/************************************************************************** * * Copyright 2009 VMware, Inc. All Rights Reserved. * * Permission is hereby granted, free of charge, to any person obtaining a * copy of this software and associated documentation files (the * "Software"), to deal in the Software without restriction, including * without limitation the rights to use, copy, modify, merge, publish, * distribute, sub license, and/or sell copies of the Software, and to * permit persons to whom the Software is furnished to do so, subject to * the following conditions: * * The above copyright notice and this permission notice (including the * next paragraph) 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 NON-INFRINGEMENT. * IN NO EVENT SHALL VMWARE AND/OR ITS SUPPLIERS 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. * **************************************************************************/ #include "arc.h" #include "matrix.h" #include "bezier.h" #include "polygon.h" #include "stroker.h" #include "path.h" #include "util/u_debug.h" #include "util/u_math.h" #ifndef M_PI #define M_PI 3.14159265358979323846 #endif #define DEBUG_ARCS 0 static const VGfloat two_pi = M_PI * 2; static const double coeffs3Low[2][4][4] = { { { 3.85268, -21.229, -0.330434, 0.0127842 }, { -1.61486, 0.706564, 0.225945, 0.263682 }, { -0.910164, 0.388383, 0.00551445, 0.00671814 }, { -0.630184, 0.192402, 0.0098871, 0.0102527 } }, { { -0.162211, 9.94329, 0.13723, 0.0124084 }, { -0.253135, 0.00187735, 0.0230286, 0.01264 }, { -0.0695069, -0.0437594, 0.0120636, 0.0163087 }, { -0.0328856, -0.00926032, -0.00173573, 0.00527385 } } }; /* coefficients for error estimation while using cubic Bézier curves for approximation 1/4 <= b/a <= 1 */ static const double coeffs3High[2][4][4] = { { { 0.0899116, -19.2349, -4.11711, 0.183362 }, { 0.138148, -1.45804, 1.32044, 1.38474 }, { 0.230903, -0.450262, 0.219963, 0.414038 }, { 0.0590565, -0.101062, 0.0430592, 0.0204699 } }, { { 0.0164649, 9.89394, 0.0919496, 0.00760802 }, { 0.0191603, -0.0322058, 0.0134667, -0.0825018 }, { 0.0156192, -0.017535, 0.00326508, -0.228157 }, { -0.0236752, 0.0405821, -0.0173086, 0.176187 } } }; /* safety factor to convert the "best" error approximation into a "max bound" error */ static const double safety3[] = { 0.001, 4.98, 0.207, 0.0067 }; /* The code below is from the OpenVG 1.1 Spec * Section 18.4 */ /* Given: Points (x0, y0) and (x1, y1) * Return: TRUE if a solution exists, FALSE otherwise * Circle centers are written to (cx0, cy0) and (cx1, cy1) */ static VGboolean find_unit_circles(double x0, double y0, double x1, double y1, double *cx0, double *cy0, double *cx1, double *cy1) { /* Compute differences and averages */ double dx = x0 - x1; double dy = y0 - y1; double xm = (x0 + x1)/2; double ym = (y0 + y1)/2; double dsq, disc, s, sdx, sdy; /* Solve for intersecting unit circles */ dsq = dx*dx + dy*dy; if (dsq == 0.0) return VG_FALSE; /* Points are coincident */ disc = 1.0/dsq - 1.0/4.0; /* the precision we care about here is around float so if we're * around the float defined zero then make it official to avoid * precision problems later on */ if (floatIsZero(disc)) disc = 0.0; if (disc < 0.0) return VG_FALSE; /* Points are too far apart */ s = sqrt(disc); sdx = s*dx; sdy = s*dy; *cx0 = xm + sdy; *cy0 = ym - sdx; *cx1 = xm - sdy; *cy1 = ym + sdx; return VG_TRUE; } /* Given: Ellipse parameters rh, rv, rot (in degrees), * endpoints (x0, y0) and (x1, y1) * Return: TRUE if a solution exists, FALSE otherwise * Ellipse centers are written to (cx0, cy0) and (cx1, cy1) */ static VGboolean find_ellipses(double rh, double rv, double rot, double x0, double y0, double x1, double y1, double *cx0, double *cy0, double *cx1, double *cy1) { double COS, SIN, x0p, y0p, x1p, y1p, pcx0, pcy0, pcx1, pcy1; /* Convert rotation angle from degrees to radians */ rot *= M_PI/180.0; /* Pre-compute rotation matrix entries */ COS = cos(rot); SIN = sin(rot); /* Transform (x0, y0) and (x1, y1) into unit space */ /* using (inverse) rotate, followed by (inverse) scale */ x0p = (x0*COS + y0*SIN)/rh; y0p = (-x0*SIN + y0*COS)/rv; x1p = (x1*COS + y1*SIN)/rh; y1p = (-x1*SIN + y1*COS)/rv; if (!find_unit_circles(x0p, y0p, x1p, y1p, &pcx0, &pcy0, &pcx1, &pcy1)) { return VG_FALSE; } /* Transform back to original coordinate space */ /* using (forward) scale followed by (forward) rotate */ pcx0 *= rh; pcy0 *= rv; pcx1 *= rh; pcy1 *= rv; *cx0 = pcx0*COS - pcy0*SIN; *cy0 = pcx0*SIN + pcy0*COS; *cx1 = pcx1*COS - pcy1*SIN; *cy1 = pcx1*SIN + pcy1*COS; return VG_TRUE; } static INLINE VGboolean try_to_fix_radii(struct arc *arc) { double COS, SIN, rot, x0p, y0p, x1p, y1p; double dx, dy, dsq, scale; /* Convert rotation angle from degrees to radians */ rot = DEGREES_TO_RADIANS(arc->theta); /* Pre-compute rotation matrix entries */ COS = cos(rot); SIN = sin(rot); /* Transform (x0, y0) and (x1, y1) into unit space */ /* using (inverse) rotate, followed by (inverse) scale */ x0p = (arc->x1*COS + arc->y1*SIN)/arc->a; y0p = (-arc->x1*SIN + arc->y1*COS)/arc->b; x1p = (arc->x2*COS + arc->y2*SIN)/arc->a; y1p = (-arc->x2*SIN + arc->y2*COS)/arc->b; /* Compute differences and averages */ dx = x0p - x1p; dy = y0p - y1p; dsq = dx*dx + dy*dy; #if 0 if (dsq <= 0.001) { debug_printf("AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAaaaaa\n"); } #endif scale = 1/(2/sqrt(dsq)); arc->a *= scale; arc->b *= scale; return VG_TRUE; } static INLINE double vector_normalize(double *v) { double sq = v[0] * v[0] + v[1] * v[1]; return sqrt(sq); } static INLINE double vector_orientation(double *v) { double norm = vector_normalize(v); double cosa = v[0] / norm; double sina = v[1] / norm; return (sina>=0 ? acos(cosa) : 2*M_PI - acos(cosa)); } static INLINE double vector_dot(double *v0, double *v1) { return v0[0] * v1[0] + v0[1] * v1[1]; } static INLINE double vector_angles(double *v0, double *v1) { double dot = vector_dot(v0, v1); double norm0 = vector_normalize(v0); double norm1 = vector_normalize(v1); return acos(dot / (norm0 * norm1)); } static VGboolean find_angles(struct arc *arc) { double vec0[2], vec1[2]; double lambda1, lambda2; double angle; struct matrix matrix; if (floatIsZero(arc->a) || floatIsZero(arc->b)) { return VG_FALSE; } /* map the points to an identity circle */ matrix_load_identity(&matrix); matrix_scale(&matrix, 1.f, arc->a/arc->b); matrix_rotate(&matrix, -arc->theta); matrix_map_point(&matrix, arc->x1, arc->y1, &arc->x1, &arc->y1); matrix_map_point(&matrix, arc->x2, arc->y2, &arc->x2, &arc->y2); matrix_map_point(&matrix, arc->cx, arc->cy, &arc->cx, &arc->cy); #if DEBUG_ARCS debug_printf("Matrix 3 [%f, %f, %f| %f, %f, %f| %f, %f, %f]\n", matrix.m[0], matrix.m[1], matrix.m[2], matrix.m[3], matrix.m[4], matrix.m[5], matrix.m[6], matrix.m[7], matrix.m[8]); debug_printf("Endpoints [%f, %f], [%f, %f]\n", arc->x1, arc->y1, arc->x2, arc->y2); #endif vec0[0] = arc->x1 - arc->cx; vec0[1] = arc->y1 - arc->cy; vec1[0] = arc->x2 - arc->cx; vec1[1] = arc->y2 - arc->cy; #if DEBUG_ARCS debug_printf("Vec is [%f, %f], [%f, %f], [%f, %f]\n", vec0[0], vec0[1], vec1[0], vec1[1], arc->cx, arc->cy); #endif lambda1 = vector_orientation(vec0); if (isnan(lambda1)) lambda1 = 0.f; if (arc->type == VG_SCWARC_TO || arc->type == VG_SCCWARC_TO) angle = vector_angles(vec0, vec1); else if (arc->type == VG_LCWARC_TO || arc->type == VG_LCCWARC_TO) { angle = 2*M_PI - vector_angles(vec0, vec1); } else abort(); if (isnan(angle)) angle = M_PI; if (arc->type == VG_SCWARC_TO || arc->type == VG_LCWARC_TO) lambda2 = lambda1 - angle; else lambda2 = lambda1 + angle; #if DEBUG_ARCS debug_printf("Angle is %f and (%f, %f)\n", angle, lambda1, lambda2); #endif #if 0 arc->eta1 = atan2(sin(lambda1) / arc->b, cos(lambda1) / arc->a); arc->eta2 = atan2(sin(lambda2) / arc->b, cos(lambda2) / arc->a); /* make sure we have eta1 <= eta2 <= eta1 + 2 PI */ arc->eta2 -= two_pi * floor((arc->eta2 - arc->eta1) / two_pi); /* the preceding correction fails if we have exactly et2 - eta1 = 2 PI it reduces the interval to zero length */ if ((lambda2 - lambda1 > M_PI) && (arc->eta2 - arc->eta1 < M_PI)) { arc->eta2 += 2 * M_PI; } #else arc->eta1 = lambda1; arc->eta2 = lambda2; #endif return VG_TRUE; } #if DEBUG_ARCS static void check_endpoints(struct arc *arc) { double x1, y1, x2, y2; double a_cos_eta1 = arc->a * cos(arc->eta1); double b_sin_eta1 = arc->b * sin(arc->eta1); x1 = arc->cx + a_cos_eta1 * arc->cos_theta - b_sin_eta1 * arc->sin_theta; y1 = arc->cy + a_cos_eta1 * arc->sin_theta + b_sin_eta1 * arc->cos_theta; double a_cos_eta2 = arc->a * cos(arc->eta2); double b_sin_eta2 = arc->b * sin(arc->eta2); x2 = arc->cx + a_cos_eta2 * arc->cos_theta - b_sin_eta2 * arc->sin_theta; y2 = arc->cy + a_cos_eta2 * arc->sin_theta + b_sin_eta2 * arc->cos_theta; debug_printf("Computed (%f, %f), (%f, %f)\n", x1, y1, x2, y2); debug_printf("Real (%f, %f), (%f, %f)\n", arc->x1, arc->y1, arc->x2, arc->y2); } #endif void arc_init(struct arc *arc, VGPathSegment type, VGfloat x1, VGfloat y1, VGfloat x2, VGfloat y2, VGfloat rh, VGfloat rv, VGfloat rot) { assert(type == VG_SCCWARC_TO || type == VG_SCWARC_TO || type == VG_LCCWARC_TO || type == VG_LCWARC_TO); arc->type = type; arc->x1 = x1; arc->y1 = y1; arc->x2 = x2; arc->y2 = y2; arc->a = rh; arc->b = rv; arc->theta = rot; arc->cos_theta = cos(arc->theta); arc->sin_theta = sin(arc->theta); { double cx0, cy0, cx1, cy1; double cx, cy; arc->is_valid = find_ellipses(rh, rv, rot, x1, y1, x2, y2, &cx0, &cy0, &cx1, &cy1); if (!arc->is_valid && try_to_fix_radii(arc)) { rh = arc->a; rv = arc->b; arc->is_valid = find_ellipses(rh, rv, rot, x1, y1, x2, y2, &cx0, &cy0, &cx1, &cy1); } if (type == VG_SCWARC_TO || type == VG_LCCWARC_TO) { cx = cx1; cy = cy1; } else { cx = cx0; cy = cy0; } #if DEBUG_ARCS debug_printf("Centers are : (%f, %f) , (%f, %f). Real (%f, %f)\n", cx0, cy0, cx1, cy1, cx, cy); #endif arc->cx = cx; arc->cy = cy; if (arc->is_valid) { arc->is_valid = find_angles(arc); #if DEBUG_ARCS check_endpoints(arc); #endif /* remap a few points. find_angles requires * rot in angles, the rest of the code * will need them in radians. and find_angles * modifies the center to match an identity * circle so lets reset it */ arc->theta = DEGREES_TO_RADIANS(rot); arc->cos_theta = cos(arc->theta); arc->sin_theta = sin(arc->theta); arc->cx = cx; arc->cy = cy; } } } static INLINE double rational_function(double x, const double *c) { return (x * (x * c[0] + c[1]) + c[2]) / (x + c[3]); } static double estimate_error(struct arc *arc, double etaA, double etaB) { double eta = 0.5 * (etaA + etaB); double x = arc->b / arc->a; double dEta = etaB - etaA; double cos2 = cos(2 * eta); double cos4 = cos(4 * eta); double cos6 = cos(6 * eta); double c0, c1; /* select the right coeficients set according to degree and b/a */ const double (*coeffs)[4][4]; const double *safety; coeffs = (x < 0.25) ? coeffs3Low : coeffs3High; safety = safety3; c0 = rational_function(x, coeffs[0][0]) + cos2 * rational_function(x, coeffs[0][1]) + cos4 * rational_function(x, coeffs[0][2]) + cos6 * rational_function(x, coeffs[0][3]); c1 = rational_function(x, coeffs[1][0]) + cos2 * rational_function(x, coeffs[1][1]) + cos4 * rational_function(x, coeffs[1][2]) + cos6 * rational_function(x, coeffs[1][3]); return rational_function(x, safety) * arc->a * exp(c0 + c1 * dEta); } struct arc_cb { void (*move)(struct arc_cb *cb, VGfloat x, VGfloat y); void (*point)(struct arc_cb *cb, VGfloat x, VGfloat y); void (*bezier)(struct arc_cb *cb, struct bezier *bezier); void *user_data; }; static void cb_null_move(struct arc_cb *cb, VGfloat x, VGfloat y) { } static void polygon_point(struct arc_cb *cb, VGfloat x, VGfloat y) { struct polygon *poly = (struct polygon*)cb->user_data; polygon_vertex_append(poly, x, y); } static void polygon_bezier(struct arc_cb *cb, struct bezier *bezier) { struct polygon *poly = (struct polygon*)cb->user_data; bezier_add_to_polygon(bezier, poly); } static void stroke_point(struct arc_cb *cb, VGfloat x, VGfloat y) { struct stroker *stroker = (struct stroker*)cb->user_data; stroker_line_to(stroker, x, y); } static void stroke_curve(struct arc_cb *cb, struct bezier *bezier) { struct stroker *stroker = (struct stroker*)cb->user_data; stroker_curve_to(stroker, bezier->x2, bezier->y2, bezier->x3, bezier->y3, bezier->x4, bezier->y4); } static void stroke_emit_point(struct arc_cb *cb, VGfloat x, VGfloat y) { struct stroker *stroker = (struct stroker*)cb->user_data; stroker_emit_line_to(stroker, x, y); } static void stroke_emit_curve(struct arc_cb *cb, struct bezier *bezier) { struct stroker *stroker = (struct stroker*)cb->user_data; stroker_emit_curve_to(stroker, bezier->x2, bezier->y2, bezier->x3, bezier->y3, bezier->x4, bezier->y4); } static void arc_path_move(struct arc_cb *cb, VGfloat x, VGfloat y) { struct path *path = (struct path*)cb->user_data; path_move_to(path, x, y); } static void arc_path_point(struct arc_cb *cb, VGfloat x, VGfloat y) { struct path *path = (struct path*)cb->user_data; path_line_to(path, x, y); } static void arc_path_bezier(struct arc_cb *cb, struct bezier *bezier) { struct path *path = (struct path*)cb->user_data; path_cubic_to(path, bezier->x2, bezier->y2, bezier->x3, bezier->y3, bezier->x4, bezier->y4); } static INLINE int num_beziers_needed(struct arc *arc) { double threshold = 0.05; VGboolean found = VG_FALSE; int n = 1; double min_eta, max_eta; min_eta = MIN2(arc->eta1, arc->eta2); max_eta = MAX2(arc->eta1, arc->eta2); while ((! found) && (n < 1024)) { double d_eta = (max_eta - min_eta) / n; if (d_eta <= 0.5 * M_PI) { double eta_b = min_eta; int i; found = VG_TRUE; for (i = 0; found && (i < n); ++i) { double etaA = eta_b; eta_b += d_eta; found = (estimate_error(arc, etaA, eta_b) <= threshold); } } n = n << 1; } return n; } static void arc_to_beziers(struct arc *arc, struct arc_cb cb, struct matrix *matrix) { int i; int n = 1; double d_eta, eta_b, cos_eta_b, sin_eta_b, a_cos_eta_b, b_sin_eta_b, a_sin_eta_b, b_cos_eta_b, x_b, y_b, x_b_dot, y_b_dot, lx, ly; double t, alpha; { /* always move to the start of the arc */ VGfloat x = arc->x1; VGfloat y = arc->y1; matrix_map_point(matrix, x, y, &x, &y); cb.move(&cb, x, y); } if (!arc->is_valid) { VGfloat x = arc->x2; VGfloat y = arc->y2; matrix_map_point(matrix, x, y, &x, &y); cb.point(&cb, x, y); return; } /* find the number of Bézier curves needed */ n = num_beziers_needed(arc); d_eta = (arc->eta2 - arc->eta1) / n; eta_b = arc->eta1; cos_eta_b = cos(eta_b); sin_eta_b = sin(eta_b); a_cos_eta_b = arc->a * cos_eta_b; b_sin_eta_b = arc->b * sin_eta_b; a_sin_eta_b = arc->a * sin_eta_b; b_cos_eta_b = arc->b * cos_eta_b; x_b = arc->cx + a_cos_eta_b * arc->cos_theta - b_sin_eta_b * arc->sin_theta; y_b = arc->cy + a_cos_eta_b * arc->sin_theta + b_sin_eta_b * arc->cos_theta; x_b_dot = -a_sin_eta_b * arc->cos_theta - b_cos_eta_b * arc->sin_theta; y_b_dot = -a_sin_eta_b * arc->sin_theta + b_cos_eta_b * arc->cos_theta; { VGfloat x = x_b, y = y_b; matrix_map_point(matrix, x, y, &x, &y); cb.point(&cb, x, y); } lx = x_b; ly = y_b; t = tan(0.5 * d_eta); alpha = sin(d_eta) * (sqrt(4 + 3 * t * t) - 1) / 3; for (i = 0; i < n; ++i) { struct bezier bezier; double xA = x_b; double yA = y_b; double xADot = x_b_dot; double yADot = y_b_dot; eta_b += d_eta; cos_eta_b = cos(eta_b); sin_eta_b = sin(eta_b); a_cos_eta_b = arc->a * cos_eta_b; b_sin_eta_b = arc->b * sin_eta_b; a_sin_eta_b = arc->a * sin_eta_b; b_cos_eta_b = arc->b * cos_eta_b; x_b = arc->cx + a_cos_eta_b * arc->cos_theta - b_sin_eta_b * arc->sin_theta; y_b = arc->cy + a_cos_eta_b * arc->sin_theta + b_sin_eta_b * arc->cos_theta; x_b_dot = -a_sin_eta_b * arc->cos_theta - b_cos_eta_b * arc->sin_theta; y_b_dot = -a_sin_eta_b * arc->sin_theta + b_cos_eta_b * arc->cos_theta; bezier_init(&bezier, lx, ly, (float) (xA + alpha * xADot), (float) (yA + alpha * yADot), (float) (x_b - alpha * x_b_dot), (float) (y_b - alpha * y_b_dot), (float) x_b, (float) y_b); #if 0 debug_printf("%d) Bezier (%f, %f), (%f, %f), (%f, %f), (%f, %f)\n", i, bezier.x1, bezier.y1, bezier.x2, bezier.y2, bezier.x3, bezier.y3, bezier.x4, bezier.y4); #endif bezier_transform(&bezier, matrix); cb.bezier(&cb, &bezier); lx = x_b; ly = y_b; } } void arc_add_to_polygon(struct arc *arc, struct polygon *poly, struct matrix *matrix) { struct arc_cb cb; cb.move = cb_null_move; cb.point = polygon_point; cb.bezier = polygon_bezier; cb.user_data = poly; arc_to_beziers(arc, cb, matrix); } void arc_stroke_cb(struct arc *arc, struct stroker *stroke, struct matrix *matrix) { struct arc_cb cb; cb.move = cb_null_move; cb.point = stroke_point; cb.bezier = stroke_curve; cb.user_data = stroke; arc_to_beziers(arc, cb, matrix); } void arc_stroker_emit(struct arc *arc, struct stroker *stroker, struct matrix *matrix) { struct arc_cb cb; cb.move = cb_null_move; cb.point = stroke_emit_point; cb.bezier = stroke_emit_curve; cb.user_data = stroker; arc_to_beziers(arc, cb, matrix); } void arc_to_path(struct arc *arc, struct path *path, struct matrix *matrix) { struct arc_cb cb; cb.move = arc_path_move; cb.point = arc_path_point; cb.bezier = arc_path_bezier; cb.user_data = path; arc_to_beziers(arc, cb, matrix); }