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/* 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., 51 Franklin Street, Suite 500, Boston, MA 02110-1335, 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
*
* 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>
*/
#include "cairoint.h"
#include "cairo-error-private.h"
#include <float.h>
#define PIXMAN_MAX_INT ((pixman_fixed_1 >> 1) - pixman_fixed_e) /* need to ensure deltas also fit */
/**
* SECTION:cairo-matrix
* @Title: cairo_matrix_t
* @Short_Description: Generic matrix operations
* @See_Also: #cairo_t
*
* #cairo_matrix_t is used throughout cairo to convert between different
* coordinate spaces. A #cairo_matrix_t holds an affine transformation,
* such as a scale, rotation, shear, or a combination of these.
* The transformation of a point (<literal>x</literal>,<literal>y</literal>)
* is given by:
*
* <programlisting>
* x_new = xx * x + xy * y + x0;
* y_new = yx * x + yy * y + y0;
* </programlisting>
*
* The current transformation matrix of a #cairo_t, represented as a
* #cairo_matrix_t, defines the transformation from user-space
* coordinates to device-space coordinates. See cairo_get_matrix() and
* cairo_set_matrix().
**/
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.
*
* Since: 1.0
**/
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>
*
* Since: 1.0
**/
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 transformation 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.
*
* Since: 1.0
**/
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.
*
* Since: 1.0
**/
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);
}
slim_hidden_def (cairo_matrix_translate);
/**
* 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.
*
* Since: 1.0
**/
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 @sx, @sy 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.
*
* Since: 1.0
**/
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.
*
* Since: 1.0
**/
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.
*
* Since: 1.0
**/
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.
*
* Since: 1.0
**/
/*
* 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);
void
_cairo_matrix_multiply (cairo_matrix_t *r,
const cairo_matrix_t *a,
const cairo_matrix_t *b)
{
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;
}
/**
* 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_point() 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.
*
* Since: 1.0
**/
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.
*
* Since: 1.0
**/
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 *x1, double *y1,
double *x2, double *y2,
cairo_bool_t *is_tight)
{
int i;
double quad_x[4], quad_y[4];
double min_x, max_x;
double min_y, max_y;
if (matrix->xy == 0. && matrix->yx == 0.) {
/* non-rotation/skew matrix, just map the two extreme points */
if (matrix->xx != 1.) {
quad_x[0] = *x1 * matrix->xx;
quad_x[1] = *x2 * matrix->xx;
if (quad_x[0] < quad_x[1]) {
*x1 = quad_x[0];
*x2 = quad_x[1];
} else {
*x1 = quad_x[1];
*x2 = quad_x[0];
}
}
if (matrix->x0 != 0.) {
*x1 += matrix->x0;
*x2 += matrix->x0;
}
if (matrix->yy != 1.) {
quad_y[0] = *y1 * matrix->yy;
quad_y[1] = *y2 * matrix->yy;
if (quad_y[0] < quad_y[1]) {
*y1 = quad_y[0];
*y2 = quad_y[1];
} else {
*y1 = quad_y[1];
*y2 = quad_y[0];
}
}
if (matrix->y0 != 0.) {
*y1 += matrix->y0;
*y2 += matrix->y0;
}
if (is_tight)
*is_tight = TRUE;
return;
}
/* general matrix */
quad_x[0] = *x1;
quad_y[0] = *y1;
cairo_matrix_transform_point (matrix, &quad_x[0], &quad_y[0]);
quad_x[1] = *x2;
quad_y[1] = *y1;
cairo_matrix_transform_point (matrix, &quad_x[1], &quad_y[1]);
quad_x[2] = *x1;
quad_y[2] = *y2;
cairo_matrix_transform_point (matrix, &quad_x[2], &quad_y[2]);
quad_x[3] = *x2;
quad_y[3] = *y2;
cairo_matrix_transform_point (matrix, &quad_x[3], &quad_y[3]);
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];
}
*x1 = min_x;
*y1 = min_y;
*x2 = max_x;
*y2 = max_y;
if (is_tight) {
/* it's tight if and only if the four corner points form an axis-aligned
rectangle.
And that's true if and only if we can derive corners 0 and 3 from
corners 1 and 2 in one of two straightforward ways...
We could use a tolerance here but for now we'll fall back to FALSE in the case
of floating point error.
*/
*is_tight =
(quad_x[1] == quad_x[0] && quad_y[1] == quad_y[3] &&
quad_x[2] == quad_x[3] && quad_y[2] == quad_y[0]) ||
(quad_x[1] == quad_x[3] && quad_y[1] == quad_y[0] &&
quad_x[2] == quad_x[0] && quad_y[2] == quad_y[3]);
}
}
cairo_private void
_cairo_matrix_transform_bounding_box_fixed (const cairo_matrix_t *matrix,
cairo_box_t *bbox,
cairo_bool_t *is_tight)
{
double x1, y1, x2, y2;
_cairo_box_to_doubles (bbox, &x1, &y1, &x2, &y2);
_cairo_matrix_transform_bounding_box (matrix, &x1, &y1, &x2, &y2, is_tight);
_cairo_box_from_doubles (bbox, &x1, &y1, &x2, &y2);
}
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 its 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.
*
* Since: 1.0
**/
cairo_status_t
cairo_matrix_invert (cairo_matrix_t *matrix)
{
double det;
/* Simple scaling|translation matrices are quite common... */
if (matrix->xy == 0. && matrix->yx == 0.) {
matrix->x0 = -matrix->x0;
matrix->y0 = -matrix->y0;
if (matrix->xx != 1.) {
if (matrix->xx == 0.)
return _cairo_error (CAIRO_STATUS_INVALID_MATRIX);
matrix->xx = 1. / matrix->xx;
matrix->x0 *= matrix->xx;
}
if (matrix->yy != 1.) {
if (matrix->yy == 0.)
return _cairo_error (CAIRO_STATUS_INVALID_MATRIX);
matrix->yy = 1. / matrix->yy;
matrix->y0 *= matrix->yy;
}
return CAIRO_STATUS_SUCCESS;
}
/* inv (A) = 1/det (A) * adj (A) */
det = _cairo_matrix_compute_determinant (matrix);
if (! ISFINITE (det))
return _cairo_error (CAIRO_STATUS_INVALID_MATRIX);
if (det == 0)
return _cairo_error (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);
cairo_bool_t
_cairo_matrix_is_invertible (const cairo_matrix_t *matrix)
{
double det;
det = _cairo_matrix_compute_determinant (matrix);
return ISFINITE (det) && det != 0.;
}
cairo_bool_t
_cairo_matrix_is_scale_0 (const cairo_matrix_t *matrix)
{
return matrix->xx == 0. &&
matrix->xy == 0. &&
matrix->yx == 0. &&
matrix->yy == 0.;
}
double
_cairo_matrix_compute_determinant (const cairo_matrix_t *matrix)
{
double a, b, c, d;
a = matrix->xx; b = matrix->yx;
c = matrix->xy; d = matrix->yy;
return a*d - b*c;
}
/**
* _cairo_matrix_compute_basis_scale_factors:
* @matrix: a matrix
* @basis_scale: the scale factor in the direction of basis
* @normal_scale: the scale factor in the direction normal to the basis
* @x_basis: basis to use. X basis if true, Y basis otherwise.
*
* Computes |Mv| and det(M)/|Mv| for v=[1,0] if x_basis is true, and v=[0,1]
* otherwise, and M is @matrix.
*
* Return value: the scale factor of @matrix on the height of the font,
* or 1.0 if @matrix is %NULL.
**/
cairo_status_t
_cairo_matrix_compute_basis_scale_factors (const cairo_matrix_t *matrix,
double *basis_scale, double *normal_scale,
cairo_bool_t x_basis)
{
double det;
det = _cairo_matrix_compute_determinant (matrix);
if (! ISFINITE (det))
return _cairo_error (CAIRO_STATUS_INVALID_MATRIX);
if (det == 0)
{
*basis_scale = *normal_scale = 0;
}
else
{
double x = x_basis != 0;
double y = x == 0;
double major, minor;
cairo_matrix_transform_distance (matrix, &x, &y);
major = hypot (x, y);
/*
* ignore mirroring
*/
if (det < 0)
det = -det;
if (major)
minor = det / major;
else
minor = 0.0;
if (x_basis)
{
*basis_scale = major;
*normal_scale = minor;
}
else
{
*basis_scale = minor;
*normal_scale = major;
}
}
return CAIRO_STATUS_SUCCESS;
}
cairo_bool_t
_cairo_matrix_is_integer_translation (const cairo_matrix_t *matrix,
int *itx, int *ity)
{
if (_cairo_matrix_is_translation (matrix))
{
cairo_fixed_t x0_fixed = _cairo_fixed_from_double (matrix->x0);
cairo_fixed_t y0_fixed = _cairo_fixed_from_double (matrix->y0);
if (_cairo_fixed_is_integer (x0_fixed) &&
_cairo_fixed_is_integer (y0_fixed))
{
if (itx)
*itx = _cairo_fixed_integer_part (x0_fixed);
if (ity)
*ity = _cairo_fixed_integer_part (y0_fixed);
return TRUE;
}
}
return FALSE;
}
#define SCALING_EPSILON _cairo_fixed_to_double(1)
/* This only returns true if the matrix is 90 degree rotations or
* flips. It appears calling code is relying on this. It will return
* false for other rotations even if the scale is one. Approximations
* are allowed to handle matricies filled in using trig functions
* such as sin(M_PI_2).
*/
cairo_bool_t
_cairo_matrix_has_unity_scale (const cairo_matrix_t *matrix)
{
/* check that the determinant is near +/-1 */
double det = _cairo_matrix_compute_determinant (matrix);
if (fabs (det * det - 1.0) < SCALING_EPSILON) {
/* check that one axis is close to zero */
if (fabs (matrix->xy) < SCALING_EPSILON &&
fabs (matrix->yx) < SCALING_EPSILON)
return TRUE;
if (fabs (matrix->xx) < SCALING_EPSILON &&
fabs (matrix->yy) < SCALING_EPSILON)
return TRUE;
/* If rotations are allowed then it must instead test for
* orthogonality. This is xx*xy+yx*yy ~= 0.
*/
}
return FALSE;
}
/* By pixel exact here, we mean a matrix that is composed only of
* 90 degree rotations, flips, and integer translations and produces a 1:1
* mapping between source and destination pixels. If we transform an image
* with a pixel-exact matrix, filtering is not useful.
*/
cairo_bool_t
_cairo_matrix_is_pixel_exact (const cairo_matrix_t *matrix)
{
cairo_fixed_t x0_fixed, y0_fixed;
if (! _cairo_matrix_has_unity_scale (matrix))
return FALSE;
x0_fixed = _cairo_fixed_from_double (matrix->x0);
y0_fixed = _cairo_fixed_from_double (matrix->y0);
return _cairo_fixed_is_integer (x0_fixed) && _cairo_fixed_is_integer (y0_fixed);
}
/*
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)).
For another derivation of the same result, using Singular Value Decomposition,
see doc/tutorial/src/singular.c.
*/
/* 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 (const cairo_matrix_t *matrix,
double radius)
{
double a, b, c, d, f, g, h, i, j;
if (_cairo_matrix_has_unity_scale (matrix))
return radius;
_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 + hypot (g, h));
/*
* we don't need the minor axis length, which is
* double min = radius * sqrt (f - sqrt (g*g+h*h));
*/
}
static const pixman_transform_t pixman_identity_transform = {{
{1 << 16, 0, 0},
{ 0, 1 << 16, 0},
{ 0, 0, 1 << 16}
}};
static cairo_status_t
_cairo_matrix_to_pixman_matrix (const cairo_matrix_t *matrix,
pixman_transform_t *pixman_transform,
double xc,
double yc)
{
cairo_matrix_t inv;
unsigned max_iterations;
pixman_transform->matrix[0][0] = _cairo_fixed_16_16_from_double (matrix->xx);
pixman_transform->matrix[0][1] = _cairo_fixed_16_16_from_double (matrix->xy);
pixman_transform->matrix[0][2] = _cairo_fixed_16_16_from_double (matrix->x0);
pixman_transform->matrix[1][0] = _cairo_fixed_16_16_from_double (matrix->yx);
pixman_transform->matrix[1][1] = _cairo_fixed_16_16_from_double (matrix->yy);
pixman_transform->matrix[1][2] = _cairo_fixed_16_16_from_double (matrix->y0);
pixman_transform->matrix[2][0] = 0;
pixman_transform->matrix[2][1] = 0;
pixman_transform->matrix[2][2] = 1 << 16;
/* The conversion above breaks cairo's translation invariance:
* a translation of (a, b) in device space translates to
* a translation of (xx * a + xy * b, yx * a + yy * b)
* for cairo, while pixman uses rounded versions of xx ... yy.
* This error increases as a and b get larger.
*
* To compensate for this, we fix the point (xc, yc) in pattern
* space and adjust pixman's transform to agree with cairo's at
* that point.
*/
if (_cairo_matrix_has_unity_scale (matrix))
return CAIRO_STATUS_SUCCESS;
if (unlikely (fabs (matrix->xx) > PIXMAN_MAX_INT ||
fabs (matrix->xy) > PIXMAN_MAX_INT ||
fabs (matrix->x0) > PIXMAN_MAX_INT ||
fabs (matrix->yx) > PIXMAN_MAX_INT ||
fabs (matrix->yy) > PIXMAN_MAX_INT ||
fabs (matrix->y0) > PIXMAN_MAX_INT))
{
return _cairo_error (CAIRO_STATUS_INVALID_MATRIX);
}
/* Note: If we can't invert the transformation, skip the adjustment. */
inv = *matrix;
if (cairo_matrix_invert (&inv) != CAIRO_STATUS_SUCCESS)
return CAIRO_STATUS_SUCCESS;
/* find the pattern space coordinate that maps to (xc, yc) */
max_iterations = 5;
do {
double x,y;
pixman_vector_t vector;
cairo_fixed_16_16_t dx, dy;
vector.vector[0] = _cairo_fixed_16_16_from_double (xc);
vector.vector[1] = _cairo_fixed_16_16_from_double (yc);
vector.vector[2] = 1 << 16;
/* If we can't transform the reference point, skip the adjustment. */
if (! pixman_transform_point_3d (pixman_transform, &vector))
return CAIRO_STATUS_SUCCESS;
x = pixman_fixed_to_double (vector.vector[0]);
y = pixman_fixed_to_double (vector.vector[1]);
cairo_matrix_transform_point (&inv, &x, &y);
/* Ideally, the vector should now be (xc, yc).
* We can now compensate for the resulting error.
*/
x -= xc;
y -= yc;
cairo_matrix_transform_distance (matrix, &x, &y);
dx = _cairo_fixed_16_16_from_double (x);
dy = _cairo_fixed_16_16_from_double (y);
pixman_transform->matrix[0][2] -= dx;
pixman_transform->matrix[1][2] -= dy;
if (dx == 0 && dy == 0)
return CAIRO_STATUS_SUCCESS;
} while (--max_iterations);
/* We didn't find an exact match between cairo and pixman, but
* the matrix should be mostly correct */
return CAIRO_STATUS_SUCCESS;
}
static inline double
_pixman_nearest_sample (double d)
{
return ceil (d - .5);
}
/**
* _cairo_matrix_is_pixman_translation:
* @matrix: a matrix
* @filter: the filter to be used on the pattern transformed by @matrix
* @x_offset: the translation in the X direction
* @y_offset: the translation in the Y direction
*
* Checks if @matrix translated by (x_offset, y_offset) can be
* represented using just an offset (within the range pixman can
* accept) and an identity matrix.
*
* Passing a non-zero value in x_offset/y_offset has the same effect
* as applying cairo_matrix_translate(matrix, x_offset, y_offset) and
* setting x_offset and y_offset to 0.
*
* Upon return x_offset and y_offset contain the translation vector if
* the return value is %TRUE. If the return value is %FALSE, they will
* not be modified.
*
* Return value: %TRUE if @matrix can be represented as a pixman
* translation, %FALSE otherwise.
**/
cairo_bool_t
_cairo_matrix_is_pixman_translation (const cairo_matrix_t *matrix,
cairo_filter_t filter,
int *x_offset,
int *y_offset)
{
double tx, ty;
if (!_cairo_matrix_is_translation (matrix))
return FALSE;
if (matrix->x0 == 0. && matrix->y0 == 0.)
return TRUE;
tx = matrix->x0 + *x_offset;
ty = matrix->y0 + *y_offset;
if (filter == CAIRO_FILTER_FAST || filter == CAIRO_FILTER_NEAREST) {
tx = _pixman_nearest_sample (tx);
ty = _pixman_nearest_sample (ty);
} else if (tx != floor (tx) || ty != floor (ty)) {
return FALSE;
}
if (fabs (tx) > PIXMAN_MAX_INT || fabs (ty) > PIXMAN_MAX_INT)
return FALSE;
*x_offset = _cairo_lround (tx);
*y_offset = _cairo_lround (ty);
return TRUE;
}
/**
* _cairo_matrix_to_pixman_matrix_offset:
* @matrix: a matrix
* @filter: the filter to be used on the pattern transformed by @matrix
* @xc: the X coordinate of the point to fix in pattern space
* @yc: the Y coordinate of the point to fix in pattern space
* @out_transform: the transformation which best approximates @matrix
* @x_offset: the translation in the X direction
* @y_offset: the translation in the Y direction
*
* This function tries to represent @matrix translated by (x_offset,
* y_offset) as a %pixman_transform_t and an translation.
*
* Passing a non-zero value in x_offset/y_offset has the same effect
* as applying cairo_matrix_translate(matrix, x_offset, y_offset) and
* setting x_offset and y_offset to 0.
*
* If it is possible to represent the matrix with an identity
* %pixman_transform_t and a translation within the valid range for
* pixman, this function will set @out_transform to be the identity,
* @x_offset and @y_offset to be the translation vector and will
* return %CAIRO_INT_STATUS_NOTHING_TO_DO. Otherwise it will try to
* evenly divide the translational component of @matrix between
* @out_transform and (@x_offset, @y_offset).
*
* Upon return x_offset and y_offset contain the translation vector.
*
* Return value: %CAIRO_INT_STATUS_NOTHING_TO_DO if the out_transform
* is the identity, %CAIRO_STATUS_INVALID_MATRIX if it was not
* possible to represent @matrix as a pixman_transform_t without
* overflows, %CAIRO_STATUS_SUCCESS otherwise.
**/
cairo_status_t
_cairo_matrix_to_pixman_matrix_offset (const cairo_matrix_t *matrix,
cairo_filter_t filter,
double xc,
double yc,
pixman_transform_t *out_transform,
int *x_offset,
int *y_offset)
{
cairo_bool_t is_pixman_translation;
is_pixman_translation = _cairo_matrix_is_pixman_translation (matrix,
filter,
x_offset,
y_offset);
if (is_pixman_translation) {
*out_transform = pixman_identity_transform;
return CAIRO_INT_STATUS_NOTHING_TO_DO;
} else {
cairo_matrix_t m;
m = *matrix;
cairo_matrix_translate (&m, *x_offset, *y_offset);
if (m.x0 != 0.0 || m.y0 != 0.0) {
double tx, ty, norm;
int i, j;
/* pixman also limits the [xy]_offset to 16 bits so evenly
* spread the bits between the two.
*
* To do this, find the solutions of:
* |x| = |x*m.xx + y*m.xy + m.x0|
* |y| = |x*m.yx + y*m.yy + m.y0|
*
* and select the one whose maximum norm is smallest.
*/
tx = m.x0;
ty = m.y0;
norm = MAX (fabs (tx), fabs (ty));
for (i = -1; i < 2; i+=2) {
for (j = -1; j < 2; j+=2) {
double x, y, den, new_norm;
den = (m.xx + i) * (m.yy + j) - m.xy * m.yx;
if (fabs (den) < DBL_EPSILON)
continue;
x = m.y0 * m.xy - m.x0 * (m.yy + j);
y = m.x0 * m.yx - m.y0 * (m.xx + i);
den = 1 / den;
x *= den;
y *= den;
new_norm = MAX (fabs (x), fabs (y));
if (norm > new_norm) {
norm = new_norm;
tx = x;
ty = y;
}
}
}
tx = floor (tx);
ty = floor (ty);
*x_offset = -tx;
*y_offset = -ty;
cairo_matrix_translate (&m, tx, ty);
} else {
*x_offset = 0;
*y_offset = 0;
}
return _cairo_matrix_to_pixman_matrix (&m, out_transform, xc, yc);
}
}