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gimp/app/core/gimp-transform-utils.c

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/* GIMP - The GNU Image Manipulation Program
* Copyright (C) 1995-2001 Spencer Kimball, Peter Mattis, and others
*
* This program is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program. If not, see <https://www.gnu.org/licenses/>.
*/
#include "config.h"
#include <string.h>
#include <glib-object.h>
#include "libgimpmath/gimpmath.h"
#include "core-types.h"
#include "gimp-transform-utils.h"
#include "gimpcoords.h"
#include "gimpcoords-interpolate.h"
#define EPSILON 1e-6
void
gimp_transform_get_rotate_center (gint x,
gint y,
gint width,
gint height,
gboolean auto_center,
gdouble *center_x,
gdouble *center_y)
{
g_return_if_fail (center_x != NULL);
g_return_if_fail (center_y != NULL);
if (auto_center)
{
*center_x = (gdouble) x + (gdouble) width / 2.0;
*center_y = (gdouble) y + (gdouble) height / 2.0;
}
}
void
gimp_transform_get_flip_axis (gint x,
gint y,
gint width,
gint height,
GimpOrientationType flip_type,
gboolean auto_center,
gdouble *axis)
{
g_return_if_fail (axis != NULL);
if (auto_center)
{
switch (flip_type)
{
case GIMP_ORIENTATION_HORIZONTAL:
*axis = ((gdouble) x + (gdouble) width / 2.0);
break;
case GIMP_ORIENTATION_VERTICAL:
*axis = ((gdouble) y + (gdouble) height / 2.0);
break;
default:
g_return_if_reached ();
break;
}
}
}
void
gimp_transform_matrix_flip (GimpMatrix3 *matrix,
GimpOrientationType flip_type,
gdouble axis)
{
g_return_if_fail (matrix != NULL);
switch (flip_type)
{
case GIMP_ORIENTATION_HORIZONTAL:
gimp_matrix3_translate (matrix, - axis, 0.0);
gimp_matrix3_scale (matrix, -1.0, 1.0);
gimp_matrix3_translate (matrix, axis, 0.0);
break;
case GIMP_ORIENTATION_VERTICAL:
gimp_matrix3_translate (matrix, 0.0, - axis);
gimp_matrix3_scale (matrix, 1.0, -1.0);
gimp_matrix3_translate (matrix, 0.0, axis);
break;
case GIMP_ORIENTATION_UNKNOWN:
break;
}
}
void
gimp_transform_matrix_flip_free (GimpMatrix3 *matrix,
gdouble x1,
gdouble y1,
gdouble x2,
gdouble y2)
{
gdouble angle;
g_return_if_fail (matrix != NULL);
angle = atan2 (y2 - y1, x2 - x1);
gimp_matrix3_identity (matrix);
gimp_matrix3_translate (matrix, -x1, -y1);
gimp_matrix3_rotate (matrix, -angle);
gimp_matrix3_scale (matrix, 1.0, -1.0);
gimp_matrix3_rotate (matrix, angle);
gimp_matrix3_translate (matrix, x1, y1);
}
void
gimp_transform_matrix_rotate (GimpMatrix3 *matrix,
GimpRotationType rotate_type,
gdouble center_x,
gdouble center_y)
{
gdouble angle = 0;
switch (rotate_type)
{
case GIMP_ROTATE_DEGREES90:
angle = G_PI_2;
break;
case GIMP_ROTATE_DEGREES180:
angle = G_PI;
break;
case GIMP_ROTATE_DEGREES270:
angle = - G_PI_2;
break;
}
gimp_transform_matrix_rotate_center (matrix, center_x, center_y, angle);
}
void
gimp_transform_matrix_rotate_rect (GimpMatrix3 *matrix,
gint x,
gint y,
gint width,
gint height,
gdouble angle)
{
gdouble center_x;
gdouble center_y;
g_return_if_fail (matrix != NULL);
center_x = (gdouble) x + (gdouble) width / 2.0;
center_y = (gdouble) y + (gdouble) height / 2.0;
gimp_matrix3_translate (matrix, -center_x, -center_y);
gimp_matrix3_rotate (matrix, angle);
gimp_matrix3_translate (matrix, +center_x, +center_y);
}
void
gimp_transform_matrix_rotate_center (GimpMatrix3 *matrix,
gdouble center_x,
gdouble center_y,
gdouble angle)
{
g_return_if_fail (matrix != NULL);
gimp_matrix3_translate (matrix, -center_x, -center_y);
gimp_matrix3_rotate (matrix, angle);
gimp_matrix3_translate (matrix, +center_x, +center_y);
}
void
gimp_transform_matrix_scale (GimpMatrix3 *matrix,
gint x,
gint y,
gint width,
gint height,
gdouble t_x,
gdouble t_y,
gdouble t_width,
gdouble t_height)
{
gdouble scale_x = 1.0;
gdouble scale_y = 1.0;
g_return_if_fail (matrix != NULL);
if (width > 0)
scale_x = t_width / (gdouble) width;
if (height > 0)
scale_y = t_height / (gdouble) height;
gimp_matrix3_identity (matrix);
gimp_matrix3_translate (matrix, -x, -y);
gimp_matrix3_scale (matrix, scale_x, scale_y);
gimp_matrix3_translate (matrix, t_x, t_y);
}
void
gimp_transform_matrix_shear (GimpMatrix3 *matrix,
gint x,
gint y,
gint width,
gint height,
GimpOrientationType orientation,
gdouble amount)
{
gdouble center_x;
gdouble center_y;
g_return_if_fail (matrix != NULL);
if (width == 0)
width = 1;
if (height == 0)
height = 1;
center_x = (gdouble) x + (gdouble) width / 2.0;
center_y = (gdouble) y + (gdouble) height / 2.0;
gimp_matrix3_identity (matrix);
gimp_matrix3_translate (matrix, -center_x, -center_y);
if (orientation == GIMP_ORIENTATION_HORIZONTAL)
gimp_matrix3_xshear (matrix, amount / height);
else
gimp_matrix3_yshear (matrix, amount / width);
gimp_matrix3_translate (matrix, +center_x, +center_y);
}
void
gimp_transform_matrix_perspective (GimpMatrix3 *matrix,
gint x,
gint y,
gint width,
gint height,
gdouble t_x1,
gdouble t_y1,
gdouble t_x2,
gdouble t_y2,
gdouble t_x3,
gdouble t_y3,
gdouble t_x4,
gdouble t_y4)
{
GimpMatrix3 trafo;
gdouble scalex;
gdouble scaley;
g_return_if_fail (matrix != NULL);
scalex = scaley = 1.0;
if (width > 0)
scalex = 1.0 / (gdouble) width;
if (height > 0)
scaley = 1.0 / (gdouble) height;
gimp_matrix3_translate (matrix, -x, -y);
gimp_matrix3_scale (matrix, scalex, scaley);
/* Determine the perspective transform that maps from
* the unit cube to the transformed coordinates
*/
{
gdouble dx1, dx2, dx3, dy1, dy2, dy3;
dx1 = t_x2 - t_x4;
dx2 = t_x3 - t_x4;
dx3 = t_x1 - t_x2 + t_x4 - t_x3;
dy1 = t_y2 - t_y4;
dy2 = t_y3 - t_y4;
dy3 = t_y1 - t_y2 + t_y4 - t_y3;
/* Is the mapping affine? */
if ((dx3 == 0.0) && (dy3 == 0.0))
{
trafo.coeff[0][0] = t_x2 - t_x1;
trafo.coeff[0][1] = t_x4 - t_x2;
trafo.coeff[0][2] = t_x1;
trafo.coeff[1][0] = t_y2 - t_y1;
trafo.coeff[1][1] = t_y4 - t_y2;
trafo.coeff[1][2] = t_y1;
trafo.coeff[2][0] = 0.0;
trafo.coeff[2][1] = 0.0;
}
else
{
gdouble det1, det2;
det1 = dx3 * dy2 - dy3 * dx2;
det2 = dx1 * dy2 - dy1 * dx2;
trafo.coeff[2][0] = (det2 == 0.0) ? 1.0 : det1 / det2;
det1 = dx1 * dy3 - dy1 * dx3;
trafo.coeff[2][1] = (det2 == 0.0) ? 1.0 : det1 / det2;
trafo.coeff[0][0] = t_x2 - t_x1 + trafo.coeff[2][0] * t_x2;
trafo.coeff[0][1] = t_x3 - t_x1 + trafo.coeff[2][1] * t_x3;
trafo.coeff[0][2] = t_x1;
trafo.coeff[1][0] = t_y2 - t_y1 + trafo.coeff[2][0] * t_y2;
trafo.coeff[1][1] = t_y3 - t_y1 + trafo.coeff[2][1] * t_y3;
trafo.coeff[1][2] = t_y1;
}
trafo.coeff[2][2] = 1.0;
}
gimp_matrix3_mult (&trafo, matrix);
}
/* modified gaussian algorithm
* solves a system of linear equations
*
* Example:
* 1x + 2y + 4z = 25
* 2x + 1y = 4
* 3x + 5y + 2z = 23
* Solution: x=1, y=2, z=5
*
* Input:
* matrix = { 1,2,4,25,2,1,0,4,3,5,2,23 }
* s = 3 (Number of variables)
* Output:
* return value == TRUE (TRUE, if there is a single unique solution)
* solution == { 1,2,5 } (if the return value is FALSE, the content
* of solution is of no use)
*/
static gboolean
mod_gauss (gdouble matrix[],
gdouble solution[],
gint s)
{
gint p[s]; /* row permutation */
gint i, j, r, temp;
gdouble q;
gint t = s + 1;
for (i = 0; i < s; i++)
{
p[i] = i;
}
for (r = 0; r < s; r++)
{
/* make sure that (r,r) is not 0 */
if (fabs (matrix[p[r] * t + r]) <= EPSILON)
{
/* we need to permutate rows */
for (i = r + 1; i <= s; i++)
{
if (i == s)
{
/* if this happens, the linear system has zero or
* more than one solutions.
*/
return FALSE;
}
if (fabs (matrix[p[i] * t + r]) > EPSILON)
break;
}
temp = p[r];
p[r] = p[i];
p[i] = temp;
}
/* make (r,r) == 1 */
q = 1.0 / matrix[p[r] * t + r];
matrix[p[r] * t + r] = 1.0;
for (j = r + 1; j < t; j++)
{
matrix[p[r] * t + j] *= q;
}
/* make that all entries in column r are 0 (except (r,r)) */
for (i = 0; i < s; i++)
{
if (i == r)
continue;
for (j = r + 1; j < t ; j++)
{
matrix[p[i] * t + j] -= matrix[p[r] * t + j] * matrix[p[i] * t + r];
}
/* we don't need to execute the following line
* since we won't access this element again:
*
* matrix[p[i] * t + r] = 0.0;
*/
}
}
for (i = 0; i < s; i++)
{
solution[i] = matrix[p[i] * t + s];
}
return TRUE;
}
/* multiplies 'matrix' by the matrix that transforms a set of 4 'input_points'
* to corresponding 'output_points', if such matrix exists, and is valid (i.e.,
* keeps the output points in front of the camera).
*
* returns TRUE if successful.
*/
gboolean
gimp_transform_matrix_generic (GimpMatrix3 *matrix,
const GimpVector2 input_points[4],
const GimpVector2 output_points[4])
{
GimpMatrix3 trafo;
gdouble coeff[8 * 9];
gboolean negative = -1;
gint i;
gboolean result = TRUE;
g_return_val_if_fail (matrix != NULL, FALSE);
g_return_val_if_fail (input_points != NULL, FALSE);
g_return_val_if_fail (output_points != NULL, FALSE);
/* find the matrix that transforms 'input_points' to 'output_points', whose
* (3, 3) coefficient is 1, by solving a system of linear equations whose
* solution is the remaining 8 coefficients.
*/
for (i = 0; i < 4; i++)
{
coeff[i * 9 + 0] = input_points[i].x;
coeff[i * 9 + 1] = input_points[i].y;
coeff[i * 9 + 2] = 1.0;
coeff[i * 9 + 3] = 0.0;
coeff[i * 9 + 4] = 0.0;
coeff[i * 9 + 5] = 0.0;
coeff[i * 9 + 6] = -input_points[i].x * output_points[i].x;
coeff[i * 9 + 7] = -input_points[i].y * output_points[i].x;
coeff[i * 9 + 8] = output_points[i].x;
coeff[(i + 4) * 9 + 0] = 0.0;
coeff[(i + 4) * 9 + 1] = 0.0;
coeff[(i + 4) * 9 + 2] = 0.0;
coeff[(i + 4) * 9 + 3] = input_points[i].x;
coeff[(i + 4) * 9 + 4] = input_points[i].y;
coeff[(i + 4) * 9 + 5] = 1.0;
coeff[(i + 4) * 9 + 6] = -input_points[i].x * output_points[i].y;
coeff[(i + 4) * 9 + 7] = -input_points[i].y * output_points[i].y;
coeff[(i + 4) * 9 + 8] = output_points[i].y;
}
/* if there is no solution, bail */
if (! mod_gauss (coeff, (gdouble *) trafo.coeff, 8))
return FALSE;
trafo.coeff[2][2] = 1.0;
/* make sure that none of the input points maps to a point at infinity, and
* that all output points are on the same side of the camera.
*/
for (i = 0; i < 4; i++)
{
gdouble w;
gboolean neg;
w = trafo.coeff[2][0] * input_points[i].x +
trafo.coeff[2][1] * input_points[i].y +
trafo.coeff[2][2];
if (fabs (w) <= EPSILON)
result = FALSE;
neg = (w < 0.0);
if (negative < 0)
{
negative = neg;
}
else if (neg != negative)
{
result = FALSE;
break;
}
}
/* if the output points are all behind the camera, negate the matrix, which
* would map the input points to the corresponding points in front of the
* camera.
*/
if (negative > 0)
{
gint r;
gint c;
for (r = 0; r < 3; r++)
{
for (c = 0; c < 3; c++)
{
trafo.coeff[r][c] = -trafo.coeff[r][c];
}
}
}
/* append the transformation to 'matrix' */
gimp_matrix3_mult (&trafo, matrix);
return result;
}
gboolean
gimp_transform_polygon_is_convex (gdouble x1,
gdouble y1,
gdouble x2,
gdouble y2,
gdouble x3,
gdouble y3,
gdouble x4,
gdouble y4)
{
gdouble z1, z2, z3, z4;
/* We test if the transformed polygon is convex. if z1 and z2 have
* the same sign as well as z3 and z4 the polygon is convex.
*/
z1 = ((x2 - x1) * (y4 - y1) -
(x4 - x1) * (y2 - y1));
z2 = ((x4 - x1) * (y3 - y1) -
(x3 - x1) * (y4 - y1));
z3 = ((x4 - x2) * (y3 - y2) -
(x3 - x2) * (y4 - y2));
z4 = ((x3 - x2) * (y1 - y2) -
(x1 - x2) * (y3 - y2));
return (z1 * z2 > 0) && (z3 * z4 > 0);
}
/* transforms the polygon or polyline, whose vertices are given by 'vertices',
* by 'matrix', performing clipping by the near plane. 'closed' indicates
* whether the vertices represent a polygon ('closed == TRUE') or a polyline
* ('closed == FALSE').
*
* returns the transformed vertices in 't_vertices', and their count in
* 'n_t_vertices'. the minimal possible number of transformed vertices is 0,
* which happens when the entire input is clipped. in general, the maximal
* possible number of transformed vertices is '3 * n_vertices / 2' (rounded
* down), however, for convex polygons the number is 'n_vertices + 1', and for
* a single line segment ('n_vertices == 2' and 'closed == FALSE') the number
* is 2.
*
* 't_vertices' may not alias 'vertices', except when transforming a single
* line segment.
*/
void
gimp_transform_polygon (const GimpMatrix3 *matrix,
const GimpVector2 *vertices,
gint n_vertices,
gboolean closed,
GimpVector2 *t_vertices,
gint *n_t_vertices)
{
GimpVector3 curr;
gboolean curr_visible;
gint i;
g_return_if_fail (matrix != NULL);
g_return_if_fail (vertices != NULL);
g_return_if_fail (n_vertices >= 0);
g_return_if_fail (t_vertices != NULL);
g_return_if_fail (n_t_vertices != NULL);
*n_t_vertices = 0;
if (n_vertices == 0)
return;
curr.x = matrix->coeff[0][0] * vertices[0].x +
matrix->coeff[0][1] * vertices[0].y +
matrix->coeff[0][2];
curr.y = matrix->coeff[1][0] * vertices[0].x +
matrix->coeff[1][1] * vertices[0].y +
matrix->coeff[1][2];
curr.z = matrix->coeff[2][0] * vertices[0].x +
matrix->coeff[2][1] * vertices[0].y +
matrix->coeff[2][2];
curr_visible = (curr.z >= GIMP_TRANSFORM_NEAR_Z);
for (i = 0; i < n_vertices; i++)
{
if (curr_visible)
{
t_vertices[(*n_t_vertices)++] = (GimpVector2) { curr.x / curr.z,
curr.y / curr.z };
}
if (i < n_vertices - 1 || closed)
{
GimpVector3 next;
gboolean next_visible;
gint j = (i + 1) % n_vertices;
next.x = matrix->coeff[0][0] * vertices[j].x +
matrix->coeff[0][1] * vertices[j].y +
matrix->coeff[0][2];
next.y = matrix->coeff[1][0] * vertices[j].x +
matrix->coeff[1][1] * vertices[j].y +
matrix->coeff[1][2];
next.z = matrix->coeff[2][0] * vertices[j].x +
matrix->coeff[2][1] * vertices[j].y +
matrix->coeff[2][2];
next_visible = (next.z >= GIMP_TRANSFORM_NEAR_Z);
if (next_visible != curr_visible)
{
gdouble ratio = (curr.z - GIMP_TRANSFORM_NEAR_Z) / (curr.z - next.z);
t_vertices[(*n_t_vertices)++] =
(GimpVector2) { (curr.x + (next.x - curr.x) * ratio) / GIMP_TRANSFORM_NEAR_Z,
(curr.y + (next.y - curr.y) * ratio) / GIMP_TRANSFORM_NEAR_Z };
}
curr = next;
curr_visible = next_visible;
}
}
}
/* same as gimp_transform_polygon(), but using GimpCoords as the vertex type,
* instead of GimpVector2.
*/
void
gimp_transform_polygon_coords (const GimpMatrix3 *matrix,
const GimpCoords *vertices,
gint n_vertices,
gboolean closed,
GimpCoords *t_vertices,
gint *n_t_vertices)
{
GimpVector3 curr;
gboolean curr_visible;
gint i;
g_return_if_fail (matrix != NULL);
g_return_if_fail (vertices != NULL);
g_return_if_fail (n_vertices >= 0);
g_return_if_fail (t_vertices != NULL);
g_return_if_fail (n_t_vertices != NULL);
*n_t_vertices = 0;
if (n_vertices == 0)
return;
curr.x = matrix->coeff[0][0] * vertices[0].x +
matrix->coeff[0][1] * vertices[0].y +
matrix->coeff[0][2];
curr.y = matrix->coeff[1][0] * vertices[0].x +
matrix->coeff[1][1] * vertices[0].y +
matrix->coeff[1][2];
curr.z = matrix->coeff[2][0] * vertices[0].x +
matrix->coeff[2][1] * vertices[0].y +
matrix->coeff[2][2];
curr_visible = (curr.z >= GIMP_TRANSFORM_NEAR_Z);
for (i = 0; i < n_vertices; i++)
{
if (curr_visible)
{
t_vertices[*n_t_vertices] = vertices[i];
t_vertices[*n_t_vertices].x = curr.x / curr.z;
t_vertices[*n_t_vertices].y = curr.y / curr.z;
(*n_t_vertices)++;
}
if (i < n_vertices - 1 || closed)
{
GimpVector3 next;
gboolean next_visible;
gint j = (i + 1) % n_vertices;
next.x = matrix->coeff[0][0] * vertices[j].x +
matrix->coeff[0][1] * vertices[j].y +
matrix->coeff[0][2];
next.y = matrix->coeff[1][0] * vertices[j].x +
matrix->coeff[1][1] * vertices[j].y +
matrix->coeff[1][2];
next.z = matrix->coeff[2][0] * vertices[j].x +
matrix->coeff[2][1] * vertices[j].y +
matrix->coeff[2][2];
next_visible = (next.z >= GIMP_TRANSFORM_NEAR_Z);
if (next_visible != curr_visible)
{
gdouble ratio = (curr.z - GIMP_TRANSFORM_NEAR_Z) / (curr.z - next.z);
gimp_coords_mix (1.0 - ratio, &vertices[i],
ratio, &vertices[j],
&t_vertices[*n_t_vertices]);
t_vertices[*n_t_vertices].x = (curr.x + (next.x - curr.x) * ratio) /
GIMP_TRANSFORM_NEAR_Z;
t_vertices[*n_t_vertices].y = (curr.y + (next.y - curr.y) * ratio) /
GIMP_TRANSFORM_NEAR_Z;
(*n_t_vertices)++;
}
curr = next;
curr_visible = next_visible;
}
}
}
/* returns the value of the polynomial 'poly', of degree 'degree', at 'x'. the
* coefficients of 'poly' should be specified in descending-degree order.
*/
static gdouble
polynomial_eval (const gdouble *poly,
gint degree,
gdouble x)
{
gdouble y = poly[0];
gint i;
for (i = 1; i <= degree; i++)
y = y * x + poly[i];
return y;
}
/* derives the polynomial 'poly', of degree 'degree'.
*
* returns the derivative in 'result'.
*/
static void
polynomial_derive (const gdouble *poly,
gint degree,
gdouble *result)
{
while (degree)
*result++ = *poly++ * degree--;
}
/* finds the real odd-multiplicity root of the polynomial 'poly', of degree
* 'degree', inside the range '(x1, x2)'.
*
* returns TRUE if such a root exists, and stores its value in '*root'.
*
* 'poly' shall be monotonic in the range '(x1, x2)'.
*/
static gboolean
polynomial_odd_root (const gdouble *poly,
gint degree,
gdouble x1,
gdouble x2,
gdouble *root)
{
gdouble y1;
gdouble y2;
gint i;
y1 = polynomial_eval (poly, degree, x1);
y2 = polynomial_eval (poly, degree, x2);
if (y1 * y2 > -EPSILON)
{
/* the two endpoints have the same sign, or one of them is zero. there's
* no root inside the range.
*/
return FALSE;
}
else if (y1 > 0.0)
{
gdouble t;
/* if the first endpoint is positive, swap the endpoints, so that the
* first endpoint is always negative, and the second endpoint is always
* positive.
*/
t = x1;
x1 = x2;
x2 = t;
}
/* approximate the root using binary search */
for (i = 0; i < 53; i++)
{
gdouble x = (x1 + x2) / 2.0;
gdouble y = polynomial_eval (poly, degree, x);
if (y > 0.0)
x2 = x;
else
x1 = x;
}
*root = (x1 + x2) / 2.0;
return TRUE;
}
/* finds the real odd-multiplicity roots of the polynomial 'poly', of degree
* 'degree', inside the range '(x1, x2)'.
*
* returns the roots in 'roots', in ascending order, and their count in
* 'n_roots'.
*/
static void
polynomial_odd_roots (const gdouble *poly,
gint degree,
gdouble x1,
gdouble x2,
gdouble *roots,
gint *n_roots)
{
*n_roots = 0;
/* find the real degree of the polynomial (skip any leading coefficients that
* are 0)
*/
for (; degree && fabs (*poly) < EPSILON; poly++, degree--);
#define ADD_ROOT(root) \
do \
{ \
gdouble r = (root); \
\
if (r > x1 && r < x2) \
roots[(*n_roots)++] = r; \
} \
while (FALSE)
switch (degree)
{
/* constant case */
case 0:
break;
/* linear case */
case 1:
ADD_ROOT (-poly[1] / poly[0]);
break;
/* quadratic case */
case 2:
{
gdouble s = SQR (poly[1]) - 4 * poly[0] * poly[2];
if (s > EPSILON)
{
s = sqrt (s);
if (poly[0] < 0.0)
s = -s;
ADD_ROOT ((-poly[1] - s) / (2.0 * poly[0]));
ADD_ROOT ((-poly[1] + s) / (2.0 * poly[0]));
}
break;
}
/* general case */
default:
{
gdouble deriv[degree];
gdouble deriv_roots[degree - 1];
gint n_deriv_roots;
gdouble a;
gdouble b;
gint i;
/* find the odd roots of the derivative, i.e., the local extrema of the
* polynomial
*/
polynomial_derive (poly, degree, deriv);
polynomial_odd_roots (deriv, degree - 1, x1, x2,
deriv_roots, &n_deriv_roots);
/* search for roots between each consecutive pair of extrema, including
* the endpoints
*/
a = x1;
for (i = 0; i <= n_deriv_roots; i++)
{
if (i < n_deriv_roots)
b = deriv_roots[i];
else
b = x2;
*n_roots += polynomial_odd_root (poly, degree, a, b,
&roots[*n_roots]);
a = b;
}
break;
}
}
#undef ADD_ROOT
}
/* clips the cubic bezier segment, defined by the four control points 'bezier',
* to the halfplane 'ax + by + c >= 0'.
*
* returns the clipped set of bezier segments in 'c_bezier', and their count in
* 'n_c_bezier'. the minimal possible number of clipped segments is 0, which
* happens when the entire segment is clipped. the maximal possible number of
* clipped segments is 2.
*
* if the first clipped segment is an initial segment of 'bezier', sets
* '*start_in' to TRUE, otherwise to FALSE. if the last clipped segment is a
* final segment of 'bezier', sets '*end_in' to TRUE, otherwise to FALSE.
*
* 'c_bezier' may not alias 'bezier'.
*/
static void
clip_bezier (const GimpCoords bezier[4],
gdouble a,
gdouble b,
gdouble c,
GimpCoords c_bezier[2][4],
gint *n_c_bezier,
gboolean *start_in,
gboolean *end_in)
{
gdouble dot[4];
gdouble poly[4];
gdouble roots[5];
gint n_roots;
gint n_positive;
gint i;
n_positive = 0;
for (i = 0; i < 4; i++)
{
dot[i] = a * bezier[i].x + b * bezier[i].y + c;
n_positive += (dot[i] >= 0.0);
}
if (n_positive == 0)
{
/* all points are out -- the entire segment is out */
*n_c_bezier = 0;
*start_in = FALSE;
*end_in = FALSE;
return;
}
else if (n_positive == 4)
{
/* all points are in -- the entire segment is in */
memcpy (c_bezier[0], bezier, sizeof (GimpCoords[4]));
*n_c_bezier = 1;
*start_in = TRUE;
*end_in = TRUE;
return;
}
/* find the points of intersection of the segment with the 'ax + by + c = 0'
* line
*/
poly[0] = dot[3] - 3.0 * dot[2] + 3.0 * dot[1] - dot[0];
poly[1] = 3.0 * (dot[2] - 2.0 * dot[1] + dot[0]);
poly[2] = 3.0 * (dot[1] - dot[0]);
poly[3] = dot[0];
roots[0] = 0.0;
polynomial_odd_roots (poly, 3, 0.0, 1.0, roots + 1, &n_roots);
roots[++n_roots] = 1.0;
/* construct the list of segments that are inside the halfplane */
*n_c_bezier = 0;
*start_in = (polynomial_eval (poly, 3, roots[1] / 2.0) > 0.0);
*end_in = (*start_in + n_roots + 1) % 2;
for (i = ! *start_in; i < n_roots; i += 2)
{
gdouble t0 = roots[i];
gdouble t1 = roots[i + 1];
gimp_coords_interpolate_bezier_at (bezier, t0,
&c_bezier[*n_c_bezier][0],
&c_bezier[*n_c_bezier][1]);
gimp_coords_interpolate_bezier_at (bezier, t1,
&c_bezier[*n_c_bezier][3],
&c_bezier[*n_c_bezier][2]);
gimp_coords_mix (1.0, &c_bezier[*n_c_bezier][0],
(t1 - t0) / 3.0, &c_bezier[*n_c_bezier][1],
&c_bezier[*n_c_bezier][1]);
gimp_coords_mix (1.0, &c_bezier[*n_c_bezier][3],
(t0 - t1) / 3.0, &c_bezier[*n_c_bezier][2],
&c_bezier[*n_c_bezier][2]);
(*n_c_bezier)++;
}
}
/* transforms the cubic bezier segment, defined by the four control points
* 'bezier', by 'matrix', subdividing it as necessary to avoid diverging too
* much from the real transformed curve. at most 'depth' subdivisions are
* performed.
*
* appends the transformed sequence of bezier segments to 't_beziers'.
*
* 'bezier' shall be fully clipped to the near plane.
*/
static void
transform_bezier_coords (const GimpMatrix3 *matrix,
const GimpCoords bezier[4],
GQueue *t_beziers,
gint depth)
{
GimpCoords *t_bezier;
gint n;
/* check if we need to split the segment */
if (depth > 0)
{
GimpVector2 v[4];
GimpVector2 c[2];
GimpVector2 b;
gint i;
for (i = 0; i < 4; i++)
v[i] = (GimpVector2) { bezier[i].x, bezier[i].y };
gimp_vector2_sub (&c[0], &v[1], &v[0]);
gimp_vector2_sub (&c[1], &v[2], &v[3]);
gimp_vector2_sub (&b, &v[3], &v[0]);
gimp_vector2_mul (&b, 1.0 / gimp_vector2_inner_product (&b, &b));
for (i = 0; i < 2; i++)
{
/* split the segment if one of the control points is too far from the
* line connecting the anchors
*/
if (fabs (gimp_vector2_cross_product (&c[i], &b).x) > 0.5)
{
GimpCoords mid_position;
GimpCoords mid_velocity;
GimpCoords sub[4];
gimp_coords_interpolate_bezier_at (bezier, 0.5,
&mid_position, &mid_velocity);
/* first half */
sub[0] = bezier[0];
sub[3] = mid_position;
gimp_coords_mix (0.5, &sub[0],
0.5, &bezier[1],
&sub[1]);
gimp_coords_mix (1.0, &sub[3],
-1.0 / 6.0, &mid_velocity,
&sub[2]);
transform_bezier_coords (matrix, sub, t_beziers, depth - 1);
/* second half */
sub[0] = mid_position;
sub[3] = bezier[3];
gimp_coords_mix (1.0, &sub[0],
+1.0 / 6.0, &mid_velocity,
&sub[1]);
gimp_coords_mix (0.5, &sub[3],
0.5, &bezier[2],
&sub[2]);
transform_bezier_coords (matrix, sub, t_beziers, depth - 1);
return;
}
}
}
/* transform the segment by transforming each of the individual points. note
* that, for non-affine transforms, this is only an approximation of the real
* transformed curve, but due to subdivision it should be good enough.
*/
t_bezier = g_new (GimpCoords, 4);
/* note that while the segments themselves are clipped to the near plane,
* their control points may still get transformed behind the camera. we
* therefore clip the control points to the near plane as well, which is not
* too meaningful, but avoids erroneously transforming them behind the
* camera.
*/
gimp_transform_polygon_coords (matrix, bezier, 2, FALSE,
t_bezier, &n);
gimp_transform_polygon_coords (matrix, bezier + 2, 2, FALSE,
t_bezier + 2, &n);
g_queue_push_tail (t_beziers, t_bezier);
}
/* transforms the cubic bezier segment, defined by the four control points
* 'bezier', by 'matrix', performing clipping by the near plane and subdividing
* as necessary.
*
* returns the transformed set of bezier-segment sequences in 't_beziers', as
* GQueues of GimpCoords[4] bezier-segments, and the number of sequences in
* 'n_t_beziers'. the minimal possible number of transformed sequences is 0,
* which happens when the entire segment is clipped. the maximal possible
* number of transformed sequences is 2. each sequence has at least one
* segment.
*
* if the first transformed segment is an initial segment of 'bezier', sets
* '*start_in' to TRUE, otherwise to FALSE. if the last transformed segment is
* a final segment of 'bezier', sets '*end_in' to TRUE, otherwise to FALSE.
*/
void
gimp_transform_bezier_coords (const GimpMatrix3 *matrix,
const GimpCoords bezier[4],
GQueue *t_beziers[2],
gint *n_t_beziers,
gboolean *start_in,
gboolean *end_in)
{
GimpCoords c_bezier[2][4];
gint i;
g_return_if_fail (matrix != NULL);
g_return_if_fail (bezier != NULL);
g_return_if_fail (t_beziers != NULL);
g_return_if_fail (n_t_beziers != NULL);
g_return_if_fail (start_in != NULL);
g_return_if_fail (end_in != NULL);
/* if the matrix is affine, transform the easy way */
if (gimp_matrix3_is_affine (matrix))
{
GimpCoords *t_bezier;
t_beziers[0] = g_queue_new ();
*n_t_beziers = 1;
t_bezier = g_new (GimpCoords, 1);
g_queue_push_tail (t_beziers[0], t_bezier);
for (i = 0; i < 4; i++)
{
t_bezier[i] = bezier[i];
gimp_matrix3_transform_point (matrix,
bezier[i].x, bezier[i].y,
&t_bezier[i].x, &t_bezier[i].y);
}
return;
}
/* clip the segment to the near plane */
clip_bezier (bezier,
matrix->coeff[2][0],
matrix->coeff[2][1],
matrix->coeff[2][2] - GIMP_TRANSFORM_NEAR_Z,
c_bezier, n_t_beziers,
start_in, end_in);
/* transform each of the resulting segments */
for (i = 0; i < *n_t_beziers; i++)
{
t_beziers[i] = g_queue_new ();
transform_bezier_coords (matrix, c_bezier[i], t_beziers[i], 3);
}
}
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