/* cairo - a vector graphics library with display and print output * * Copyright © 2002 University of Southern California * * This library is free software; you can redistribute it and/or * modify it either under the terms of the GNU Lesser General Public * License version 2.1 as published by the Free Software Foundation * (the "LGPL") or, at your option, under the terms of the Mozilla * Public License Version 1.1 (the "MPL"). If you do not alter this * notice, a recipient may use your version of this file under either * the MPL or the LGPL. * * You should have received a copy of the LGPL along with this library * in the file COPYING-LGPL-2.1; if not, write to the Free Software * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA * You should have received a copy of the MPL along with this library * in the file COPYING-MPL-1.1 * * The contents of this file are subject to the Mozilla Public License * Version 1.1 (the "License"); you may not use this file except in * compliance with the License. You may obtain a copy of the License at * http://www.mozilla.org/MPL/ * * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY * OF ANY KIND, either express or implied. See the LGPL or the MPL for * the specific language governing rights and limitations. * * The Original Code is the cairo graphics library. * * The Initial Developer of the Original Code is University of Southern * California. * * Contributor(s): * Carl D. Worth <cworth@cworth.org> */ #define _GNU_SOURCE #include <stdlib.h> #include "cairoint.h" static void _cairo_matrix_scalar_multiply (cairo_matrix_t *matrix, double scalar); static void _cairo_matrix_compute_adjoint (cairo_matrix_t *matrix); /** * cairo_matrix_init_identity: * @matrix: a #cairo_matrix_t * * Modifies @matrix to be an identity transformation. **/ void cairo_matrix_init_identity (cairo_matrix_t *matrix) { cairo_matrix_init (matrix, 1, 0, 0, 1, 0, 0); } slim_hidden_def(cairo_matrix_init_identity); /** * cairo_matrix_init: * @matrix: a cairo_matrix_t * @xx: xx component of the affine transformation * @yx: yx component of the affine transformation * @xy: xy component of the affine transformation * @yy: yy component of the affine transformation * @x0: X translation component of the affine transformation * @y0: Y translation component of the affine transformation * * Sets @matrix to be the affine transformation given by * @xx, @yx, @xy, @yy, @x0, @y0. The transformation is given * by: * <programlisting> * x_new = xx * x + xy * y + x0; * y_new = yx * x + yy * y + y0; * </programlisting> **/ void cairo_matrix_init (cairo_matrix_t *matrix, double xx, double yx, double xy, double yy, double x0, double y0) { matrix->xx = xx; matrix->yx = yx; matrix->xy = xy; matrix->yy = yy; matrix->x0 = x0; matrix->y0 = y0; } slim_hidden_def(cairo_matrix_init); /** * _cairo_matrix_get_affine: * @matrix: a @cairo_matrix_t * @xx: location to store xx component of matrix * @yx: location to store yx component of matrix * @xy: location to store xy component of matrix * @yy: location to store yy component of matrix * @x0: location to store x0 (X-translation component) of matrix, or %NULL * @y0: location to store y0 (Y-translation component) of matrix, or %NULL * * Gets the matrix values for the affine tranformation that @matrix represents. * See cairo_matrix_init(). * * * This function is a leftover from the old public API, but is still * mildly useful as an internal means for getting at the matrix * members in a positional way. For example, when reassigning to some * external matrix type, or when renaming members to more meaningful * names (such as a,b,c,d,e,f) for particular manipulations. **/ void _cairo_matrix_get_affine (const cairo_matrix_t *matrix, double *xx, double *yx, double *xy, double *yy, double *x0, double *y0) { *xx = matrix->xx; *yx = matrix->yx; *xy = matrix->xy; *yy = matrix->yy; if (x0) *x0 = matrix->x0; if (y0) *y0 = matrix->y0; } /** * cairo_matrix_init_translate: * @matrix: a cairo_matrix_t * @tx: amount to translate in the X direction * @ty: amount to translate in the Y direction * * Initializes @matrix to a transformation that translates by @tx and * @ty in the X and Y dimensions, respectively. **/ void cairo_matrix_init_translate (cairo_matrix_t *matrix, double tx, double ty) { cairo_matrix_init (matrix, 1, 0, 0, 1, tx, ty); } slim_hidden_def(cairo_matrix_init_translate); /** * cairo_matrix_translate: * @matrix: a cairo_matrix_t * @tx: amount to translate in the X direction * @ty: amount to translate in the Y direction * * Applies a translation by @tx, @ty to the transformation in * @matrix. The effect of the new transformation is to first translate * the coordinates by @tx and @ty, then apply the original transformation * to the coordinates. **/ void cairo_matrix_translate (cairo_matrix_t *matrix, double tx, double ty) { cairo_matrix_t tmp; cairo_matrix_init_translate (&tmp, tx, ty); cairo_matrix_multiply (matrix, &tmp, matrix); } /** * cairo_matrix_init_scale: * @matrix: a cairo_matrix_t * @sx: scale factor in the X direction * @sy: scale factor in the Y direction * * Initializes @matrix to a transformation that scales by @sx and @sy * in the X and Y dimensions, respectively. **/ void cairo_matrix_init_scale (cairo_matrix_t *matrix, double sx, double sy) { cairo_matrix_init (matrix, sx, 0, 0, sy, 0, 0); } slim_hidden_def(cairo_matrix_init_scale); /** * cairo_matrix_scale: * @matrix: a #cairo_matrix_t * @sx: scale factor in the X direction * @sy: scale factor in the Y direction * * Applies scaling by @tx, @ty to the transformation in @matrix. The * effect of the new transformation is to first scale the coordinates * by @sx and @sy, then apply the original transformation to the coordinates. **/ void cairo_matrix_scale (cairo_matrix_t *matrix, double sx, double sy) { cairo_matrix_t tmp; cairo_matrix_init_scale (&tmp, sx, sy); cairo_matrix_multiply (matrix, &tmp, matrix); } slim_hidden_def(cairo_matrix_scale); /** * cairo_matrix_init_rotate: * @matrix: a cairo_matrix_t * @radians: angle of rotation, in radians. The direction of rotation * is defined such that positive angles rotate in the direction from * the positive X axis toward the positive Y axis. With the default * axis orientation of cairo, positive angles rotate in a clockwise * direction. * * Initialized @matrix to a transformation that rotates by @radians. **/ void cairo_matrix_init_rotate (cairo_matrix_t *matrix, double radians) { double s; double c; s = sin (radians); c = cos (radians); cairo_matrix_init (matrix, c, s, -s, c, 0, 0); } slim_hidden_def(cairo_matrix_init_rotate); /** * cairo_matrix_rotate: * @matrix: a @cairo_matrix_t * @radians: angle of rotation, in radians. The direction of rotation * is defined such that positive angles rotate in the direction from * the positive X axis toward the positive Y axis. With the default * axis orientation of cairo, positive angles rotate in a clockwise * direction. * * Applies rotation by @radians to the transformation in * @matrix. The effect of the new transformation is to first rotate the * coordinates by @radians, then apply the original transformation * to the coordinates. **/ void cairo_matrix_rotate (cairo_matrix_t *matrix, double radians) { cairo_matrix_t tmp; cairo_matrix_init_rotate (&tmp, radians); cairo_matrix_multiply (matrix, &tmp, matrix); } /** * cairo_matrix_multiply: * @result: a @cairo_matrix_t in which to store the result * @a: a @cairo_matrix_t * @b: a @cairo_matrix_t * * Multiplies the affine transformations in @a and @b together * and stores the result in @result. The effect of the resulting * transformation is to first apply the transformation in @a to the * coordinates and then apply the transformation in @b to the * coordinates. * * It is allowable for @result to be identical to either @a or @b. **/ /* * XXX: The ordering of the arguments to this function corresponds * to [row_vector]*A*B. If we want to use column vectors instead, * then we need to switch the two arguments and fix up all * uses. */ void cairo_matrix_multiply (cairo_matrix_t *result, const cairo_matrix_t *a, const cairo_matrix_t *b) { cairo_matrix_t r; r.xx = a->xx * b->xx + a->yx * b->xy; r.yx = a->xx * b->yx + a->yx * b->yy; r.xy = a->xy * b->xx + a->yy * b->xy; r.yy = a->xy * b->yx + a->yy * b->yy; r.x0 = a->x0 * b->xx + a->y0 * b->xy + b->x0; r.y0 = a->x0 * b->yx + a->y0 * b->yy + b->y0; *result = r; } slim_hidden_def(cairo_matrix_multiply); /** * cairo_matrix_transform_distance: * @matrix: a @cairo_matrix_t * @dx: X component of a distance vector. An in/out parameter * @dy: Y component of a distance vector. An in/out parameter * * Transforms the distance vector (@dx,@dy) by @matrix. This is * similar to cairo_matrix_transform() except that the translation * components of the transformation are ignored. The calculation of * the returned vector is as follows: * * <programlisting> * dx2 = dx1 * a + dy1 * c; * dy2 = dx1 * b + dy1 * d; * </programlisting> * * Affine transformations are position invariant, so the same vector * always transforms to the same vector. If (@x1,@y1) transforms * to (@x2,@y2) then (@x1+@dx1,@y1+@dy1) will transform to * (@x1+@dx2,@y1+@dy2) for all values of @x1 and @x2. **/ void cairo_matrix_transform_distance (const cairo_matrix_t *matrix, double *dx, double *dy) { double new_x, new_y; new_x = (matrix->xx * *dx + matrix->xy * *dy); new_y = (matrix->yx * *dx + matrix->yy * *dy); *dx = new_x; *dy = new_y; } slim_hidden_def(cairo_matrix_transform_distance); /** * cairo_matrix_transform_point: * @matrix: a @cairo_matrix_t * @x: X position. An in/out parameter * @y: Y position. An in/out parameter * * Transforms the point (@x, @y) by @matrix. **/ void cairo_matrix_transform_point (const cairo_matrix_t *matrix, double *x, double *y) { cairo_matrix_transform_distance (matrix, x, y); *x += matrix->x0; *y += matrix->y0; } slim_hidden_def(cairo_matrix_transform_point); void _cairo_matrix_transform_bounding_box (const cairo_matrix_t *matrix, double *x, double *y, double *width, double *height) { int i; double quad_x[4], quad_y[4]; double dx1, dy1; double dx2, dy2; double min_x, max_x; double min_y, max_y; quad_x[0] = *x; quad_y[0] = *y; cairo_matrix_transform_point (matrix, &quad_x[0], &quad_y[0]); dx1 = *width; dy1 = 0; cairo_matrix_transform_distance (matrix, &dx1, &dy1); quad_x[1] = quad_x[0] + dx1; quad_y[1] = quad_y[0] + dy1; dx2 = 0; dy2 = *height; cairo_matrix_transform_distance (matrix, &dx2, &dy2); quad_x[2] = quad_x[0] + dx2; quad_y[2] = quad_y[0] + dy2; quad_x[3] = quad_x[0] + dx1 + dx2; quad_y[3] = quad_y[0] + dy1 + dy2; min_x = max_x = quad_x[0]; min_y = max_y = quad_y[0]; for (i=1; i < 4; i++) { if (quad_x[i] < min_x) min_x = quad_x[i]; if (quad_x[i] > max_x) max_x = quad_x[i]; if (quad_y[i] < min_y) min_y = quad_y[i]; if (quad_y[i] > max_y) max_y = quad_y[i]; } *x = min_x; *y = min_y; *width = max_x - min_x; *height = max_y - min_y; } static void _cairo_matrix_scalar_multiply (cairo_matrix_t *matrix, double scalar) { matrix->xx *= scalar; matrix->yx *= scalar; matrix->xy *= scalar; matrix->yy *= scalar; matrix->x0 *= scalar; matrix->y0 *= scalar; } /* This function isn't a correct adjoint in that the implicit 1 in the homogeneous result should actually be ad-bc instead. But, since this adjoint is only used in the computation of the inverse, which divides by det (A)=ad-bc anyway, everything works out in the end. */ static void _cairo_matrix_compute_adjoint (cairo_matrix_t *matrix) { /* adj (A) = transpose (C:cofactor (A,i,j)) */ double a, b, c, d, tx, ty; _cairo_matrix_get_affine (matrix, &a, &b, &c, &d, &tx, &ty); cairo_matrix_init (matrix, d, -b, -c, a, c*ty - d*tx, b*tx - a*ty); } /** * cairo_matrix_invert: * @matrix: a @cairo_matrix_t * * Changes @matrix to be the inverse of it's original value. Not * all transformation matrices have inverses; if the matrix * collapses points together (it is <firstterm>degenerate</firstterm>), * then it has no inverse and this function will fail. * * Returns: If @matrix has an inverse, modifies @matrix to * be the inverse matrix and returns %CAIRO_STATUS_SUCCESS. Otherwise, * returns %CAIRO_STATUS_INVALID_MATRIX. **/ cairo_status_t cairo_matrix_invert (cairo_matrix_t *matrix) { /* inv (A) = 1/det (A) * adj (A) */ double det; _cairo_matrix_compute_determinant (matrix, &det); if (det == 0) return CAIRO_STATUS_INVALID_MATRIX; _cairo_matrix_compute_adjoint (matrix); _cairo_matrix_scalar_multiply (matrix, 1 / det); return CAIRO_STATUS_SUCCESS; } slim_hidden_def(cairo_matrix_invert); void _cairo_matrix_compute_determinant (const cairo_matrix_t *matrix, double *det) { double a, b, c, d; a = matrix->xx; b = matrix->yx; c = matrix->xy; d = matrix->yy; *det = a*d - b*c; } /* Compute the amount that each basis vector is scaled by. */ cairo_status_t _cairo_matrix_compute_scale_factors (const cairo_matrix_t *matrix, double *sx, double *sy, int x_major) { double det; _cairo_matrix_compute_determinant (matrix, &det); if (det == 0) { *sx = *sy = 0; } else { double x = x_major != 0; double y = x == 0; double major, minor; cairo_matrix_transform_distance (matrix, &x, &y); major = sqrt(x*x + y*y); /* * ignore mirroring */ if (det < 0) det = -det; if (major) minor = det / major; else minor = 0.0; if (x_major) { *sx = major; *sy = minor; } else { *sx = minor; *sy = major; } } return CAIRO_STATUS_SUCCESS; } cairo_bool_t _cairo_matrix_is_integer_translation(const cairo_matrix_t *m, int *itx, int *ity) { cairo_bool_t is_integer_translation; cairo_fixed_t x0_fixed, y0_fixed; x0_fixed = _cairo_fixed_from_double (m->x0); y0_fixed = _cairo_fixed_from_double (m->y0); is_integer_translation = ((m->xx == 1.0) && (m->yx == 0.0) && (m->xy == 0.0) && (m->yy == 1.0) && (_cairo_fixed_is_integer(x0_fixed)) && (_cairo_fixed_is_integer(y0_fixed))); if (! is_integer_translation) return FALSE; if (itx) *itx = _cairo_fixed_integer_part(x0_fixed); if (ity) *ity = _cairo_fixed_integer_part(y0_fixed); return TRUE; } /* A circle in user space is transformed into an ellipse in device space. The following is a derivation of a formula to calculate the length of the major axis for this ellipse; this is useful for error bounds calculations. Thanks to Walter Brisken <wbrisken@aoc.nrao.edu> for this derivation: 1. First some notation: All capital letters represent vectors in two dimensions. A prime ' represents a transformed coordinate. Matrices are written in underlined form, ie _R_. Lowercase letters represent scalar real values. 2. The question has been posed: What is the maximum expansion factor achieved by the linear transformation X' = X _R_ where _R_ is a real-valued 2x2 matrix with entries: _R_ = [a b] [c d] . In other words, what is the maximum radius, MAX[ |X'| ], reached for any X on the unit circle ( |X| = 1 ) ? 3. Some useful formulae (A) through (C) below are standard double-angle formulae. (D) is a lesser known result and is derived below: (A) sin²(θ) = (1 - cos(2*θ))/2 (B) cos²(θ) = (1 + cos(2*θ))/2 (C) sin(θ)*cos(θ) = sin(2*θ)/2 (D) MAX[a*cos(θ) + b*sin(θ)] = sqrt(a² + b²) Proof of (D): find the maximum of the function by setting the derivative to zero: -a*sin(θ)+b*cos(θ) = 0 From this it follows that tan(θ) = b/a and hence sin(θ) = b/sqrt(a² + b²) and cos(θ) = a/sqrt(a² + b²) Thus the maximum value is MAX[a*cos(θ) + b*sin(θ)] = (a² + b²)/sqrt(a² + b²) = sqrt(a² + b²) 4. Derivation of maximum expansion To find MAX[ |X'| ] we search brute force method using calculus. The unit circle on which X is constrained is to be parameterized by t: X(θ) = (cos(θ), sin(θ)) Thus X'(θ) = X(θ) * _R_ = (cos(θ), sin(θ)) * [a b] [c d] = (a*cos(θ) + c*sin(θ), b*cos(θ) + d*sin(θ)). Define r(θ) = |X'(θ)| Thus r²(θ) = (a*cos(θ) + c*sin(θ))² + (b*cos(θ) + d*sin(θ))² = (a² + b²)*cos²(θ) + (c² + d²)*sin²(θ) + 2*(a*c + b*d)*cos(θ)*sin(θ) Now apply the double angle formulae (A) to (C) from above: r²(θ) = (a² + b² + c² + d²)/2 + (a² + b² - c² - d²)*cos(2*θ)/2 + (a*c + b*d)*sin(2*θ) = f + g*cos(φ) + h*sin(φ) Where f = (a² + b² + c² + d²)/2 g = (a² + b² - c² - d²)/2 h = (a*c + d*d) φ = 2*θ It is clear that MAX[ |X'| ] = sqrt(MAX[ r² ]). Here we determine MAX[ r² ] using (D) from above: MAX[ r² ] = f + sqrt(g² + h²) And finally MAX[ |X'| ] = sqrt( f + sqrt(g² + h²) ) Which is the solution to this problem. Walter Brisken 2004/10/08 (Note that the minor axis length is at the minimum of the above solution, which is just sqrt ( f - sqrt(g² + h²) ) given the symmetry of (D)). */ /* determine the length of the major axis of a circle of the given radius after applying the transformation matrix. */ double _cairo_matrix_transformed_circle_major_axis (cairo_matrix_t *matrix, double radius) { double a, b, c, d, f, g, h, i, j; _cairo_matrix_get_affine (matrix, &a, &b, &c, &d, NULL, NULL); i = a*a + b*b; j = c*c + d*d; f = 0.5 * (i + j); g = 0.5 * (i - j); h = a*c + b*d; return radius * sqrt (f + sqrt (g*g+h*h)); /* * we don't need the minor axis length, which is * double min = radius * sqrt (f - sqrt (g*g+h*h)); */ } void _cairo_matrix_to_pixman_matrix (const cairo_matrix_t *matrix, pixman_transform_t *pixman_transform) { pixman_transform->matrix[0][0] = _cairo_fixed_from_double (matrix->xx); pixman_transform->matrix[0][1] = _cairo_fixed_from_double (matrix->xy); pixman_transform->matrix[0][2] = _cairo_fixed_from_double (matrix->x0); pixman_transform->matrix[1][0] = _cairo_fixed_from_double (matrix->yx); pixman_transform->matrix[1][1] = _cairo_fixed_from_double (matrix->yy); pixman_transform->matrix[1][2] = _cairo_fixed_from_double (matrix->y0); pixman_transform->matrix[2][0] = 0; pixman_transform->matrix[2][1] = 0; pixman_transform->matrix[2][2] = _cairo_fixed_from_double (1); }