/* * 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) 1994-1997 Sun Microsystems, Inc. * * See the file "license.terms" for information on usage and redistribution of * this file, and for a DISCLAIMER OF ALL WARRANTIES. */ #include "tkInt.h" #include "tkCanvas.h" #undef MIN #define MIN(a,b) (((a) < (b)) ? (a) : (b)) #undef MAX #define MAX(a,b) (((a) > (b)) ? (a) : (b)) /* *-------------------------------------------------------------- * * 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( 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( 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 function 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( 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( 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( 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( 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( register double *ovalPtr, /* Points to coordinates defining 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( 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 * curve 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( 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 * curve 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( 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: the name of this function should *not* be taken to mean that it * interprets the input points as directly defining Bezier curves. * Rather, it internally computes a Bezier curve representation of each * parabolic spline segment. (These Bezier curves are then flattened to * produce the points filled into the output arrays.) * * 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( 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. */ if (!pointPtr) { /* * Of pointPtr == NULL, this function returns an upper limit of the * array size to store the coordinates. This can be used to allocate * storage, before the actual coordinates are calculated. */ return 1 + numPoints * numSteps; } 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; } /* *-------------------------------------------------------------- * * TkMakeRawCurve -- * * Interpret the given set of points as the raw knots and control points * defining a sequence of cubic Bezier curves. Create a new set of points * that fit these Bezier curves. Output points are produced in either of * two forms. * * 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. * * Side effects: * None. * *-------------------------------------------------------------- */ int TkMakeRawCurve( 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 curve * segment (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 outputPoints, i; int numSegments = (numPoints+1)/3; double *segPtr; /* * The input describes a curve with s Bezier curve segments if there are * 3s+1, 3s, or 3s-1 input points. In the last two cases, 1 or 2 initial * points from the first curve segment are reused as defining points also * for the last curve segment. In the case of 3s input points, this will * automatically close the curve. */ if (!pointPtr) { /* * If pointPtr == NULL, this function returns an upper limit of the * array size to store the coordinates. This can be used to allocate * storage, before the actual coordinates are calculated. */ return 1 + numSegments * numSteps; } outputPoints = 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; /* * The next loop handles all curve segments except one that overlaps the * end of the list of coordinates. */ for (i=numPoints,segPtr=pointPtr ; i>=4 ; i-=3,segPtr+=6) { if (segPtr[0]==segPtr[2] && segPtr[1]==segPtr[3] && segPtr[4]==segPtr[6] && segPtr[5]==segPtr[7]) { /* * The control points on this segment are equal to their * neighbouring knots, so this segment is just a straight line. A * single point is sufficient. */ if (xPoints != NULL) { Tk_CanvasDrawableCoords(canvas, segPtr[6], segPtr[7], &xPoints->x, &xPoints->y); xPoints += 1; } if (dblPoints != NULL) { dblPoints[0] = segPtr[6]; dblPoints[1] = segPtr[7]; dblPoints += 2; } outputPoints += 1; } else { /* * This is a generic Bezier curve segment. */ if (xPoints != NULL) { TkBezierScreenPoints(canvas, segPtr, numSteps, xPoints); xPoints += numSteps; } if (dblPoints != NULL) { TkBezierPoints(segPtr, numSteps, dblPoints); dblPoints += 2*numSteps; } outputPoints += numSteps; } } /* * If at this point i>1, then there is some point which has not yet been * used. Make another curve segment. */ if (i > 1) { int j; double control[8]; /* * Copy the relevant coordinates to control[], so that it can be * passed as a unit to e.g. TkBezierPoints. */ for (j=0; j<2*i; j++) { control[j] = segPtr[j]; } for (; j<8; j++) { control[j] = pointPtr[j-2*i]; } /* * Then we just do the same things as above. */ if (control[0]==control[2] && control[1]==control[3] && control[4]==control[6] && control[5]==control[7]) { /* * The control points on this segment are equal to their * neighbouring knots, so this segment is just a straight line. A * single point is sufficient. */ if (xPoints != NULL) { Tk_CanvasDrawableCoords(canvas, control[6], control[7], &xPoints->x, &xPoints->y); xPoints += 1; } if (dblPoints != NULL) { dblPoints[0] = control[6]; dblPoints[1] = control[7]; dblPoints += 2; } outputPoints += 1; } else { /* * This is a generic Bezier curve segment. */ 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 function generates Postscript commands that create a path * corresponding to a given Bezier curve. * * Results: * None. Postscript commands to generate the path are appended to the * interp's result. * * Side effects: * None. * *-------------------------------------------------------------- */ void TkMakeBezierPostscript( 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]; Tcl_Obj *psObj; /* * 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]; psObj = Tcl_ObjPrintf( "%.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]; psObj = Tcl_ObjPrintf("%.15g %.15g moveto\n", control[6], Tk_CanvasPsY(canvas, control[7])); } /* * 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]; Tcl_AppendPrintfToObj(psObj, "%.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_AppendObjToObj(Tcl_GetObjResult(interp), psObj); Tcl_DecrRefCount(psObj); } /* *-------------------------------------------------------------- * * TkMakeRawCurvePostscript -- * * This function interprets the input points as the raw knot and control * points for a curve composed of Bezier curve segments, just like * TkMakeRawCurve. It generates Postscript commands that create a path * corresponding to this given curve. * * Results: * None. Postscript commands to generate the path are appended to the * interp's result. * * Side effects: * None. * *-------------------------------------------------------------- */ void TkMakeRawCurvePostscript( 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 i; double *segPtr; Tcl_Obj *psObj; /* * Put the first point into the path. */ psObj = Tcl_ObjPrintf("%.15g %.15g moveto\n", pointPtr[0], Tk_CanvasPsY(canvas, pointPtr[1])); /* * Loop through all the remaining points in the curve, generating a * straight line or curve section for every three of them. */ for (i=numPoints-1,segPtr=pointPtr ; i>=3 ; i-=3,segPtr+=6) { if (segPtr[0]==segPtr[2] && segPtr[1]==segPtr[3] && segPtr[4]==segPtr[6] && segPtr[5]==segPtr[7]) { /* * The control points on this segment are equal to their * neighbouring knots, so this segment is just a straight line. */ Tcl_AppendPrintfToObj(psObj, "%.15g %.15g lineto\n", segPtr[6], Tk_CanvasPsY(canvas, segPtr[7])); } else { /* * This is a generic Bezier curve segment. */ Tcl_AppendPrintfToObj(psObj, "%.15g %.15g %.15g %.15g %.15g %.15g curveto\n", segPtr[2], Tk_CanvasPsY(canvas, segPtr[3]), segPtr[4], Tk_CanvasPsY(canvas, segPtr[5]), segPtr[6], Tk_CanvasPsY(canvas, segPtr[7])); } } /* * If there are any points left that haven't been used, then build the * last segment and generate Postscript in the same way for that. */ if (i > 0) { int j; double control[8]; for (j=0; j<2*i+2; j++) { control[j] = segPtr[j]; } for (; j<8; j++) { control[j] = pointPtr[j-2*i-2]; } if (control[0]==control[2] && control[1]==control[3] && control[4]==control[6] && control[5]==control[7]) { /* * Straight line. */ Tcl_AppendPrintfToObj(psObj, "%.15g %.15g lineto\n", control[6], Tk_CanvasPsY(canvas, control[7])); } else { /* * Bezier curve segment. */ Tcl_AppendPrintfToObj(psObj, "%.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_AppendObjToObj(Tcl_GetObjResult(interp), psObj); Tcl_DecrRefCount(psObj); } /* *-------------------------------------------------------------- * * 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( 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; #ifndef _MSC_VER static const double elevenDegrees = (11.0*2.0*PI)/360.0; #else /* msvc8 with -fp:strict requires it this way */ static const double elevenDegrees = 0.19198621771937624; #endif /* * 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( 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; } } } /* * Local Variables: * mode: c * c-basic-offset: 4 * fill-column: 78 * End: */