/* Bezier interpolation for inkscape drawing code.
*
* Original code published in:
* An Algorithm for Automatically Fitting Digitized Curves
* by Philip J. Schneider
* "Graphics Gems", Academic Press, 1990
*
* Authors:
* Philip J. Schneider
* Lauris Kaplinski <lauris@kaplinski.com>
* Peter Moulder <pmoulder@mail.csse.monash.edu.au>
*
* Copyright (C) 1990 Philip J. Schneider
* Copyright (C) 2001 Lauris Kaplinski
* Copyright (C) 2001 Ximian, Inc.
* Copyright (C) 2003,2004 Monash University
*
* 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.
*
*/
#define SP_HUGE 1e5
#define noBEZIER_DEBUG
#ifdef HAVE_IEEEFP_H
# include <ieeefp.h>
#endif
#include <2geom/bezier-utils.h>
#include <2geom/math-utils.h>
#include <assert.h>
namespace Geom {
/* Forward declarations */
static void generate_bezier(Point b[], Point const d[], double const u[], unsigned len,
Point const &tHat1, Point const &tHat2, double tolerance_sq);
static void estimate_lengths(Point bezier[],
Point const data[], double const u[], unsigned len,
Point const &tHat1, Point const &tHat2);
static void estimate_bi(Point b[4], unsigned ei,
Point const data[], double const u[], unsigned len);
static void reparameterize(Point const d[], unsigned len, double u[], Point const bezCurve[]);
static double NewtonRaphsonRootFind(Point const Q[], Point const &P, double u);
static Point darray_center_tangent(Point const d[], unsigned center, unsigned length);
static Point darray_right_tangent(Point const d[], unsigned const len);
static unsigned copy_without_nans_or_adjacent_duplicates(Point const src[], unsigned src_len, Point dest[]);
static void chord_length_parameterize(Point const d[], double u[], unsigned len);
static double compute_max_error_ratio(Point const d[], double const u[], unsigned len,
Point const bezCurve[], double tolerance,
unsigned *splitPoint);
static double compute_hook(Point const &a, Point const &b, double const u, Point const bezCurve[],
double const tolerance);
static Point const unconstrained_tangent(0, 0);
/*
* B0, B1, B2, B3 : Bezier multipliers
*/
#define B0(u) ( ( 1.0 - u ) * ( 1.0 - u ) * ( 1.0 - u ) )
#define B1(u) ( 3 * u * ( 1.0 - u ) * ( 1.0 - u ) )
#define B2(u) ( 3 * u * u * ( 1.0 - u ) )
#define B3(u) ( u * u * u )
#ifdef BEZIER_DEBUG
# define DOUBLE_ASSERT(x) assert( ( (x) > -SP_HUGE ) && ( (x) < SP_HUGE ) )
# define BEZIER_ASSERT(b) do { \
DOUBLE_ASSERT((b)[0][X]); DOUBLE_ASSERT((b)[0][Y]); \
DOUBLE_ASSERT((b)[1][X]); DOUBLE_ASSERT((b)[1][Y]); \
DOUBLE_ASSERT((b)[2][X]); DOUBLE_ASSERT((b)[2][Y]); \
DOUBLE_ASSERT((b)[3][X]); DOUBLE_ASSERT((b)[3][Y]); \
} while(0)
#else
# define DOUBLE_ASSERT(x) do { } while(0)
# define BEZIER_ASSERT(b) do { } while(0)
#endif
/**
* Fit a single-segment Bezier curve to a set of digitized points.
*
* \return Number of segments generated, or -1 on error.
*/
int
bezier_fit_cubic(Point *bezier, Point const *data, int len, double error)
{
return bezier_fit_cubic_r(bezier, data, len, error, 1);
}
/**
* Fit a multi-segment Bezier curve to a set of digitized points, with
* possible weedout of identical points and NaNs.
*
* \param max_beziers Maximum number of generated segments
* \param Result array, must be large enough for n. segments * 4 elements.
*
* \return Number of segments generated, or -1 on error.
*/
int
bezier_fit_cubic_r(Point bezier[], Point const data[], int const len, double const error, unsigned const max_beziers)
{
if(bezier == NULL ||
data == NULL ||
len <= 0 ||
max_beziers >= (1ul << (31 - 2 - 1 - 3)))
return -1;
Point *uniqued_data = new Point[len];
unsigned uniqued_len = copy_without_nans_or_adjacent_duplicates(data, len, uniqued_data);
if ( uniqued_len < 2 ) {
delete[] uniqued_data;
return 0;
}
/* Call fit-cubic function with recursion. */
int const ret = bezier_fit_cubic_full(bezier, NULL, uniqued_data, uniqued_len,
unconstrained_tangent, unconstrained_tangent,
error, max_beziers);
delete[] uniqued_data;
return ret;
}
/**
* Copy points from src to dest, filter out points containing NaN and
* adjacent points with equal x and y.
* \return length of dest
*/
static unsigned
copy_without_nans_or_adjacent_duplicates(Point const src[], unsigned src_len, Point dest[])
{
unsigned si = 0;
for (;;) {
if ( si == src_len ) {
return 0;
}
if (!IS_NAN(src[si][X]) &&
!IS_NAN(src[si][Y])) {
dest[0] = Point(src[si]);
++si;
break;
}
si++;
}
unsigned di = 0;
for (; si < src_len; ++si) {
Point const src_pt = Point(src[si]);
if ( src_pt != dest[di]
&& !IS_NAN(src_pt[X])
&& !IS_NAN(src_pt[Y])) {
dest[++di] = src_pt;
}
}
unsigned dest_len = di + 1;
assert( dest_len <= src_len );
return dest_len;
}
/**
* Fit a multi-segment Bezier curve to a set of digitized points, without
* possible weedout of identical points and NaNs.
*
* \pre data is uniqued, i.e. not exist i: data[i] == data[i + 1].
* \param max_beziers Maximum number of generated segments
* \param Result array, must be large enough for n. segments * 4 elements.
*/
int
bezier_fit_cubic_full(Point bezier[], int split_points[],
Point const data[], int const len,
Point const &tHat1, Point const &tHat2,
double const error, unsigned const max_beziers)
{
if(!(bezier != NULL) ||
!(data != NULL) ||
!(len > 0) ||
!(max_beziers >= 1) ||
!(error >= 0.0))
return -1;
if ( len < 2 ) return 0;
if ( len == 2 ) {
/* We have 2 points, which can be fitted trivially. */
bezier[0] = data[0];
bezier[3] = data[len - 1];
double const dist = distance(bezier[0], bezier[3]) / 3.0;
if (IS_NAN(dist)) {
/* Numerical problem, fall back to straight line segment. */
bezier[1] = bezier[0];
bezier[2] = bezier[3];
} else {
bezier[1] = ( is_zero(tHat1)
? ( 2 * bezier[0] + bezier[3] ) / 3.
: bezier[0] + dist * tHat1 );
bezier[2] = ( is_zero(tHat2)
? ( bezier[0] + 2 * bezier[3] ) / 3.
: bezier[3] + dist * tHat2 );
}
BEZIER_ASSERT(bezier);
return 1;
}
/* Parameterize points, and attempt to fit curve */
unsigned splitPoint; /* Point to split point set at. */
bool is_corner;
{
double *u = new double[len];
chord_length_parameterize(data, u, len);
if ( u[len - 1] == 0.0 ) {
/* Zero-length path: every point in data[] is the same.
*
* (Clients aren't allowed to pass such data; handling the case is defensive
* programming.)
*/
delete[] u;
return 0;
}
generate_bezier(bezier, data, u, len, tHat1, tHat2, error);
reparameterize(data, len, u, bezier);
/* Find max deviation of points to fitted curve. */
double const tolerance = sqrt(error + 1e-9);
double maxErrorRatio = compute_max_error_ratio(data, u, len, bezier, tolerance, &splitPoint);
if ( fabs(maxErrorRatio) <= 1.0 ) {
BEZIER_ASSERT(bezier);
delete[] u;
return 1;
}
/* If error not too large, then try some reparameterization and iteration. */
if ( 0.0 <= maxErrorRatio && maxErrorRatio <= 3.0 ) {
int const maxIterations = 4; /* std::max times to try iterating */
for (int i = 0; i < maxIterations; i++) {
generate_bezier(bezier, data, u, len, tHat1, tHat2, error);
reparameterize(data, len, u, bezier);
maxErrorRatio = compute_max_error_ratio(data, u, len, bezier, tolerance, &splitPoint);
if ( fabs(maxErrorRatio) <= 1.0 ) {
BEZIER_ASSERT(bezier);
delete[] u;
return 1;
}
}
}
delete[] u;
is_corner = (maxErrorRatio < 0);
}
if (is_corner) {
assert(splitPoint < unsigned(len));
if (splitPoint == 0) {
if (is_zero(tHat1)) {
/* Got spike even with unconstrained initial tangent. */
++splitPoint;
} else {
return bezier_fit_cubic_full(bezier, split_points, data, len, unconstrained_tangent, tHat2,
error, max_beziers);
}
} else if (splitPoint == unsigned(len - 1)) {
if (is_zero(tHat2)) {
/* Got spike even with unconstrained final tangent. */
--splitPoint;
} else {
return bezier_fit_cubic_full(bezier, split_points, data, len, tHat1, unconstrained_tangent,
error, max_beziers);
}
}
}
if ( 1 < max_beziers ) {
/*
* Fitting failed -- split at max error point and fit recursively
*/
unsigned const rec_max_beziers1 = max_beziers - 1;
Point recTHat2, recTHat1;
if (is_corner) {
if(!(0 < splitPoint && splitPoint < unsigned(len - 1)))
return -1;
recTHat1 = recTHat2 = unconstrained_tangent;
} else {
/* Unit tangent vector at splitPoint. */
recTHat2 = darray_center_tangent(data, splitPoint, len);
recTHat1 = -recTHat2;
}
int const nsegs1 = bezier_fit_cubic_full(bezier, split_points, data, splitPoint + 1,
tHat1, recTHat2, error, rec_max_beziers1);
if ( nsegs1 < 0 ) {
#ifdef BEZIER_DEBUG
g_print("fit_cubic[1]: recursive call failed\n");
#endif
return -1;
}
assert( nsegs1 != 0 );
if (split_points != NULL) {
split_points[nsegs1 - 1] = splitPoint;
}
unsigned const rec_max_beziers2 = max_beziers - nsegs1;
int const nsegs2 = bezier_fit_cubic_full(bezier + nsegs1*4,
( split_points == NULL
? NULL
: split_points + nsegs1 ),
data + splitPoint, len - splitPoint,
recTHat1, tHat2, error, rec_max_beziers2);
if ( nsegs2 < 0 ) {
#ifdef BEZIER_DEBUG
g_print("fit_cubic[2]: recursive call failed\n");
#endif
return -1;
}
#ifdef BEZIER_DEBUG
g_print("fit_cubic: success[nsegs: %d+%d=%d] on max_beziers:%u\n",
nsegs1, nsegs2, nsegs1 + nsegs2, max_beziers);
#endif
return nsegs1 + nsegs2;
} else {
return -1;
}
}
/**
* Fill in \a bezier[] based on the given data and tangent requirements, using
* a least-squares fit.
*
* Each of tHat1 and tHat2 should be either a zero vector or a unit vector.
* If it is zero, then bezier[1 or 2] is estimated without constraint; otherwise,
* it bezier[1 or 2] is placed in the specified direction from bezier[0 or 3].
*
* \param tolerance_sq Used only for an initial guess as to tangent directions
* when \a tHat1 or \a tHat2 is zero.
*/
static void
generate_bezier(Point bezier[],
Point const data[], double const u[], unsigned const len,
Point const &tHat1, Point const &tHat2,
double const tolerance_sq)
{
bool const est1 = is_zero(tHat1);
bool const est2 = is_zero(tHat2);
Point est_tHat1( est1
? darray_left_tangent(data, len, tolerance_sq)
: tHat1 );
Point est_tHat2( est2
? darray_right_tangent(data, len, tolerance_sq)
: tHat2 );
estimate_lengths(bezier, data, u, len, est_tHat1, est_tHat2);
/* We find that darray_right_tangent tends to produce better results
for our current freehand tool than full estimation. */
if (est1) {
estimate_bi(bezier, 1, data, u, len);
if (bezier[1] != bezier[0]) {
est_tHat1 = unit_vector(bezier[1] - bezier[0]);
}
estimate_lengths(bezier, data, u, len, est_tHat1, est_tHat2);
}
}
static void
estimate_lengths(Point bezier[],
Point const data[], double const uPrime[], unsigned const len,
Point const &tHat1, Point const &tHat2)
{
double C[2][2]; /* Matrix C. */
double X[2]; /* Matrix X. */
/* Create the C and X matrices. */
C[0][0] = 0.0;
C[0][1] = 0.0;
C[1][0] = 0.0;
C[1][1] = 0.0;
X[0] = 0.0;
X[1] = 0.0;
/* First and last control points of the Bezier curve are positioned exactly at the first and
last data points. */
bezier[0] = data[0];
bezier[3] = data[len - 1];
for (unsigned i = 0; i < len; i++) {
/* Bezier control point coefficients. */
double const b0 = B0(uPrime[i]);
double const b1 = B1(uPrime[i]);
double const b2 = B2(uPrime[i]);
double const b3 = B3(uPrime[i]);
/* rhs for eqn */
Point const a1 = b1 * tHat1;
Point const a2 = b2 * tHat2;
C[0][0] += dot(a1, a1);
C[0][1] += dot(a1, a2);
C[1][0] = C[0][1];
C[1][1] += dot(a2, a2);
/* Additional offset to the data point from the predicted point if we were to set bezier[1]
to bezier[0] and bezier[2] to bezier[3]. */
Point const shortfall
= ( data[i]
- ( ( b0 + b1 ) * bezier[0] )
- ( ( b2 + b3 ) * bezier[3] ) );
X[0] += dot(a1, shortfall);
X[1] += dot(a2, shortfall);
}
/* We've constructed a pair of equations in the form of a matrix product C * alpha = X.
Now solve for alpha. */
double alpha_l, alpha_r;
/* Compute the determinants of C and X. */
double const det_C0_C1 = C[0][0] * C[1][1] - C[1][0] * C[0][1];
if ( det_C0_C1 != 0 ) {
/* Apparently Kramer's rule. */
double const det_C0_X = C[0][0] * X[1] - C[0][1] * X[0];
double const det_X_C1 = X[0] * C[1][1] - X[1] * C[0][1];
alpha_l = det_X_C1 / det_C0_C1;
alpha_r = det_C0_X / det_C0_C1;
} else {
/* The matrix is under-determined. Try requiring alpha_l == alpha_r.
*
* One way of implementing the constraint alpha_l == alpha_r is to treat them as the same
* variable in the equations. We can do this by adding the columns of C to form a single
* column, to be multiplied by alpha to give the column vector X.
*
* We try each row in turn.
*/
double const c0 = C[0][0] + C[0][1];
if (c0 != 0) {
alpha_l = alpha_r = X[0] / c0;
} else {
double const c1 = C[1][0] + C[1][1];
if (c1 != 0) {
alpha_l = alpha_r = X[1] / c1;
} else {
/* Let the below code handle this. */
alpha_l = alpha_r = 0.;
}
}
}
/* If alpha negative, use the Wu/Barsky heuristic (see text). (If alpha is 0, you get
coincident control points that lead to divide by zero in any subsequent
NewtonRaphsonRootFind() call.) */
/// \todo Check whether this special-casing is necessary now that
/// NewtonRaphsonRootFind handles non-positive denominator.
if ( alpha_l < 1.0e-6 ||
alpha_r < 1.0e-6 )
{
alpha_l = alpha_r = distance(data[0], data[len-1]) / 3.0;
}
/* Control points 1 and 2 are positioned an alpha distance out on the tangent vectors, left and
right, respectively. */
bezier[1] = alpha_l * tHat1 + bezier[0];
bezier[2] = alpha_r * tHat2 + bezier[3];
return;
}
static double lensq(Point const p) {
return dot(p, p);
}
static void
estimate_bi(Point bezier[4], unsigned const ei,
Point const data[], double const u[], unsigned const len)
{
if(!(1 <= ei && ei <= 2))
return;
unsigned const oi = 3 - ei;
double num[2] = {0., 0.};
double den = 0.;
for (unsigned i = 0; i < len; ++i) {
double const ui = u[i];
double const b[4] = {
B0(ui),
B1(ui),
B2(ui),
B3(ui)
};
for (unsigned d = 0; d < 2; ++d) {
num[d] += b[ei] * (b[0] * bezier[0][d] +
b[oi] * bezier[oi][d] +
b[3] * bezier[3][d] +
- data[i][d]);
}
den -= b[ei] * b[ei];
}
if (den != 0.) {
for (unsigned d = 0; d < 2; ++d) {
bezier[ei][d] = num[d] / den;
}
} else {
bezier[ei] = ( oi * bezier[0] + ei * bezier[3] ) / 3.;
}
}
/**
* Given set of points and their parameterization, try to find a better assignment of parameter
* values for the points.
*
* \param d Array of digitized points.
* \param u Current parameter values.
* \param bezCurve Current fitted curve.
* \param len Number of values in both d and u arrays.
* Also the size of the array that is allocated for return.
*/
static void
reparameterize(Point const d[],
unsigned const len,
double u[],
Point const bezCurve[])
{
assert( 2 <= len );
unsigned const last = len - 1;
assert( bezCurve[0] == d[0] );
assert( bezCurve[3] == d[last] );
assert( u[0] == 0.0 );
assert( u[last] == 1.0 );
/* Otherwise, consider including 0 and last in the below loop. */
for (unsigned i = 1; i < last; i++) {
u[i] = NewtonRaphsonRootFind(bezCurve, d[i], u[i]);
}
}
/**
* Use Newton-Raphson iteration to find better root.
*
* \param Q Current fitted curve
* \param P Digitized point
* \param u Parameter value for "P"
*
* \return Improved u
*/
static double
NewtonRaphsonRootFind(Point const Q[], Point const &P, double const u)
{
assert( 0.0 <= u );
assert( u <= 1.0 );
/* Generate control vertices for Q'. */
Point Q1[3];
for (unsigned i = 0; i < 3; i++) {
Q1[i] = 3.0 * ( Q[i+1] - Q[i] );
}
/* Generate control vertices for Q''. */
Point Q2[2];
for (unsigned i = 0; i < 2; i++) {
Q2[i] = 2.0 * ( Q1[i+1] - Q1[i] );
}
/* Compute Q(u), Q'(u) and Q''(u). */
Point const Q_u = bezier_pt(3, Q, u);
Point const Q1_u = bezier_pt(2, Q1, u);
Point const Q2_u = bezier_pt(1, Q2, u);
/* Compute f(u)/f'(u), where f is the derivative wrt u of distsq(u) = 0.5 * the square of the
distance from P to Q(u). Here we're using Newton-Raphson to find a stationary point in the
distsq(u), hopefully corresponding to a local minimum in distsq (and hence a local minimum
distance from P to Q(u)). */
Point const diff = Q_u - P;
double numerator = dot(diff, Q1_u);
double denominator = dot(Q1_u, Q1_u) + dot(diff, Q2_u);
double improved_u;
if ( denominator > 0. ) {
/* One iteration of Newton-Raphson:
improved_u = u - f(u)/f'(u) */
improved_u = u - ( numerator / denominator );
} else {
/* Using Newton-Raphson would move in the wrong direction (towards a local maximum rather
than local minimum), so we move an arbitrary amount in the right direction. */
if ( numerator > 0. ) {
improved_u = u * .98 - .01;
} else if ( numerator < 0. ) {
/* Deliberately asymmetrical, to reduce the chance of cycling. */
improved_u = .031 + u * .98;
} else {
improved_u = u;
}
}
if (!IS_FINITE(improved_u)) {
improved_u = u;
} else if ( improved_u < 0.0 ) {
improved_u = 0.0;
} else if ( improved_u > 1.0 ) {
improved_u = 1.0;
}
/* Ensure that improved_u isn't actually worse. */
{
double const diff_lensq = lensq(diff);
for (double proportion = .125; ; proportion += .125) {
if ( lensq( bezier_pt(3, Q, improved_u) - P ) > diff_lensq ) {
if ( proportion > 1.0 ) {
//g_warning("found proportion %g", proportion);
improved_u = u;
break;
}
improved_u = ( ( 1 - proportion ) * improved_u +
proportion * u );
} else {
break;
}
}
}
DOUBLE_ASSERT(improved_u);
return improved_u;
}
/**
* Evaluate a Bezier curve at parameter value \a t.
*
* \param degree The degree of the Bezier curve: 3 for cubic, 2 for quadratic etc. Must be less
* than 4.
* \param V The control points for the Bezier curve. Must have (\a degree+1)
* elements.
* \param t The "parameter" value, specifying whereabouts along the curve to
* evaluate. Typically in the range [0.0, 1.0].
*
* Let s = 1 - t.
* BezierII(1, V) gives (s, t) * V, i.e. t of the way
* from V[0] to V[1].
* BezierII(2, V) gives (s**2, 2*s*t, t**2) * V.
* BezierII(3, V) gives (s**3, 3 s**2 t, 3s t**2, t**3) * V.
*
* The derivative of BezierII(i, V) with respect to t
* is i * BezierII(i-1, V'), where for all j, V'[j] =
* V[j + 1] - V[j].
*/
Point
bezier_pt(unsigned const degree, Point const V[], double const t)
{
/** Pascal's triangle. */
static int const pascal[4][4] = {{1, 0, 0, 0},
{1, 1, 0, 0},
{1, 2, 1, 0},
{1, 3, 3, 1}};
assert( degree < 4);
double const s = 1.0 - t;
/* Calculate powers of t and s. */
double spow[4];
double tpow[4];
spow[0] = 1.0; spow[1] = s;
tpow[0] = 1.0; tpow[1] = t;
for (unsigned i = 1; i < degree; ++i) {
spow[i + 1] = spow[i] * s;
tpow[i + 1] = tpow[i] * t;
}
Point ret = spow[degree] * V[0];
for (unsigned i = 1; i <= degree; ++i) {
ret += pascal[degree][i] * spow[degree - i] * tpow[i] * V[i];
}
return ret;
}
/*
* ComputeLeftTangent, ComputeRightTangent, ComputeCenterTangent :
* Approximate unit tangents at endpoints and "center" of digitized curve
*/
/**
* Estimate the (forward) tangent at point d[first + 0.5].
*
* Unlike the center and right versions, this calculates the tangent in
* the way one might expect, i.e., wrt increasing index into d.
* \pre (2 \<= len) and (d[0] != d[1]).
**/
Point
darray_left_tangent(Point const d[], unsigned const len)
{
assert( len >= 2 );
assert( d[0] != d[1] );
return unit_vector( d[1] - d[0] );
}
/**
* Estimates the (backward) tangent at d[last - 0.5].
*
* \note The tangent is "backwards", i.e. it is with respect to
* decreasing index rather than increasing index.
*
* \pre 2 \<= len.
* \pre d[len - 1] != d[len - 2].
* \pre all[p in d] in_svg_plane(p).
*/
static Point
darray_right_tangent(Point const d[], unsigned const len)
{
assert( 2 <= len );
unsigned const last = len - 1;
unsigned const prev = last - 1;
assert( d[last] != d[prev] );
return unit_vector( d[prev] - d[last] );
}
/**
* Estimate the (forward) tangent at point d[0].
*
* Unlike the center and right versions, this calculates the tangent in
* the way one might expect, i.e., wrt increasing index into d.
*
* \pre 2 \<= len.
* \pre d[0] != d[1].
* \pre all[p in d] in_svg_plane(p).
* \post is_unit_vector(ret).
**/
Point
darray_left_tangent(Point const d[], unsigned const len, double const tolerance_sq)
{
assert( 2 <= len );
assert( 0 <= tolerance_sq );
for (unsigned i = 1;;) {
Point const pi(d[i]);
Point const t(pi - d[0]);
double const distsq = dot(t, t);
if ( tolerance_sq < distsq ) {
return unit_vector(t);
}
++i;
if (i == len) {
return ( distsq == 0
? darray_left_tangent(d, len)
: unit_vector(t) );
}
}
}
/**
* Estimates the (backward) tangent at d[last].
*
* \note The tangent is "backwards", i.e. it is with respect to
* decreasing index rather than increasing index.
*
* \pre 2 \<= len.
* \pre d[len - 1] != d[len - 2].
* \pre all[p in d] in_svg_plane(p).
*/
Point
darray_right_tangent(Point const d[], unsigned const len, double const tolerance_sq)
{
assert( 2 <= len );
assert( 0 <= tolerance_sq );
unsigned const last = len - 1;
for (unsigned i = last - 1;; i--) {
Point const pi(d[i]);
Point const t(pi - d[last]);
double const distsq = dot(t, t);
if ( tolerance_sq < distsq ) {
return unit_vector(t);
}
if (i == 0) {
return ( distsq == 0
? darray_right_tangent(d, len)
: unit_vector(t) );
}
}
}
/**
* Estimates the (backward) tangent at d[center], by averaging the two
* segments connected to d[center] (and then normalizing the result).
*
* \note The tangent is "backwards", i.e. it is with respect to
* decreasing index rather than increasing index.
*
* \pre (0 \< center \< len - 1) and d is uniqued (at least in
* the immediate vicinity of \a center).
*/
static Point
darray_center_tangent(Point const d[],
unsigned const center,
unsigned const len)
{
assert( center != 0 );
assert( center < len - 1 );
Point ret;
if ( d[center + 1] == d[center - 1] ) {
/* Rotate 90 degrees in an arbitrary direction. */
Point const diff = d[center] - d[center - 1];
ret = rot90(diff);
} else {
ret = d[center - 1] - d[center + 1];
}
ret.normalize();
return ret;
}
/**
* Assign parameter values to digitized points using relative distances between points.
*
* \pre Parameter array u must have space for \a len items.
*/
static void
chord_length_parameterize(Point const d[], double u[], unsigned const len)
{
if(!( 2 <= len ))
return;
/* First let u[i] equal the distance travelled along the path from d[0] to d[i]. */
u[0] = 0.0;
for (unsigned i = 1; i < len; i++) {
double const dist = distance(d[i], d[i-1]);
u[i] = u[i-1] + dist;
}
/* Then scale to [0.0 .. 1.0]. */
double tot_len = u[len - 1];
if(!( tot_len != 0 ))
return;
if (IS_FINITE(tot_len)) {
for (unsigned i = 1; i < len; ++i) {
u[i] /= tot_len;
}
} else {
/* We could do better, but this probably never happens anyway. */
for (unsigned i = 1; i < len; ++i) {
u[i] = i / (double) ( len - 1 );
}
}
/** \todo
* It's been reported that u[len - 1] can differ from 1.0 on some
* systems (amd64), despite it having been calculated as x / x where x
* is isFinite and non-zero.
*/
if (u[len - 1] != 1) {
double const diff = u[len - 1] - 1;
if (fabs(diff) > 1e-13) {
assert(0); // No warnings in 2geom
//g_warning("u[len - 1] = %19g (= 1 + %19g), expecting exactly 1",
// u[len - 1], diff);
}
u[len - 1] = 1;
}
#ifdef BEZIER_DEBUG
assert( u[0] == 0.0 && u[len - 1] == 1.0 );
for (unsigned i = 1; i < len; i++) {
assert( u[i] >= u[i-1] );
}
#endif
}
/**
* Find the maximum squared distance of digitized points to fitted curve, and (if this maximum
* error is non-zero) set \a *splitPoint to the corresponding index.
*
* \pre 2 \<= len.
* \pre u[0] == 0.
* \pre u[len - 1] == 1.0.
* \post ((ret == 0.0)
* || ((*splitPoint \< len - 1)
* \&\& (*splitPoint != 0 || ret \< 0.0))).
*/
static double
compute_max_error_ratio(Point const d[], double const u[], unsigned const len,
Point const bezCurve[], double const tolerance,
unsigned *const splitPoint)
{
assert( 2 <= len );
unsigned const last = len - 1;
assert( bezCurve[0] == d[0] );
assert( bezCurve[3] == d[last] );
assert( u[0] == 0.0 );
assert( u[last] == 1.0 );
/* I.e. assert that the error for the first & last points is zero.
* Otherwise we should include those points in the below loop.
* The assertion is also necessary to ensure 0 < splitPoint < last.
*/
double maxDistsq = 0.0; /* Maximum error */
double max_hook_ratio = 0.0;
unsigned snap_end = 0;
Point prev = bezCurve[0];
for (unsigned i = 1; i <= last; i++) {
Point const curr = bezier_pt(3, bezCurve, u[i]);
double const distsq = lensq( curr - d[i] );
if ( distsq > maxDistsq ) {
maxDistsq = distsq;
*splitPoint = i;
}
double const hook_ratio = compute_hook(prev, curr, .5 * (u[i - 1] + u[i]), bezCurve, tolerance);
if (max_hook_ratio < hook_ratio) {
max_hook_ratio = hook_ratio;
snap_end = i;
}
prev = curr;
}
double const dist_ratio = sqrt(maxDistsq) / tolerance;
double ret;
if (max_hook_ratio <= dist_ratio) {
ret = dist_ratio;
} else {
assert(0 < snap_end);
ret = -max_hook_ratio;
*splitPoint = snap_end - 1;
}
assert( ret == 0.0
|| ( ( *splitPoint < last )
&& ( *splitPoint != 0 || ret < 0. ) ) );
return ret;
}
/**
* Whereas compute_max_error_ratio() checks for itself that each data point
* is near some point on the curve, this function checks that each point on
* the curve is near some data point (or near some point on the polyline
* defined by the data points, or something like that: we allow for a
* "reasonable curviness" from such a polyline). "Reasonable curviness"
* means we draw a circle centred at the midpoint of a..b, of radius
* proportional to the length |a - b|, and require that each point on the
* segment of bezCurve between the parameters of a and b be within that circle.
* If any point P on the bezCurve segment is outside of that allowable
* region (circle), then we return some metric that increases with the
* distance from P to the circle.
*
* Given that this is a fairly arbitrary criterion for finding appropriate
* places for sharp corners, we test only one point on bezCurve, namely
* the point on bezCurve with parameter halfway between our estimated
* parameters for a and b. (Alternatives are taking the farthest of a
* few parameters between those of a and b, or even using a variant of
* NewtonRaphsonFindRoot() for finding the maximum rather than minimum
* distance.)
*/
static double
compute_hook(Point const &a, Point const &b, double const u, Point const bezCurve[],
double const tolerance)
{
Point const P = bezier_pt(3, bezCurve, u);
double const dist = distance((a+b)*.5, P);
if (dist < tolerance) {
return 0;
}
double const allowed = distance(a, b) + tolerance;
return dist / allowed;
/** \todo
* effic: Hooks are very rare. We could start by comparing
* distsq, only resorting to the more expensive L2 in cases of
* uncertainty.
*/
}
}
/*
Local Variables:
mode:c++
c-file-style:"stroustrup"
c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
indent-tabs-mode:nil
fill-column:99
End:
*/
// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:fileencoding=utf-8:textwidth=99 :