void
_pixman_general_conical_gradient_get_scanline_32 (pixman_image_t *image,
int x,
int y,
int width,
uint32_t * buffer,
const uint32_t *mask,
uint32_t mask_bits)
{
source_image_t *source = (source_image_t *)image;
gradient_t *gradient = (gradient_t *)source;
conical_gradient_t *conical = (conical_gradient_t *)image;
uint32_t *end = buffer + width;
pixman_gradient_walker_t walker;
pixman_bool_t affine = TRUE;
double cx = 1.;
double cy = 0.;
double cz = 0.;
double rx = x + 0.5;
double ry = y + 0.5;
double rz = 1.;
double a = (conical->angle * M_PI) / (180. * 65536);
_pixman_gradient_walker_init (&walker, gradient, source->common.repeat);
if (source->common.transform)
{
pixman_vector_t v;
/* reference point is the center of the pixel */
v.vector[0] = pixman_int_to_fixed (x) + pixman_fixed_1 / 2;
v.vector[1] = pixman_int_to_fixed (y) + pixman_fixed_1 / 2;
v.vector[2] = pixman_fixed_1;
if (!pixman_transform_point_3d (source->common.transform, &v))
return;
cx = source->common.transform->matrix[0][0] / 65536.;
cy = source->common.transform->matrix[1][0] / 65536.;
cz = source->common.transform->matrix[2][0] / 65536.;
rx = v.vector[0] / 65536.;
ry = v.vector[1] / 65536.;
rz = v.vector[2] / 65536.;
affine =
source->common.transform->matrix[2][0] == 0 &&
v.vector[2] == pixman_fixed_1;
}
if (affine)
{
rx -= conical->center.x / 65536.;
ry -= conical->center.y / 65536.;
while (buffer < end)
{
double angle;
if (!mask || *mask++ & mask_bits)
{
pixman_fixed_48_16_t t;
angle = atan2 (ry, rx) + a;
t = (pixman_fixed_48_16_t) (angle * (65536. / (2 * M_PI)));
*buffer = _pixman_gradient_walker_pixel (&walker, t);
}
++buffer;
rx += cx;
ry += cy;
}
}
else
{
while (buffer < end)
{
double x, y;
double angle;
if (!mask || *mask++ & mask_bits)
{
pixman_fixed_48_16_t t;
if (rz != 0)
{
x = rx / rz;
y = ry / rz;
}
else
{
x = y = 0.;
}
x -= conical->center.x / 65536.;
y -= conical->center.y / 65536.;
angle = atan2 (y, x) + a;
t = (pixman_fixed_48_16_t) (angle * (65536. / (2 * M_PI)));
*buffer = _pixman_gradient_walker_pixel (&walker, t);
}
++buffer;
rx += cx;
ry += cy;
rz += cz;
}
}
}
static uint32_t
radial_compute_color (double a,
double b,
double c,
double inva,
double dr,
double mindr,
pixman_gradient_walker_t *walker,
pixman_repeat_t repeat)
{
/*
* In this function error propagation can lead to bad results:
* - discr can have an unbound error (if b*b-a*c is very small),
* potentially making it the opposite sign of what it should have been
* (thus clearing a pixel that would have been colored or vice-versa)
* or propagating the error to sqrtdiscr;
* if discr has the wrong sign or b is very small, this can lead to bad
* results
*
* - the algorithm used to compute the solutions of the quadratic
* equation is not numerically stable (but saves one division compared
* to the numerically stable one);
* this can be a problem if a*c is much smaller than b*b
*
* - the above problems are worse if a is small (as inva becomes bigger)
*/
double discr;
if (a == 0)
{
double t;
if (b == 0)
return 0;
t = pixman_fixed_1 / 2 * c / b;
if (repeat == PIXMAN_REPEAT_NONE)
{
if (0 <= t && t <= pixman_fixed_1)
return _pixman_gradient_walker_pixel (walker, t);
}
else
{
if (t * dr >= mindr)
return _pixman_gradient_walker_pixel (walker, t);
}
return 0;
}
discr = fdot (b, a, 0, b, -c, 0);
if (discr >= 0)
{
double sqrtdiscr, t0, t1;
sqrtdiscr = sqrt (discr);
t0 = (b + sqrtdiscr) * inva;
t1 = (b - sqrtdiscr) * inva;
/*
* The root that must be used is the biggest one that belongs
* to the valid range ([0,1] for PIXMAN_REPEAT_NONE, any
* solution that results in a positive radius otherwise).
*
* If a > 0, t0 is the biggest solution, so if it is valid, it
* is the correct result.
*
* If a < 0, only one of the solutions can be valid, so the
* order in which they are tested is not important.
*/
if (repeat == PIXMAN_REPEAT_NONE)
{
if (0 <= t0 && t0 <= pixman_fixed_1)
return _pixman_gradient_walker_pixel (walker, t0);
else if (0 <= t1 && t1 <= pixman_fixed_1)
return _pixman_gradient_walker_pixel (walker, t1);
}
else
{
if (t0 * dr >= mindr)
return _pixman_gradient_walker_pixel (walker, t0);
else if (t1 * dr >= mindr)
return _pixman_gradient_walker_pixel (walker, t1);
}
}
return 0;
}
static uint32_t
radial_compute_color (double a,
double b,
double c,
double inva,
double dr,
double mindr,
pixman_gradient_walker_t *walker,
pixman_repeat_t repeat)
{
/*
* In this function error propagation can lead to bad results:
* - det can have an unbound error (if b*b-a*c is very small),
* potentially making it the opposite sign of what it should have been
* (thus clearing a pixel that would have been colored or vice-versa)
* or propagating the error to sqrtdet;
* if det has the wrong sign or b is very small, this can lead to bad
* results
*
* - the algorithm used to compute the solutions of the quadratic
* equation is not numerically stable (but saves one division compared
* to the numerically stable one);
* this can be a problem if a*c is much smaller than b*b
*
* - the above problems are worse if a is small (as inva becomes bigger)
*/
double det;
if (a == 0)
{
double t;
if (b == 0)
return 0;
t = pixman_fixed_1 / 2 * c / b;
if (repeat == PIXMAN_REPEAT_NONE)
{
if (0 <= t && t <= pixman_fixed_1)
return _pixman_gradient_walker_pixel (walker, t);
}
else
{
if (t * dr > mindr)
return _pixman_gradient_walker_pixel (walker, t);
}
return 0;
}
det = fdot (b, a, 0, b, -c, 0);
if (det >= 0)
{
double sqrtdet, t0, t1;
sqrtdet = sqrt (det);
t0 = (b + sqrtdet) * inva;
t1 = (b - sqrtdet) * inva;
if (repeat == PIXMAN_REPEAT_NONE)
{
if (0 <= t0 && t0 <= pixman_fixed_1)
return _pixman_gradient_walker_pixel (walker, t0);
else if (0 <= t1 && t1 <= pixman_fixed_1)
return _pixman_gradient_walker_pixel (walker, t1);
}
else
{
if (t0 * dr > mindr)
return _pixman_gradient_walker_pixel (walker, t0);
else if (t1 * dr > mindr)
return _pixman_gradient_walker_pixel (walker, t1);
}
}
return 0;
}
static uint32_t *
conical_get_scanline_narrow (pixman_iter_t *iter, const uint32_t *mask)
{
pixman_image_t *image = iter->image;
int x = iter->x;
int y = iter->y;
int width = iter->width;
uint32_t *buffer = iter->buffer;
gradient_t *gradient = (gradient_t *)image;
conical_gradient_t *conical = (conical_gradient_t *)image;
uint32_t *end = buffer + width;
pixman_gradient_walker_t walker;
pixman_bool_t affine = TRUE;
double cx = 1.;
double cy = 0.;
double cz = 0.;
double rx = x + 0.5;
double ry = y + 0.5;
double rz = 1.;
_pixman_gradient_walker_init (&walker, gradient, image->common.repeat);
if (image->common.transform)
{
pixman_vector_t v;
/* reference point is the center of the pixel */
v.vector[0] = pixman_int_to_fixed (x) + pixman_fixed_1 / 2;
v.vector[1] = pixman_int_to_fixed (y) + pixman_fixed_1 / 2;
v.vector[2] = pixman_fixed_1;
if (!pixman_transform_point_3d (image->common.transform, &v))
return iter->buffer;
cx = image->common.transform->matrix[0][0] / 65536.;
cy = image->common.transform->matrix[1][0] / 65536.;
cz = image->common.transform->matrix[2][0] / 65536.;
rx = v.vector[0] / 65536.;
ry = v.vector[1] / 65536.;
rz = v.vector[2] / 65536.;
affine =
image->common.transform->matrix[2][0] == 0 &&
v.vector[2] == pixman_fixed_1;
}
if (affine)
{
rx -= conical->center.x / 65536.;
ry -= conical->center.y / 65536.;
while (buffer < end)
{
if (!mask || *mask++)
{
double t = coordinates_to_parameter (rx, ry, conical->angle);
*buffer = _pixman_gradient_walker_pixel (
&walker, (pixman_fixed_48_16_t)pixman_double_to_fixed (t));
}
++buffer;
rx += cx;
ry += cy;
}
}
else
{
while (buffer < end)
{
double x, y;
if (!mask || *mask++)
{
double t;
if (rz != 0)
{
x = rx / rz;
y = ry / rz;
}
else
{
x = y = 0.;
}
x -= conical->center.x / 65536.;
y -= conical->center.y / 65536.;
t = coordinates_to_parameter (x, y, conical->angle);
*buffer = _pixman_gradient_walker_pixel (
&walker, (pixman_fixed_48_16_t)pixman_double_to_fixed (t));
}
++buffer;
rx += cx;
ry += cy;
rz += cz;
}
}
iter->y++;
return iter->buffer;
}
void
_pixman_general_radial_gradient_get_scanline_32 (pixman_image_t *image,
int x,
int y,
int width,
uint32_t * buffer,
const uint32_t *mask,
uint32_t mask_bits)
{
/*
* In the radial gradient problem we are given two circles (c₁,r₁) and
* (c₂,r₂) that define the gradient itself. Then, for any point p, we
* must compute the value(s) of t within [0.0, 1.0] representing the
* circle(s) that would color the point.
*
* There are potentially two values of t since the point p can be
* colored by both sides of the circle, (which happens whenever one
* circle is not entirely contained within the other).
*
* If we solve for a value of t that is outside of [0.0, 1.0] then we
* use the extend mode (NONE, REPEAT, REFLECT, or PAD) to map to a
* value within [0.0, 1.0].
*
* Here is an illustration of the problem:
*
* p₂
* p •
* • ╲
* · ╲r₂
* p₁ · ╲
* • θ╲
* ╲ ╌╌•
* ╲r₁ · c₂
* θ╲ ·
* ╌╌•
* c₁
*
* Given (c₁,r₁), (c₂,r₂) and p, we must find an angle θ such that two
* points p₁ and p₂ on the two circles are collinear with p. Then, the
* desired value of t is the ratio of the length of p₁p to the length
* of p₁p₂.
*
* So, we have six unknown values: (p₁x, p₁y), (p₂x, p₂y), θ and t.
* We can also write six equations that constrain the problem:
*
* Point p₁ is a distance r₁ from c₁ at an angle of θ:
*
* 1. p₁x = c₁x + r₁·cos θ
* 2. p₁y = c₁y + r₁·sin θ
*
* Point p₂ is a distance r₂ from c₂ at an angle of θ:
*
* 3. p₂x = c₂x + r2·cos θ
* 4. p₂y = c₂y + r2·sin θ
*
* Point p lies at a fraction t along the line segment p₁p₂:
*
* 5. px = t·p₂x + (1-t)·p₁x
* 6. py = t·p₂y + (1-t)·p₁y
*
* To solve, first subtitute 1-4 into 5 and 6:
*
* px = t·(c₂x + r₂·cos θ) + (1-t)·(c₁x + r₁·cos θ)
* py = t·(c₂y + r₂·sin θ) + (1-t)·(c₁y + r₁·sin θ)
*
* Then solve each for cos θ and sin θ expressed as a function of t:
*
* cos θ = (-(c₂x - c₁x)·t + (px - c₁x)) / ((r₂-r₁)·t + r₁)
* sin θ = (-(c₂y - c₁y)·t + (py - c₁y)) / ((r₂-r₁)·t + r₁)
*
* To simplify this a bit, we define new variables for several of the
* common terms as shown below:
*
* p₂
* p •
* • ╲
* · ┆ ╲r₂
* p₁ · ┆ ╲
* • pdy┆ ╲
* ╲ ┆ •c₂
* ╲r₁ ┆ · ┆
* ╲ ·┆ ┆cdy
* •╌╌╌╌┴╌╌╌╌╌╌╌┘
* c₁ pdx cdx
*
* cdx = (c₂x - c₁x)
* cdy = (c₂y - c₁y)
* dr = r₂-r₁
* pdx = px - c₁x
* pdy = py - c₁y
*
* Note that cdx, cdy, and dr do not depend on point p at all, so can
* be pre-computed for the entire gradient. The simplifed equations
* are now:
*
* cos θ = (-cdx·t + pdx) / (dr·t + r₁)
* sin θ = (-cdy·t + pdy) / (dr·t + r₁)
*
* Finally, to get a single function of t and eliminate the last
* unknown θ, we use the identity sin²θ + cos²θ = 1. First, square
* each equation, (we knew a quadratic was coming since it must be
* possible to obtain two solutions in some cases):
*
* cos²θ = (cdx²t² - 2·cdx·pdx·t + pdx²) / (dr²·t² + 2·r₁·dr·t + r₁²)
* sin²θ = (cdy²t² - 2·cdy·pdy·t + pdy²) / (dr²·t² + 2·r₁·dr·t + r₁²)
*
* Then add both together, set the result equal to 1, and express as a
* standard quadratic equation in t of the form At² + Bt + C = 0
*
* (cdx² + cdy² - dr²)·t² - 2·(cdx·pdx + cdy·pdy + r₁·dr)·t + (pdx² + pdy² - r₁²) = 0
*
* In other words:
*
* A = cdx² + cdy² - dr²
* B = -2·(pdx·cdx + pdy·cdy + r₁·dr)
* C = pdx² + pdy² - r₁²
*
* And again, notice that A does not depend on p, so can be
* precomputed. From here we just use the quadratic formula to solve
* for t:
*
* t = (-2·B ± ⎷(B² - 4·A·C)) / 2·A
*/
gradient_t *gradient = (gradient_t *)image;
source_image_t *source = (source_image_t *)image;
radial_gradient_t *radial = (radial_gradient_t *)image;
uint32_t *end = buffer + width;
pixman_gradient_walker_t walker;
pixman_bool_t affine = TRUE;
double cx = 1.;
double cy = 0.;
double cz = 0.;
double rx = x + 0.5;
double ry = y + 0.5;
double rz = 1.;
_pixman_gradient_walker_init (&walker, gradient, source->common.repeat);
if (source->common.transform)
{
pixman_vector_t v;
/* reference point is the center of the pixel */
v.vector[0] = pixman_int_to_fixed (x) + pixman_fixed_1 / 2;
v.vector[1] = pixman_int_to_fixed (y) + pixman_fixed_1 / 2;
v.vector[2] = pixman_fixed_1;
if (!pixman_transform_point_3d (source->common.transform, &v))
return;
cx = source->common.transform->matrix[0][0] / 65536.;
cy = source->common.transform->matrix[1][0] / 65536.;
cz = source->common.transform->matrix[2][0] / 65536.;
rx = v.vector[0] / 65536.;
ry = v.vector[1] / 65536.;
rz = v.vector[2] / 65536.;
affine =
source->common.transform->matrix[2][0] == 0 &&
v.vector[2] == pixman_fixed_1;
}
if (affine)
{
/* When computing t over a scanline, we notice that some expressions
* are constant so we can compute them just once. Given:
*
* t = (-2·B ± ⎷(B² - 4·A·C)) / 2·A
*
* where
*
* A = cdx² + cdy² - dr² [precomputed as radial->A]
* B = -2·(pdx·cdx + pdy·cdy + r₁·dr)
* C = pdx² + pdy² - r₁²
*
* Since we have an affine transformation, we know that (pdx, pdy)
* increase linearly with each pixel,
*
* pdx = pdx₀ + n·cx,
* pdy = pdy₀ + n·cy,
*
* we can then express B in terms of an linear increment along
* the scanline:
*
* B = B₀ + n·cB, with
* B₀ = -2·(pdx₀·cdx + pdy₀·cdy + r₁·dr) and
* cB = -2·(cx·cdx + cy·cdy)
*
* Thus we can replace the full evaluation of B per-pixel (4 multiplies,
* 2 additions) with a single addition.
*/
float r1 = radial->c1.radius / 65536.;
float r1sq = r1 * r1;
float pdx = rx - radial->c1.x / 65536.;
float pdy = ry - radial->c1.y / 65536.;
float A = radial->A;
float invA = -65536. / (2. * A);
float A4 = -4. * A;
float B = -2. * (pdx*radial->cdx + pdy*radial->cdy + r1*radial->dr);
float cB = -2. * (cx*radial->cdx + cy*radial->cdy);
pixman_bool_t invert = A * radial->dr < 0;
while (buffer < end)
{
if (!mask || *mask++ & mask_bits)
{
pixman_fixed_t t;
double det = B * B + A4 * (pdx * pdx + pdy * pdy - r1sq);
if (det <= 0.)
t = (pixman_fixed_t) (B * invA);
else if (invert)
t = (pixman_fixed_t) ((B + sqrt (det)) * invA);
else
t = (pixman_fixed_t) ((B - sqrt (det)) * invA);
*buffer = _pixman_gradient_walker_pixel (&walker, t);
}
++buffer;
pdx += cx;
pdy += cy;
B += cB;
}
}
else
{
/* projective */
while (buffer < end)
{
if (!mask || *mask++ & mask_bits)
{
double pdx, pdy;
double B, C;
double det;
double c1x = radial->c1.x / 65536.0;
double c1y = radial->c1.y / 65536.0;
double r1 = radial->c1.radius / 65536.0;
pixman_fixed_48_16_t t;
double x, y;
if (rz != 0)
{
x = rx / rz;
y = ry / rz;
}
else
{
x = y = 0.;
}
pdx = x - c1x;
pdy = y - c1y;
B = -2 * (pdx * radial->cdx +
pdy * radial->cdy +
r1 * radial->dr);
C = (pdx * pdx + pdy * pdy - r1 * r1);
det = (B * B) - (4 * radial->A * C);
if (det < 0.0)
det = 0.0;
if (radial->A * radial->dr < 0)
t = (pixman_fixed_48_16_t) ((-B - sqrt (det)) / (2.0 * radial->A) * 65536);
else
t = (pixman_fixed_48_16_t) ((-B + sqrt (det)) / (2.0 * radial->A) * 65536);
*buffer = _pixman_gradient_walker_pixel (&walker, t);
}
++buffer;
rx += cx;
ry += cy;
rz += cz;
}
}
}
static uint32_t *
linear_get_scanline_narrow (pixman_iter_t *iter,
const uint32_t *mask)
{
pixman_image_t *image = iter->image;
int x = iter->x;
int y = iter->y;
int width = iter->width;
uint32_t * buffer = iter->buffer;
pixman_vector_t v, unit;
pixman_fixed_32_32_t l;
pixman_fixed_48_16_t dx, dy;
gradient_t *gradient = (gradient_t *)image;
linear_gradient_t *linear = (linear_gradient_t *)image;
uint32_t *end = buffer + width;
pixman_gradient_walker_t walker;
_pixman_gradient_walker_init (&walker, gradient, image->common.repeat);
/* reference point is the center of the pixel */
v.vector[0] = pixman_int_to_fixed (x) + pixman_fixed_1 / 2;
v.vector[1] = pixman_int_to_fixed (y) + pixman_fixed_1 / 2;
v.vector[2] = pixman_fixed_1;
if (image->common.transform)
{
if (!pixman_transform_point_3d (image->common.transform, &v))
return iter->buffer;
unit.vector[0] = image->common.transform->matrix[0][0];
unit.vector[1] = image->common.transform->matrix[1][0];
unit.vector[2] = image->common.transform->matrix[2][0];
}
else
{
unit.vector[0] = pixman_fixed_1;
unit.vector[1] = 0;
unit.vector[2] = 0;
}
dx = linear->p2.x - linear->p1.x;
dy = linear->p2.y - linear->p1.y;
l = dx * dx + dy * dy;
if (l == 0 || unit.vector[2] == 0)
{
/* affine transformation only */
pixman_fixed_32_32_t t, next_inc;
double inc;
if (l == 0 || v.vector[2] == 0)
{
t = 0;
inc = 0;
}
else
{
double invden, v2;
invden = pixman_fixed_1 * (double) pixman_fixed_1 /
(l * (double) v.vector[2]);
v2 = v.vector[2] * (1. / pixman_fixed_1);
t = ((dx * v.vector[0] + dy * v.vector[1]) -
(dx * linear->p1.x + dy * linear->p1.y) * v2) * invden;
inc = (dx * unit.vector[0] + dy * unit.vector[1]) * invden;
}
next_inc = 0;
if (((pixman_fixed_32_32_t )(inc * width)) == 0)
{
register uint32_t color;
color = _pixman_gradient_walker_pixel (&walker, t);
while (buffer < end)
*buffer++ = color;
}
else
{
int i;
i = 0;
while (buffer < end)
{
if (!mask || *mask++)
{
*buffer = _pixman_gradient_walker_pixel (&walker,
t + next_inc);
}
i++;
next_inc = inc * i;
buffer++;
}
}
}
else
{
/* projective transformation */
double t;
t = 0;
while (buffer < end)
{
if (!mask || *mask++)
{
if (v.vector[2] != 0)
{
double invden, v2;
invden = pixman_fixed_1 * (double) pixman_fixed_1 /
(l * (double) v.vector[2]);
v2 = v.vector[2] * (1. / pixman_fixed_1);
t = ((dx * v.vector[0] + dy * v.vector[1]) -
(dx * linear->p1.x + dy * linear->p1.y) * v2) * invden;
}
*buffer = _pixman_gradient_walker_pixel (&walker, t);
}
++buffer;
v.vector[0] += unit.vector[0];
v.vector[1] += unit.vector[1];
v.vector[2] += unit.vector[2];
}
}
iter->y++;
return iter->buffer;
}
