3
* Copyright © 2000 Keith Packard, member of The XFree86 Project, Inc.
4
* Copyright © 2000 SuSE, Inc.
5
* 2005 Lars Knoll & Zack Rusin, Trolltech
6
* Copyright © 2007 Red Hat, Inc.
9
* Permission to use, copy, modify, distribute, and sell this software and its
10
* documentation for any purpose is hereby granted without fee, provided that
11
* the above copyright notice appear in all copies and that both that
12
* copyright notice and this permission notice appear in supporting
13
* documentation, and that the name of Keith Packard not be used in
14
* advertising or publicity pertaining to distribution of the software without
15
* specific, written prior permission. Keith Packard makes no
16
* representations about the suitability of this software for any purpose. It
17
* is provided "as is" without express or implied warranty.
19
* THE COPYRIGHT HOLDERS DISCLAIM ALL WARRANTIES WITH REGARD TO THIS
20
* SOFTWARE, INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND
21
* FITNESS, IN NO EVENT SHALL THE COPYRIGHT HOLDERS BE LIABLE FOR ANY
22
* SPECIAL, INDIRECT OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
23
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN
24
* AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING
25
* OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS
34
#include "pixman-private.h"
37
radial_gradient_get_scanline_32 (pixman_image_t *image,
46
* In the radial gradient problem we are given two circles (c₁,r₁) and
47
* (c₂,r₂) that define the gradient itself. Then, for any point p, we
48
* must compute the value(s) of t within [0.0, 1.0] representing the
49
* circle(s) that would color the point.
51
* There are potentially two values of t since the point p can be
52
* colored by both sides of the circle, (which happens whenever one
53
* circle is not entirely contained within the other).
55
* If we solve for a value of t that is outside of [0.0, 1.0] then we
56
* use the extend mode (NONE, REPEAT, REFLECT, or PAD) to map to a
57
* value within [0.0, 1.0].
59
* Here is an illustration of the problem:
73
* Given (c₁,r₁), (c₂,r₂) and p, we must find an angle θ such that two
74
* points p₁ and p₂ on the two circles are collinear with p. Then, the
75
* desired value of t is the ratio of the length of p₁p to the length
78
* So, we have six unknown values: (p₁x, p₁y), (p₂x, p₂y), θ and t.
79
* We can also write six equations that constrain the problem:
81
* Point p₁ is a distance r₁ from c₁ at an angle of θ:
83
* 1. p₁x = c₁x + r₁·cos θ
84
* 2. p₁y = c₁y + r₁·sin θ
86
* Point p₂ is a distance r₂ from c₂ at an angle of θ:
88
* 3. p₂x = c₂x + r2·cos θ
89
* 4. p₂y = c₂y + r2·sin θ
91
* Point p lies at a fraction t along the line segment p₁p₂:
93
* 5. px = t·p₂x + (1-t)·p₁x
94
* 6. py = t·p₂y + (1-t)·p₁y
96
* To solve, first subtitute 1-4 into 5 and 6:
98
* px = t·(c₂x + r₂·cos θ) + (1-t)·(c₁x + r₁·cos θ)
99
* py = t·(c₂y + r₂·sin θ) + (1-t)·(c₁y + r₁·sin θ)
101
* Then solve each for cos θ and sin θ expressed as a function of t:
103
* cos θ = (-(c₂x - c₁x)·t + (px - c₁x)) / ((r₂-r₁)·t + r₁)
104
* sin θ = (-(c₂y - c₁y)·t + (py - c₁y)) / ((r₂-r₁)·t + r₁)
106
* To simplify this a bit, we define new variables for several of the
107
* common terms as shown below:
127
* Note that cdx, cdy, and dr do not depend on point p at all, so can
128
* be pre-computed for the entire gradient. The simplifed equations
131
* cos θ = (-cdx·t + pdx) / (dr·t + r₁)
132
* sin θ = (-cdy·t + pdy) / (dr·t + r₁)
134
* Finally, to get a single function of t and eliminate the last
135
* unknown θ, we use the identity sin²θ + cos²θ = 1. First, square
136
* each equation, (we knew a quadratic was coming since it must be
137
* possible to obtain two solutions in some cases):
139
* cos²θ = (cdx²t² - 2·cdx·pdx·t + pdx²) / (dr²·t² + 2·r₁·dr·t + r₁²)
140
* sin²θ = (cdy²t² - 2·cdy·pdy·t + pdy²) / (dr²·t² + 2·r₁·dr·t + r₁²)
142
* Then add both together, set the result equal to 1, and express as a
143
* standard quadratic equation in t of the form At² + Bt + C = 0
145
* (cdx² + cdy² - dr²)·t² - 2·(cdx·pdx + cdy·pdy + r₁·dr)·t + (pdx² + pdy² - r₁²) = 0
149
* A = cdx² + cdy² - dr²
150
* B = -2·(pdx·cdx + pdy·cdy + r₁·dr)
151
* C = pdx² + pdy² - r₁²
153
* And again, notice that A does not depend on p, so can be
154
* precomputed. From here we just use the quadratic formula to solve
157
* t = (-2·B ± ⎷(B² - 4·A·C)) / 2·A
160
gradient_t *gradient = (gradient_t *)image;
161
source_image_t *source = (source_image_t *)image;
162
radial_gradient_t *radial = (radial_gradient_t *)image;
163
uint32_t *end = buffer + width;
164
pixman_gradient_walker_t walker;
165
pixman_bool_t affine = TRUE;
173
_pixman_gradient_walker_init (&walker, gradient, source->common.repeat);
175
if (source->common.transform)
178
/* reference point is the center of the pixel */
179
v.vector[0] = pixman_int_to_fixed (x) + pixman_fixed_1 / 2;
180
v.vector[1] = pixman_int_to_fixed (y) + pixman_fixed_1 / 2;
181
v.vector[2] = pixman_fixed_1;
183
if (!pixman_transform_point_3d (source->common.transform, &v))
186
cx = source->common.transform->matrix[0][0] / 65536.;
187
cy = source->common.transform->matrix[1][0] / 65536.;
188
cz = source->common.transform->matrix[2][0] / 65536.;
190
rx = v.vector[0] / 65536.;
191
ry = v.vector[1] / 65536.;
192
rz = v.vector[2] / 65536.;
195
source->common.transform->matrix[2][0] == 0 &&
196
v.vector[2] == pixman_fixed_1;
201
/* When computing t over a scanline, we notice that some expressions
202
* are constant so we can compute them just once. Given:
204
* t = (-2·B ± ⎷(B² - 4·A·C)) / 2·A
208
* A = cdx² + cdy² - dr² [precomputed as radial->A]
209
* B = -2·(pdx·cdx + pdy·cdy + r₁·dr)
210
* C = pdx² + pdy² - r₁²
212
* Since we have an affine transformation, we know that (pdx, pdy)
213
* increase linearly with each pixel,
218
* we can then express B in terms of an linear increment along
221
* B = B₀ + n·cB, with
222
* B₀ = -2·(pdx₀·cdx + pdy₀·cdy + r₁·dr) and
223
* cB = -2·(cx·cdx + cy·cdy)
225
* Thus we can replace the full evaluation of B per-pixel (4 multiplies,
226
* 2 additions) with a single addition.
228
double r1 = radial->c1.radius / 65536.;
229
double r1sq = r1 * r1;
230
double pdx = rx - radial->c1.x / 65536.;
231
double pdy = ry - radial->c1.y / 65536.;
232
double A = radial->A;
233
double invA = -65536. / (2. * A);
235
double B = -2. * (pdx*radial->cdx + pdy*radial->cdy + r1*radial->dr);
236
double cB = -2. * (cx*radial->cdx + cy*radial->cdy);
237
pixman_bool_t invert = A * radial->dr < 0;
241
if (!mask || *mask++ & mask_bits)
243
pixman_fixed_48_16_t t;
244
double det = B * B + A4 * (pdx * pdx + pdy * pdy - r1sq);
246
t = (pixman_fixed_48_16_t) (B * invA);
248
t = (pixman_fixed_48_16_t) ((B + sqrt (det)) * invA);
250
t = (pixman_fixed_48_16_t) ((B - sqrt (det)) * invA);
252
*buffer = _pixman_gradient_walker_pixel (&walker, t);
266
if (!mask || *mask++ & mask_bits)
271
double c1x = radial->c1.x / 65536.0;
272
double c1y = radial->c1.y / 65536.0;
273
double r1 = radial->c1.radius / 65536.0;
274
pixman_fixed_48_16_t t;
290
B = -2 * (pdx * radial->cdx +
293
C = (pdx * pdx + pdy * pdy - r1 * r1);
295
det = (B * B) - (4 * radial->A * C);
299
if (radial->A * radial->dr < 0)
300
t = (pixman_fixed_48_16_t) ((-B - sqrt (det)) / (2.0 * radial->A) * 65536);
302
t = (pixman_fixed_48_16_t) ((-B + sqrt (det)) / (2.0 * radial->A) * 65536);
304
*buffer = _pixman_gradient_walker_pixel (&walker, t);
317
radial_gradient_property_changed (pixman_image_t *image)
319
image->common.get_scanline_32 = radial_gradient_get_scanline_32;
320
image->common.get_scanline_64 = _pixman_image_get_scanline_generic_64;
323
PIXMAN_EXPORT pixman_image_t *
324
pixman_image_create_radial_gradient (pixman_point_fixed_t * inner,
325
pixman_point_fixed_t * outer,
326
pixman_fixed_t inner_radius,
327
pixman_fixed_t outer_radius,
328
const pixman_gradient_stop_t *stops,
331
pixman_image_t *image;
332
radial_gradient_t *radial;
334
return_val_if_fail (n_stops >= 2, NULL);
336
image = _pixman_image_allocate ();
341
radial = &image->radial;
343
if (!_pixman_init_gradient (&radial->common, stops, n_stops))
349
image->type = RADIAL;
351
radial->c1.x = inner->x;
352
radial->c1.y = inner->y;
353
radial->c1.radius = inner_radius;
354
radial->c2.x = outer->x;
355
radial->c2.y = outer->y;
356
radial->c2.radius = outer_radius;
357
radial->cdx = pixman_fixed_to_double (radial->c2.x - radial->c1.x);
358
radial->cdy = pixman_fixed_to_double (radial->c2.y - radial->c1.y);
359
radial->dr = pixman_fixed_to_double (radial->c2.radius - radial->c1.radius);
360
radial->A = (radial->cdx * radial->cdx +
361
radial->cdy * radial->cdy -
362
radial->dr * radial->dr);
364
image->common.property_changed = radial_gradient_property_changed;
added
removed
pixman/pixman-radial-gradient.c
