/**************************************************************************
*
* 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);
}