[1]; high = end1Ptr[1];
      }
      x = end1Ptr[0] + (rectPtr[1] - end1Ptr[1])/m;
      if ((x >= rectPtr[0]) && (x <= rectPtr[2])
            && (rectPtr[1] >= low) && (rectPtr[1] <= high)) {
          return 0;
      }
    
      /*
       * Top edge.
       */
    
      x += (rectPtr[3] - rectPtr[1])/m;
      if ((x >= rectPtr[0]) && (x <= rectPtr[2])
            && (rectPtr[3] >= low) && (rectPtr[3] <= high)) {
          return 0;
      }
    }
    return -1;
}

/*
 *--------------------------------------------------------------
 *
 * TkThickPolyLineToArea --
 *
 *    This procedure is called to determine whether a connected
 *    series of line segments lies entirely inside, entirely
 *    outside, or overlapping a given rectangular area.
 *
 * Results:
 *    -1 is returned if the lines are entirely outside the area,
 *    0 if they overlap, and 1 if they are entirely inside the
 *    given area.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

      /* ARGSUSED */
int
TkThickPolyLineToArea(coordPtr, numPoints, width, capStyle, joinStyle, rectPtr)
    double *coordPtr;         /* Points to an array of coordinates for
                         * the polyline:  x0, y0, x1, y1, ... */
    int numPoints;            /* Total number of points at *coordPtr. */
    double width;       /* Width of each line segment. */
    int capStyle;       /* How are end-points of polyline drawn?
                         * CapRound, CapButt, or CapProjecting. */
    int joinStyle;            /* How are joints in polyline drawn?
                         * JoinMiter, JoinRound, or JoinBevel. */
    double *rectPtr;          /* Rectangular area to check against. */
{
    double radius, poly[10];
    int count;
    int changedMiterToBevel;  /* Non-zero means that a mitered corner
                         * had to be treated as beveled after all
                         * because the angle was < 11 degrees. */
    int inside;               /* Tentative guess about what to return,
                         * based on all points seen so far:  one
                         * means everything seen so far was
                         * inside the area;  -1 means everything
                         * was outside the area.  0 means overlap
                         * has been found. */ 

    radius = width/2.0;
    inside = -1;

    if ((coordPtr[0] >= rectPtr[0]) && (coordPtr[0] <= rectPtr[2])
          && (coordPtr[1] >= rectPtr[1]) && (coordPtr[1] <= rectPtr[3])) {
      inside = 1;
    }

    /*
     * Iterate through all of the edges of the line, computing a polygon
     * for each edge and testing the area against that polygon.  In
     * addition, there are additional tests to deal with rounded joints
     * and caps.
     */

    changedMiterToBevel = 0;
    for (count = numPoints; count >= 2; count--, coordPtr += 2) {

      /*
       * If rounding is done around the first point of the edge
       * then test a circular region around the point with the
       * area.
       */

      if (((capStyle == CapRound) && (count == numPoints))
            || ((joinStyle == JoinRound) && (count != numPoints))) {
          poly[0] = coordPtr[0] - radius;
          poly[1] = coordPtr[1] - radius;
          poly[2] = coordPtr[0] + radius;
          poly[3] = coordPtr[1] + radius;
          if (TkOvalToArea(poly, rectPtr) != inside) {
            return 0;
          }
      }

      /*
       * Compute the polygonal shape corresponding to this edge,
       * consisting of two points for the first point of the edge
       * and two points for the last point of the edge.
       */

      if (count == numPoints) {
          TkGetButtPoints(coordPtr+2, coordPtr, width,
                capStyle == CapProjecting, poly, poly+2);
      } else if ((joinStyle == JoinMiter) && !changedMiterToBevel) {
          poly[0] = poly[6];
          poly[1] = poly[7];
          poly[2] = poly[4];
          poly[3] = poly[5];
      } else {
          TkGetButtPoints(coordPtr+2, coordPtr, width, 0, poly, poly+2);

          /*
           * If the last joint was beveled, then also check a
           * polygon comprising the last two points of the previous
           * polygon and the first two from this polygon;  this checks
           * the wedges that fill the beveled joint.
           */

          if ((joinStyle == JoinBevel) || changedMiterToBevel) {
            poly[8] = poly[0];
            poly[9] = poly[1];
            if (TkPolygonToArea(poly, 5, rectPtr) != inside) {
                return 0;
            }
            changedMiterToBevel = 0;
          }
      }
      if (count == 2) {
          TkGetButtPoints(coordPtr, coordPtr+2, width,
                capStyle == CapProjecting, poly+4, poly+6);
      } else if (joinStyle == JoinMiter) {
          if (TkGetMiterPoints(coordPtr, coordPtr+2, coordPtr+4,
                (double) width, poly+4, poly+6) == 0) {
            changedMiterToBevel = 1;
            TkGetButtPoints(coordPtr, coordPtr+2, width, 0, poly+4,
                  poly+6);
          }
      } else {
          TkGetButtPoints(coordPtr, coordPtr+2, width, 0, poly+4, poly+6);
      }
      poly[8] = poly[0];
      poly[9] = poly[1];
      if (TkPolygonToArea(poly, 5, rectPtr) != inside) {
          return 0;
      }
    }

    /*
     * If caps are rounded, check the cap around the final point
     * of the line.
     */

    if (capStyle == CapRound) {
      poly[0] = coordPtr[0] - radius;
      poly[1] = coordPtr[1] - radius;
      poly[2] = coordPtr[0] + radius;
      poly[3] = coordPtr[1] + radius;
      if (TkOvalToArea(poly, rectPtr) != inside) {
          return 0;
      }
    }

    return inside;
}

/*
 *--------------------------------------------------------------
 *
 * TkPolygonToPoint --
 *
 *    Compute the distance from a point to a polygon.
 *
 * Results:
 *    The return value is 0.0 if the point referred to by
 *    pointPtr is within the polygon referred to by polyPtr
 *    and numPoints.  Otherwise the return value is the
 *    distance of the point from the polygon.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

double
TkPolygonToPoint(polyPtr, numPoints, pointPtr)
    double *polyPtr;          /* Points to an array coordinates for
                         * closed polygon:  x0, y0, x1, y1, ...
                         * The polygon may be self-intersecting. */
    int numPoints;            /* Total number of points at *polyPtr. */
    double *pointPtr;         /* Points to coords for point. */
{
    double bestDist;          /* Closest distance between point and
                         * any edge in polygon. */
    int intersections;        /* Number of edges in the polygon that
                         * intersect a ray extending vertically
                         * upwards from the point to infinity. */
    int count;
    register double *pPtr;

    /*
     * Iterate through all of the edges in the polygon, updating
     * bestDist and intersections.
     *
     * TRICKY POINT:  when computing intersections, include left
     * x-coordinate of line within its range, but not y-coordinate.
     * Otherwise if the point lies exactly below a vertex we'll
     * count it as two intersections.
     */

    bestDist = 1.0e36;
    intersections = 0;

    for (count = numPoints, pPtr = polyPtr; count > 1; count--, pPtr += 2) {
      double x, y, dist;

      /*
       * Compute the point on the current edge closest to the point
       * and update the intersection count.  This must be done
       * separately for vertical edges, horizontal edges, and
       * other edges.
       */

      if (pPtr[2] == pPtr[0]) {

          /*
           * Vertical edge.
           */

          x = pPtr[0];
          if (pPtr[1] >= pPtr[3]) {
            y = MIN(pPtr[1], pointPtr[1]);
            y = MAX(y, pPtr[3]);
          } else {
            y = MIN(pPtr[3], pointPtr[1]);
            y = MAX(y, pPtr[1]);
          }
      } else if (pPtr[3] == pPtr[1]) {

          /*
           * Horizontal edge.
           */

          y = pPtr[1];
          if (pPtr[0] >= pPtr[2]) {
            x = MIN(pPtr[0], pointPtr[0]);
            x = MAX(x, pPtr[2]);
            if ((pointPtr[1] < y) && (pointPtr[0] < pPtr[0])
                  && (pointPtr[0] >= pPtr[2])) {
                intersections++;
            }
          } else {
            x = MIN(pPtr[2], pointPtr[0]);
            x = MAX(x, pPtr[0]);
            if ((pointPtr[1] < y) && (pointPtr[0] < pPtr[2])
                  && (pointPtr[0] >= pPtr[0])) {
                intersections++;
            }
          }
      } else {
          double m1, b1, m2, b2;
          int lower;                /* Non-zero means point below line. */

          /*
           * The edge is neither horizontal nor vertical.  Convert the
           * edge to a line equation of the form y = m1*x + b1.  Then
           * compute a line perpendicular to this edge but passing
           * through the point, also in the form y = m2*x + b2.
           */

          m1 = (pPtr[3] - pPtr[1])/(pPtr[2] - pPtr[0]);
          b1 = pPtr[1] - m1*pPtr[0];
          m2 = -1.0/m1;
          b2 = pointPtr[1] - m2*pointPtr[0];
          x = (b2 - b1)/(m1 - m2);
          y = m1*x + b1;
          if (pPtr[0] > pPtr[2]) {
            if (x > pPtr[0]) {
                x = pPtr[0];
                y = pPtr[1];
            } else if (x < pPtr[2]) {
                x = pPtr[2];
                y = pPtr[3];
            }
          } else {
            if (x > pPtr[2]) {
                x = pPtr[2];
                y = pPtr[3];
            } else if (x < pPtr[0]) {
                x = pPtr[0];
                y = pPtr[1];
            }
          }
          lower = (m1*pointPtr[0] + b1) > pointPtr[1];
          if (lower && (pointPtr[0] >= MIN(pPtr[0], pPtr[2]))
                && (pointPtr[0] < MAX(pPtr[0], pPtr[2]))) {
            intersections++;
          }
      }

      /*
       * Compute the distance to the closest point, and see if that
       * is the best distance seen so far.
       */

      dist = hypot(pointPtr[0] - x, pointPtr[1] - y);
      if (dist < bestDist) {
          bestDist = dist;
      }
    }

    /*
     * We've processed all of the points.  If the number of intersections
     * is odd, the point is inside the polygon.
     */

    if (intersections & 0x1) {
      return 0.0;
    }
    return bestDist;
}

/*
 *--------------------------------------------------------------
 *
 * TkPolygonToArea --
 *
 *    Determine whether a polygon lies entirely inside, entirely
 *    outside, or overlapping a given rectangular area.
 *
 * Results:
 *    -1 is returned if the polygon given by polyPtr and numPoints
 *    is entirely outside the rectangle given by rectPtr.  0 is
 *    returned if the polygon overlaps the rectangle, and 1 is
 *    returned if the polygon is entirely inside the rectangle.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

int
TkPolygonToArea(polyPtr, numPoints, rectPtr)
    double *polyPtr;          /* Points to an array coordinates for
                         * closed polygon:  x0, y0, x1, y1, ...
                         * The polygon may be self-intersecting. */
    int numPoints;            /* Total number of points at *polyPtr. */
    register double *rectPtr; /* Points to coords for rectangle, in the
                         * order x1, y1, x2, y2.  X1 and y1 must
                         * be lower-left corner. */
{
    int state;                /* State of all edges seen so far (-1 means
                         * outside, 1 means inside, won't ever be
                         * 0). */
    int count;
    register double *pPtr;

    /*
     * Iterate over all of the edges of the polygon and test them
     * against the rectangle.  Can quit as soon as the state becomes
     * "intersecting".
     */

    state = TkLineToArea(polyPtr, polyPtr+2, rectPtr);
    if (state == 0) {
      return 0;
    }
    for (pPtr = polyPtr+2, count = numPoints-1; count >= 2;
          pPtr += 2, count--) {
      if (TkLineToArea(pPtr, pPtr+2, rectPtr) != state) {
          return 0;
      }
    }

    /*
     * If all of the edges were inside the rectangle we're done.
     * If all of the edges were outside, then the rectangle could
     * still intersect the polygon (if it's entirely enclosed).
     * Call TkPolygonToPoint to figure this out.
     */

    if (state == 1) {
      return 1;
    }
    if (TkPolygonToPoint(polyPtr, numPoints, rectPtr) == 0.0) {
      return 0;
    }
    return -1;
}

/*
 *--------------------------------------------------------------
 *
 * TkOvalToPoint --
 *
 *    Computes the distance from a given point to a given
 *    oval, in canvas units.
 *
 * Results:
 *    The return value is 0 if the point given by *pointPtr is
 *    inside the oval.  If the point isn't inside the
 *    oval then the return value is approximately the distance
 *    from the point to the oval.  If the oval is filled, then
 *    anywhere in the interior is considered "inside";  if
 *    the oval isn't filled, then "inside" means only the area
 *    occupied by the outline.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

      /* ARGSUSED */
double
TkOvalToPoint(ovalPtr, width, filled, pointPtr)
    double ovalPtr[4];        /* Pointer to array of four coordinates
                         * (x1, y1, x2, y2) defining oval's bounding
                         * box. */
    double width;       /* Width of outline for oval. */
    int filled;               /* Non-zero means oval should be treated as
                         * filled;  zero means only consider outline. */
    double pointPtr[2];       /* Coordinates of point. */
{
    double xDelta, yDelta, scaledDistance, distToOutline, distToCenter;
    double xDiam, yDiam;

    /*
     * Compute the distance between the center of the oval and the
     * point in question, using a coordinate system where the oval
     * has been transformed to a circle with unit radius.
     */

    xDelta = (pointPtr[0] - (ovalPtr[0] + ovalPtr[2])/2.0);
    yDelta = (pointPtr[1] - (ovalPtr[1] + ovalPtr[3])/2.0);
    distToCenter = hypot(xDelta, yDelta);
    scaledDistance = hypot(xDelta / ((ovalPtr[2] + width - ovalPtr[0])/2.0),
          yDelta / ((ovalPtr[3] + width - ovalPtr[1])/2.0));


    /*
     * If the scaled distance is greater than 1 then it means no
     * hit.  Compute the distance from the point to the edge of
     * the circle, then scale this distance back to the original
     * coordinate system.
     *
     * Note: this distance isn't completely accurate.  It's only
     * an approximation, and it can overestimate the correct
     * distance when the oval is eccentric.
     */

    if (scaledDistance > 1.0) {
      return (distToCenter/scaledDistance) * (scaledDistance - 1.0);
    }

    /*
     * Scaled distance less than 1 means the point is inside the
     * outer edge of the oval.  If this is a filled oval, then we
     * have a hit.  Otherwise, do the same computation as above
     * (scale back to original coordinate system), but also check
     * to see if the point is within the width of the outline.
     */

    if (filled) {
      return 0.0;
    }
    if (scaledDistance > 1E-10) {
      distToOutline = (distToCenter/scaledDistance) * (1.0 - scaledDistance)
            - width;
    } else {
      /*
       * Avoid dividing by a very small number (it could cause an
       * arithmetic overflow).  This problem occurs if the point is
       * very close to the center of the oval.
       */

      xDiam = ovalPtr[2] - ovalPtr[0];
      yDiam = ovalPtr[3] - ovalPtr[1];
      if (xDiam < yDiam) {
          distToOutline = (xDiam - width)/2;
      } else {
          distToOutline = (yDiam - width)/2;
      }
    }

    if (distToOutline < 0.0) {
      return 0.0;
    }
    return distToOutline;
}

/*
 *--------------------------------------------------------------
 *
 * TkOvalToArea --
 *
 *    Determine whether an oval lies entirely inside, entirely
 *    outside, or overlapping a given rectangular area.
 *
 * Results:
 *    -1 is returned if the oval described by ovalPtr is entirely
 *    outside the rectangle given by rectPtr.  0 is returned if the
 *    oval overlaps the rectangle, and 1 is returned if the oval
 *    is entirely inside the rectangle.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

int
TkOvalToArea(ovalPtr, rectPtr)
    register double *ovalPtr; /* Points to coordinates definining the
                         * bounding rectangle for the oval: x1, y1,
                         * x2, y2.  X1 must be less than x2 and y1
                         * less than y2. */
    register double *rectPtr; /* Points to coords for rectangle, in the
                         * order x1, y1, x2, y2.  X1 and y1 must
                         * be lower-left corner. */
{
    double centerX, centerY, radX, radY, deltaX, deltaY;

    /*
     * First, see if oval is entirely inside rectangle or entirely
     * outside rectangle.
     */

    if ((rectPtr[0] <= ovalPtr[0]) && (rectPtr[2] >= ovalPtr[2])
          && (rectPtr[1] <= ovalPtr[1]) && (rectPtr[3] >= ovalPtr[3])) {
      return 1;
    }
    if ((rectPtr[2] < ovalPtr[0]) || (rectPtr[0] > ovalPtr[2])
          || (rectPtr[3] < ovalPtr[1]) || (rectPtr[1] > ovalPtr[3])) {
      return -1;
    }

    /*
     * Next, go through the rectangle side by side.  For each side
     * of the rectangle, find the point on the side that is closest
     * to the oval's center, and see if that point is inside the
     * oval.  If at least one such point is inside the oval, then
     * the rectangle intersects the oval.
     */

    centerX = (ovalPtr[0] + ovalPtr[2])/2;
    centerY = (ovalPtr[1] + ovalPtr[3])/2;
    radX = (ovalPtr[2] - ovalPtr[0])/2;
    radY = (ovalPtr[3] - ovalPtr[1])/2;

    deltaY = rectPtr[1] - centerY;
    if (deltaY < 0.0) {
      deltaY = centerY - rectPtr[3];
      if (deltaY < 0.0) {
          deltaY = 0;
      }
    }
    deltaY /= radY;
    deltaY *= deltaY;

    /*
     * Left side:
     */

    deltaX = (rectPtr[0] - centerX)/radX;
    deltaX *= deltaX;
    if ((deltaX + deltaY) <= 1.0) {
      return 0;
    }

    /*
     * Right side:
     */

    deltaX = (rectPtr[2] - centerX)/radX;
    deltaX *= deltaX;
    if ((deltaX + deltaY) <= 1.0) {
      return 0;
    }

    deltaX = rectPtr[0] - centerX;
    if (deltaX < 0.0) {
      deltaX = centerX - rectPtr[2];
      if (deltaX < 0.0) {
          deltaX = 0;
      }
    }
    deltaX /= radX;
    deltaX *= deltaX;

    /*
     * Bottom side:
     */

    deltaY = (rectPtr[1] - centerY)/radY;
    deltaY *= deltaY;
    if ((deltaX + deltaY) < 1.0) {
      return 0;
    }

    /*
     * Top side:
     */

    deltaY = (rectPtr[3] - centerY)/radY;
    deltaY *= deltaY;
    if ((deltaX + deltaY) < 1.0) {
      return 0;
    }

    return -1;
}

/*
 *--------------------------------------------------------------
 *
 * TkIncludePoint --
 *
 *    Given a point and a generic canvas item header, expand
 *    the item's bounding box if needed to include the point.
 *
 * Results:
 *    None.
 *
 * Side effects:
 *    The boudn.
 *
 *--------------------------------------------------------------
 */

      /* ARGSUSED */
void
TkIncludePoint(itemPtr, pointPtr)
    register Tk_Item *itemPtr;            /* Item whose bounding box is
                               * being calculated. */
    double *pointPtr;               /* Address of two doubles giving
                               * x and y coordinates of point. */
{
    int tmp;

    tmp = (int) (pointPtr[0] + 0.5);
    if (tmp < itemPtr->x1) {
      itemPtr->x1 = tmp;
    }
    if (tmp > itemPtr->x2) {
      itemPtr->x2 = tmp;
    }
    tmp = (int) (pointPtr[1] + 0.5);
    if (tmp < itemPtr->y1) {
      itemPtr->y1 = tmp;
    }
    if (tmp > itemPtr->y2) {
      itemPtr->y2 = tmp;
    }
}

/*
 *--------------------------------------------------------------
 *
 * TkBezierScreenPoints --
 *
 *    Given four control points, create a larger set of XPoints
 *    for a Bezier spline based on the points.
 *
 * Results:
 *    The array at *xPointPtr gets filled in with numSteps XPoints
 *    corresponding to the Bezier spline defined by the four 
 *    control points.  Note:  no output point is generated for the
 *    first input point, but an output point *is* generated for
 *    the last input point.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

void
TkBezierScreenPoints(canvas, control, numSteps, xPointPtr)
    Tk_Canvas canvas;               /* Canvas in which curve is to be
                               * drawn. */
    double control[];               /* Array of coordinates for four
                               * control points:  x0, y0, x1, y1,
                               * ... x3 y3. */
    int numSteps;             /* Number of curve points to
                               * generate.  */
    register XPoint *xPointPtr;           /* Where to put new points. */
{
    int i;
    double u, u2, u3, t, t2, t3;

    for (i = 1; i <= numSteps; i++, xPointPtr++) {
      t = ((double) i)/((double) numSteps);
      t2 = t*t;
      t3 = t2*t;
      u = 1.0 - t;
      u2 = u*u;
      u3 = u2*u;
      Tk_CanvasDrawableCoords(canvas,
            (control[0]*u3 + 3.0 * (control[2]*t*u2 + control[4]*t2*u)
                + control[6]*t3),
            (control[1]*u3 + 3.0 * (control[3]*t*u2 + control[5]*t2*u)
                + control[7]*t3),
            &xPointPtr->x, &xPointPtr->y);
    }
}

/*
 *--------------------------------------------------------------
 *
 * TkBezierPoints --
 *
 *    Given four control points, create a larger set of points
 *    for a Bezier spline based on the points.
 *
 * Results:
 *    The array at *coordPtr gets filled in with 2*numSteps
 *    coordinates, which correspond to the Bezier spline defined
 *    by the four control points.  Note:  no output point is
 *    generated for the first input point, but an output point
 *    *is* generated for the last input point.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

void
TkBezierPoints(control, numSteps, coordPtr)
    double control[];               /* Array of coordinates for four
                               * control points:  x0, y0, x1, y1,
                               * ... x3 y3. */
    int numSteps;             /* Number of curve points to
                               * generate.  */
    register double *coordPtr;            /* Where to put new points. */
{
    int i;
    double u, u2, u3, t, t2, t3;

    for (i = 1; i <= numSteps; i++, coordPtr += 2) {
      t = ((double) i)/((double) numSteps);
      t2 = t*t;
      t3 = t2*t;
      u = 1.0 - t;
      u2 = u*u;
      u3 = u2*u;
      coordPtr[0] = control[0]*u3
            + 3.0 * (control[2]*t*u2 + control[4]*t2*u) + control[6]*t3;
      coordPtr[1] = control[1]*u3
            + 3.0 * (control[3]*t*u2 + control[5]*t2*u) + control[7]*t3;
    }
}

/*
 *--------------------------------------------------------------
 *
 * TkMakeBezierCurve --
 *
 *    Given a set of points, create a new set of points that fit
 *    parabolic splines to the line segments connecting the original
 *    points.  Produces output points in either of two forms.
 *
 *    Note: in spite of this procedure's name, it does *not* generate
 *    Bezier curves.  Since only three control points are used for
 *    each curve segment, not four, the curves are actually just
 *    parabolic.
 *
 * Results:
 *    Either or both of the xPoints or dblPoints arrays are filled
 *    in.  The return value is the number of points placed in the
 *    arrays.  Note:  if the first and last points are the same, then
 *    a closed curve is generated.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

int
TkMakeBezierCurve(canvas, pointPtr, numPoints, numSteps, xPoints, dblPoints)
    Tk_Canvas canvas;               /* Canvas in which curve is to be
                               * drawn. */
    double *pointPtr;               /* Array of input coordinates:  x0,
                               * y0, x1, y1, etc.. */
    int numPoints;                  /* Number of points at pointPtr. */
    int numSteps;             /* Number of steps to use for each
                               * spline segments (determines
                               * smoothness of curve). */
    XPoint xPoints[];               /* Array of XPoints to fill in (e.g.
                               * for display.  NULL means don't
                               * fill in any XPoints. */
    double dblPoints[];             /* Array of points to fill in as
                               * doubles, in the form x0, y0,
                               * x1, y1, ....  NULL means don't
                               * fill in anything in this form. 
                               * Caller must make sure that this
                               * array has enough space. */
{
    int closed, outputPoints, i;
    int numCoords = numPoints*2;
    double control[8];

    /*
     * If the curve is a closed one then generate a special spline
     * that spans the last points and the first ones.  Otherwise
     * just put the first point into the output.
     */

    outputPoints = 0;
    if ((pointPtr[0] == pointPtr[numCoords-2])
          && (pointPtr[1] == pointPtr[numCoords-1])) {
      closed = 1;
      control[0] = 0.5*pointPtr[numCoords-4] + 0.5*pointPtr[0];
      control[1] = 0.5*pointPtr[numCoords-3] + 0.5*pointPtr[1];
      control[2] = 0.167*pointPtr[numCoords-4] + 0.833*pointPtr[0];
      control[3] = 0.167*pointPtr[numCoords-3] + 0.833*pointPtr[1];
      control[4] = 0.833*pointPtr[0] + 0.167*pointPtr[2];
      control[5] = 0.833*pointPtr[1] + 0.167*pointPtr[3];
      control[6] = 0.5*pointPtr[0] + 0.5*pointPtr[2];
      control[7] = 0.5*pointPtr[1] + 0.5*pointPtr[3];
      if (xPoints != NULL) {
          Tk_CanvasDrawableCoords(canvas, control[0], control[1],
                &xPoints->x, &xPoints->y);
          TkBezierScreenPoints(canvas, control, numSteps, xPoints+1);
          xPoints += numSteps+1;
      }
      if (dblPoints != NULL) {
          dblPoints[0] = control[0];
          dblPoints[1] = control[1];
          TkBezierPoints(control, numSteps, dblPoints+2);
          dblPoints += 2*(numSteps+1);
      }
      outputPoints += numSteps+1;
    } else {
      closed = 0;
      if (xPoints != NULL) {
          Tk_CanvasDrawableCoords(canvas, pointPtr[0], pointPtr[1],
                &xPoints->x, &xPoints->y);
          xPoints += 1;
      }
      if (dblPoints != NULL) {
          dblPoints[0] = pointPtr[0];
          dblPoints[1] = pointPtr[1];
          dblPoints += 2;
      }
      outputPoints += 1;
    }

    for (i = 2; i < numPoints; i++, pointPtr += 2) {
      /*
       * Set up the first two control points.  This is done
       * differently for the first spline of an open curve
       * than for other cases.
       */

      if ((i == 2) && !closed) {
          control[0] = pointPtr[0];
          control[1] = pointPtr[1];
          control[2] = 0.333*pointPtr[0] + 0.667*pointPtr[2];
          control[3] = 0.333*pointPtr[1] + 0.667*pointPtr[3];
      } else {
          control[0] = 0.5*pointPtr[0] + 0.5*pointPtr[2];
          control[1] = 0.5*pointPtr[1] + 0.5*pointPtr[3];
          control[2] = 0.167*pointPtr[0] + 0.833*pointPtr[2];
          control[3] = 0.167*pointPtr[1] + 0.833*pointPtr[3];
      }

      /*
       * Set up the last two control points.  This is done
       * differently for the last spline of an open curve
       * than for other cases.
       */

      if ((i == (numPoints-1)) && !closed) {
          control[4] = .667*pointPtr[2] + .333*pointPtr[4];
          control[5] = .667*pointPtr[3] + .333*pointPtr[5];
          control[6] = pointPtr[4];
          control[7] = pointPtr[5];
      } else {
          control[4] = .833*pointPtr[2] + .167*pointPtr[4];
          control[5] = .833*pointPtr[3] + .167*pointPtr[5];
          control[6] = 0.5*pointPtr[2] + 0.5*pointPtr[4];
          control[7] = 0.5*pointPtr[3] + 0.5*pointPtr[5];
      }

      /*
       * If the first two points coincide, or if the last
       * two points coincide, then generate a single
       * straight-line segment by outputting the last control
       * point.
       */

      if (((pointPtr[0] == pointPtr[2]) && (pointPtr[1] == pointPtr[3]))
            || ((pointPtr[2] == pointPtr[4])
            && (pointPtr[3] == pointPtr[5]))) {
          if (xPoints != NULL) {
            Tk_CanvasDrawableCoords(canvas, control[6], control[7],
                  &xPoints[0].x, &xPoints[0].y);
            xPoints++;
          }
          if (dblPoints != NULL) {
            dblPoints[0] = control[6];
            dblPoints[1] = control[7];
            dblPoints += 2;
          }
          outputPoints += 1;
          continue;
      }

      /*
       * Generate a Bezier spline using the control points.
       */


      if (xPoints != NULL) {
          TkBezierScreenPoints(canvas, control, numSteps, xPoints);
          xPoints += numSteps;
      }
      if (dblPoints != NULL) {
          TkBezierPoints(control, numSteps, dblPoints);
          dblPoints += 2*numSteps;
      }
      outputPoints += numSteps;
    }
    return outputPoints;
}

/*
 *--------------------------------------------------------------
 *
 * TkMakeBezierPostscript --
 *
 *    This procedure generates Postscript commands that create
 *    a path corresponding to a given Bezier curve.
 *
 * Results:
 *    None.  Postscript commands to generate the path are appended
 *    to interp->result.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

void
TkMakeBezierPostscript(interp, canvas, pointPtr, numPoints)
    Tcl_Interp *interp;             /* Interpreter in whose result the
                               * Postscript is to be stored. */
    Tk_Canvas canvas;               /* Canvas widget for which the
                               * Postscript is being generated. */
    double *pointPtr;               /* Array of input coordinates:  x0,
                               * y0, x1, y1, etc.. */
    int numPoints;                  /* Number of points at pointPtr. */
{
    int closed, i;
    int numCoords = numPoints*2;
    double control[8];
    char buffer[200];

    /*
     * If the curve is a closed one then generate a special spline
     * that spans the last points and the first ones.  Otherwise
     * just put the first point into the path.
     */

    if ((pointPtr[0] == pointPtr[numCoords-2])
          && (pointPtr[1] == pointPtr[numCoords-1])) {
      closed = 1;
      control[0] = 0.5*pointPtr[numCoords-4] + 0.5*pointPtr[0];
      control[1] = 0.5*pointPtr[numCoords-3] + 0.5*pointPtr[1];
      control[2] = 0.167*pointPtr[numCoords-4] + 0.833*pointPtr[0];
      control[3] = 0.167*pointPtr[numCoords-3] + 0.833*pointPtr[1];
      control[4] = 0.833*pointPtr[0] + 0.167*pointPtr[2];
      control[5] = 0.833*pointPtr[1] + 0.167*pointPtr[3];
      control[6] = 0.5*pointPtr[0] + 0.5*pointPtr[2];
      control[7] = 0.5*pointPtr[1] + 0.5*pointPtr[3];
      sprintf(buffer, "%.15g %.15g moveto\n%.15g %.15g %.15g %.15g %.15g %.15g curveto\n",
            control[0], Tk_CanvasPsY(canvas, control[1]),
            control[2], Tk_CanvasPsY(canvas, control[3]),
            control[4], Tk_CanvasPsY(canvas, control[5]),
            control[6], Tk_CanvasPsY(canvas, control[7]));
    } else {
      closed = 0;
      control[6] = pointPtr[0];
      control[7] = pointPtr[1];
      sprintf(buffer, "%.15g %.15g moveto\n",
            control[6], Tk_CanvasPsY(canvas, control[7]));
    }
    Tcl_AppendResult(interp, buffer, (char *) NULL);

    /*
     * Cycle through all the remaining points in the curve, generating
     * a curve section for each vertex in the linear path.
     */

    for (i = numPoints-2, pointPtr += 2; i > 0; i--, pointPtr += 2) {
      control[2] = 0.333*control[6] + 0.667*pointPtr[0];
      control[3] = 0.333*control[7] + 0.667*pointPtr[1];

      /*
       * Set up the last two control points.  This is done
       * differently for the last spline of an open curve
       * than for other cases.
       */

      if ((i == 1) && !closed) {
          control[6] = pointPtr[2];
          control[7] = pointPtr[3];
      } else {
          control[6] = 0.5*pointPtr[0] + 0.5*pointPtr[2];
          control[7] = 0.5*pointPtr[1] + 0.5*pointPtr[3];
      }
      control[4] = 0.333*control[6] + 0.667*pointPtr[0];
      control[5] = 0.333*control[7] + 0.667*pointPtr[1];

      sprintf(buffer, "%.15g %.15g %.15g %.15g %.15g %.15g curveto\n",
            control[2], Tk_CanvasPsY(canvas, control[3]),
            control[4], Tk_CanvasPsY(canvas, control[5]),
            control[6], Tk_CanvasPsY(canvas, control[7]));
      Tcl_AppendResult(interp, buffer, (char *) NULL);
    }
}

/*
 *--------------------------------------------------------------
 *
 * TkGetMiterPoints --
 *
 *    Given three points forming an angle, compute the
 *    coordinates of the inside and outside points of
 *    the mitered corner formed by a line of a given
 *    width at that angle.
 *
 * Results:
 *    If the angle formed by the three points is less than
 *    11 degrees then 0 is returned and m1 and m2 aren't
 *    modified.  Otherwise 1 is returned and the points at
 *    m1 and m2 are filled in with the positions of the points
 *    of the mitered corner.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

int
TkGetMiterPoints(p1, p2, p3, width, m1, m2)
    double p1[];        /* Points to x- and y-coordinates of point
                         * before vertex. */
    double p2[];        /* Points to x- and y-coordinates of vertex
                         * for mitered joint. */
    double p3[];        /* Points to x- and y-coordinates of point
                         * after vertex. */
    double width;       /* Width of line.  */
    double m1[];        /* Points to place to put "left" vertex
                         * point (see as you face from p1 to p2). */
    double m2[];        /* Points to place to put "right" vertex
                         * point. */
{
    double theta1;            /* Angle of segment p2-p1. */
    double theta2;            /* Angle of segment p2-p3. */
    double theta;       /* Angle between line segments (angle
                         * of joint). */
    double theta3;            /* Angle that bisects theta1 and
                         * theta2 and points to m1. */
    double dist;        /* Distance of miter points from p2. */
    double deltaX, deltaY;    /* X and y offsets cooresponding to
                         * dist (fudge factors for bounding
                         * box). */
    double p1x, p1y, p2x, p2y, p3x, p3y;
    static double elevenDegrees = (11.0*2.0*PI)/360.0;

    /*
     * Round the coordinates to integers to mimic what happens when the
     * line segments are displayed; without this code, the bounding box
     * of a mitered line can be miscomputed greatly.
     */

    p1x = floor(p1[0]+0.5);
    p1y = floor(p1[1]+0.5);
    p2x = floor(p2[0]+0.5);
    p2y = floor(p2[1]+0.5);
    p3x = floor(p3[0]+0.5);
    p3y = floor(p3[1]+0.5);

    if (p2y == p1y) {
      theta1 = (p2x < p1x) ? 0 : PI;
    } else if (p2x == p1x) {
      theta1 = (p2y < p1y) ? PI/2.0 : -PI/2.0;
    } else {
      theta1 = atan2(p1y - p2y, p1x - p2x);
    }
    if (p3y == p2y) {
      theta2 = (p3x > p2x) ? 0 : PI;
    } else if (p3x == p2x) {
      theta2 = (p3y > p2y) ? PI/2.0 : -PI/2.0;
    } else {
      theta2 = atan2(p3y - p2y, p3x - p2x);
    }
    theta = theta1 - theta2;
    if (theta > PI) {
      theta -= 2*PI;
    } else if (theta < -PI) {
      theta += 2*PI;
    }
    if ((theta < elevenDegrees) && (theta > -elevenDegrees)) {
      return 0;
    }
    dist = 0.5*width/sin(0.5*theta);
    if (dist < 0.0) {
      dist = -dist;
    }

    /*
     * Compute theta3 (make sure that it points to the left when
     * looking from p1 to p2).
     */

    theta3 = (theta1 + theta2)/2.0;
    if (sin(theta3 - (theta1 + PI)) < 0.0) {
      theta3 += PI;
    }
    deltaX = dist*cos(theta3);
    m1[0] = p2x + deltaX;
    m2[0] = p2x - deltaX;
    deltaY = dist*sin(theta3);
    m1[1] = p2y + deltaY;
    m2[1] = p2y - deltaY;
    return 1;
}

/*
 *--------------------------------------------------------------
 *
 * TkGetButtPoints --
 *
 *    Given two points forming a line segment, compute the
 *    coordinates of two endpoints of a rectangle formed by
 *    bloating the line segment until it is width units wide.
 *
 * Results:
 *    There is no return value.  M1 and m2 are filled in to
 *    correspond to m1 and m2 in the diagram below:
 *
 *             ----------------* m1
 *                         |
 *          p1 *---------------* p2
 *                         |
 *             ----------------* m2
 *
 *    M1 and m2 will be W units apart, with p2 centered between
 *    them and m1-m2 perpendicular to p1-p2.  However, if
 *    "project" is true then m1 and m2 will be as follows:
 *
 *             -------------------* m1
 *                        p2  |
 *          p1 *---------------*  |
 *                            |
 *             -------------------* m2
 *
 *    In this case p2 will be width/2 units from the segment m1-m2.
 *
 * Side effects:
 *    None.
 *
 *--------------------------------------------------------------
 */

void
TkGetButtPoints(p1, p2, width, project, m1, m2)
    double p1[];        /* Points to x- and y-coordinates of point
                         * before vertex. */
    double p2[];        /* Points to x- and y-coordinates of vertex
                         * for mitered joint. */
    double width;       /* Width of line.  */
    int project;        /* Non-zero means project p2 by an additional
                         * width/2 before computing m1 and m2. */
    double m1[];        /* Points to place to put "left" result
                         * point, as you face from p1 to p2. */
    double m2[];        /* Points to place to put "right" result
                         * point. */
{
    double length;            /* Length of p1-p2 segment. */
    double deltaX, deltaY;    /* Increments in coords. */

    width *= 0.5;
    length = hypot(p2[0] - p1[0], p2[1] - p1[1]);
    if (length == 0.0) {
      m1[0] = m2[0] = p2[0];
      m1[1] = m2[1] = p2[1];
    } else {
      deltaX = -width * (p2[1] - p1[1]) / length;
      deltaY = width * (p2[0] - p1[0]) / length;
      m1[0] = p2[0] + deltaX;
      m2[0] = p2[0] - deltaX;
      m1[1] = p2[1] + deltaY;
      m2[1] = p2[1] - deltaY;
      if (project) {
          m1[0] += deltaY;
          m2[0] += deltaY;
          m1[1] -= deltaX;
          m2[1] -= deltaX;
      }
    }
}

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