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thbezier.cxx
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thbezier.cxx
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/**
* @file thbezier.cxx
*/
/*
* 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
*
*/
/* Copyright (C) 2007 Stacho Mudrak
*
* $Date: $
* $RCSfile: $
* $Revision: $
*
* --------------------------------------------------------------------
* 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 2 of the License, or
* 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, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
* --------------------------------------------------------------------
*/
#include "thbezier.h"
#include <cassert>
#include <cstring>
#include <cstdio>
#include <cmath>
#define SP_HUGE 1e5
#define noBEZIER_DEBUG
#ifdef HAVE_CONFIG_H
# include <config.h>
#endif
#ifdef HAVE_IEEEFP_H
# include <ieeefp.h>
#endif
//#include <glib/gmessages.h>
//#include <glib/gmem.h>
//#include <glib/gtypes.h>
//#include <libnr/nr-forward.h>
//#include <libnr/nr-point-fns.h>
#include <cmath>
#define g_return_val_if_fail(C,V) if (!(C)) return (V);
#define g_return_if_fail(C) if (!(C)) return;
#define g_new(T,L) new T [L];
#define g_free(V) delete [] V
#define g_assert(X) assert(X)
/* You might try changing the above to <cmath> if you have problems.
* Whether you use math.h or cmath, you may need to edit the .cpp file
* and/or other .h files to use the same header file.
*/
#if defined(__isnan)
# define isNaN(_a) (__isnan(_a)) /* MacOSX/Darwin definition < 10.4 */
#elif defined(WIN32) || defined(_isnan)
# define isNaN(_a) (_isnan(_a)) /* Win32 definition */
#elif defined(isnan) || defined(__FreeBSD__) || defined(__osf__)
# define isNaN(_a) (isnan(_a)) /* GNU definition */
#else
# define isNaN(_a) (std::isnan(_a))
#endif
/* If the above doesn't work, then try (a != a).
* Also, please report a bug as per http://www.inkscape.org/report_bugs.php,
* giving information about what platform and compiler version you're using.
*/
#if defined(__isfinite)
# define isFinite(_a) (__isfinite(_a)) /* MacOSX/Darwin definition < 10.4 */
#elif defined(__sgi)
# define isFinite(_a) (_isfinite(_a))
#elif defined(isfinite)
# define isFinite(_a) (isfinite(_a))
#elif defined(__osf__)
# define isFinite(_a) (finite(_a) && !isNaN(_a))
#else
# define isFinite(_a) (std::isfinite(_a))
#endif
/* If the above doesn't work, then try (finite(_a) && !isNaN(_a)) or (!isNaN((_a) - (_a))).
* Also, please report a bug as per http://www.inkscape.org/report_bugs.php,
* giving information about what platform and compiler version you're using.
*/
class NR_Point {
private:
double _pt[2] = {};
public:
inline NR_Point()
{ _pt[0] = _pt[1] = 0; }
inline NR_Point(double x, double y) {
_pt[0] = x;
_pt[1] = y;
}
inline NR_Point(NR_Point const &p) {
for (unsigned i = 0; i < 2; ++i) {
_pt[i] = p._pt[i];
}
}
inline NR_Point &operator=(NR_Point const &p) {
for (unsigned i = 0; i < 2; ++i) {
_pt[i] = p._pt[i];
}
return *this;
}
inline double operator[](unsigned i) const {
return _pt[i];
}
inline double &operator[](unsigned i) {
return _pt[i];
}
//double operator[](size_t d) const throw() { return _pt[d]; }
//double &operator[](size_t d) throw() { return _pt[d]; }
/** Return a point like this point but rotated -90 degrees.
(If the y axis grows downwards and the x axis grows to the
right, then this is 90 degrees counter-clockwise.)
**/
NR_Point ccw() const {
return NR_Point(_pt[1], -_pt[0]);
}
/** Return a point like this point but rotated +90 degrees.
(If the y axis grows downwards and the x axis grows to the
right, then this is 90 degrees clockwise.)
**/
NR_Point cw() const {
return NR_Point(-_pt[1], _pt[0]);
}
void normalize();
inline NR_Point &operator+=(NR_Point const &o) {
for ( unsigned i = 0 ; i < 2 ; ++i ) {
_pt[i] += o._pt[i];
}
return *this;
}
inline NR_Point &operator-=(NR_Point const &o) {
for ( unsigned i = 0 ; i < 2 ; ++i ) {
_pt[i] -= o._pt[i];
}
return *this;
}
inline NR_Point &operator/=(double const s) {
for ( unsigned i = 0 ; i < 2 ; ++i ) {
_pt[i] /= s;
}
return *this;
}
inline NR_Point &operator*=(double const s) {
for ( unsigned i = 0 ; i < 2 ; ++i ) {
_pt[i] *= s;
}
return *this;
}
//inline bool operator == (const NR_Point &in_pnt) const {
// return ((_pt[0] == in_pnt[0]) && (_pt[1] == in_pnt[1]));
//}
};
inline NR_Point rot90(NR_Point const &p)
{
return NR_Point(-p[1], p[0]);
}
inline NR_Point operator+(NR_Point const &a, NR_Point const &b)
{
NR_Point ret;
for (int i = 0; i < 2; i++) {
ret[i] = a[i] + b[i];
}
return ret;
}
inline double dot(NR_Point const &a, NR_Point const &b)
{
double ret = 0;
for ( int i = 0 ; i < 2 ; i++ ) {
ret += a[i] * b[i];
}
return ret;
}
void NR_Point::normalize() {
double len = std::hypot(_pt[0], _pt[1]);
g_return_if_fail(len != 0);
g_return_if_fail(!isNaN(len));
static double const inf = 1e300;
if(len != inf) {
*this /= len;
} else {
unsigned n_inf_coords = 0;
/* Delay updating pt in case neither coord is infinite. */
NR_Point tmp;
for ( unsigned i = 0 ; i < 2 ; ++i ) {
if ( _pt[i] == inf ) {
++n_inf_coords;
tmp[i] = 1.0;
} else if ( _pt[i] == -inf ) {
++n_inf_coords;
tmp[i] = -1.0;
} else {
tmp[i] = 0.0;
}
}
switch (n_inf_coords) {
case 0:
/* Can happen if both coords are near +/-DBL_MAX. */
*this /= 4.0;
len = std::hypot(_pt[0], _pt[1]);
g_assert(len != inf);
*this /= len;
break;
case 1:
*this = tmp;
break;
case 2:
*this = NR_Point(sqrt(0.5) * tmp[0], sqrt(0.5) * tmp[1]);
break;
}
}
}
inline NR_Point operator-(NR_Point const &a, NR_Point const &b)
{
NR_Point ret;
for (int i = 0; i < 2; i++) {
ret[i] = a[i] - b[i];
}
return ret;
}
/** This is a rotation (sort of). */
inline NR_Point operator^(NR_Point const &a, NR_Point const &b)
{
NR_Point const ret(a[0] * b[0] - a[1] * b[1],
a[1] * b[0] + a[0] * b[1]);
return ret;
}
inline NR_Point operator-(NR_Point const &a)
{
NR_Point ret;
for(unsigned i = 0; i < 2; i++) {
ret[i] = -a[i];
}
return ret;
}
inline NR_Point operator*(double const s, NR_Point const &b)
{
NR_Point ret;
for(int i = 0; i < 2; i++) {
ret[i] = s * b[i];
}
return ret;
}
inline NR_Point operator/(NR_Point const &b, double const d)
{
NR_Point ret;
for(int i = 0; i < 2; i++) {
ret[i] = b[i] / d;
}
return ret;
}
inline bool operator==(NR_Point const &a, NR_Point const &b)
{
return ( ( a[0] == b[0] ) && ( a[1] == b[1] ) );
}
inline bool operator!=(NR_Point const &a, NR_Point const &b)
{
return ( ( a[0] != b[0] ) || ( a[1] != b[1] ) );
}
inline double L2(NR_Point const &p) {
return std::hypot(p[0], p[1]);
}
bool is_zero(NR_Point const &p)
{
return ( p[0] == 0 &&
p[1] == 0 );
}
NR_Point unit_vector(NR_Point const &a)
{
NR_Point ret(a);
ret.normalize();
return ret;
}
NR_Point bezier_pt(unsigned degree, NR_Point const V[], double t);
long sp_bezier_fit_cubic(NR_Point bezier[], NR_Point const data[], long len, double error);
long sp_bezier_fit_cubic_r(NR_Point bezier[], NR_Point const data[], long len, double error,
unsigned max_beziers);
long sp_bezier_fit_cubic_full(NR_Point bezier[], int split_points[], NR_Point const data[], long len,
NR_Point const &tHat1, NR_Point const &tHat2,
double error, unsigned max_beziers);
NR_Point sp_darray_left_tangent(NR_Point const d[], unsigned const len);
NR_Point sp_darray_left_tangent(NR_Point const d[], unsigned const len, double const tolerance_sq);
NR_Point sp_darray_right_tangent(NR_Point const d[], unsigned const length, double const tolerance_sq);
/*
* Temporary fix for various misdefinitions of isnan().
* isnan() is becoming undef'd in some .h files.
* #include this last in your .cpp file to get it right.
*
* The problem is that isnan and isfinite are part of C99 but aren't part of
* the C++ standard (which predates C99).
*
* Authors:
* Inkscape groupies and obsessive-compulsives
*
* Copyright (C) 2004 authors
*
* Released under GNU GPL, read the file 'COPYING' for more information
*
* 2005 modification hereby placed in public domain. Probably supercedes the 2004 copyright
* for the code itself.
*/
typedef NR_Point BezierCurve[];
/* Forward declarations */
static void generate_bezier(NR_Point b[], NR_Point const d[], double const u[], unsigned len,
NR_Point const &tHat1, NR_Point const &tHat2, double tolerance_sq);
static void estimate_lengths(NR_Point bezier[],
NR_Point const data[], double const u[], unsigned len,
NR_Point const &tHat1, NR_Point const &tHat2);
static void estimate_bi(NR_Point b[4], unsigned ei,
NR_Point const data[], double const u[], unsigned len);
static void reparameterize(NR_Point const d[], unsigned len, double u[], BezierCurve const bezCurve);
static double NewtonRaphsonRootFind(BezierCurve const Q, NR_Point const &P, double u);
static NR_Point sp_darray_center_tangent(NR_Point const d[], unsigned center, unsigned length);
static NR_Point sp_darray_right_tangent(NR_Point const d[], unsigned const len);
static unsigned copy_without_nans_or_adjacent_duplicates(NR_Point const src[], unsigned src_len, NR_Point dest[]);
static void chord_length_parameterize(NR_Point const d[], double u[], unsigned len);
static double compute_max_error_ratio(NR_Point const d[], double const u[], unsigned len,
BezierCurve const bezCurve, double tolerance,
unsigned *splitPoint);
static double compute_hook(NR_Point const &a, NR_Point const &b, double const u, BezierCurve const bezCurve,
double const tolerance);
static NR_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) g_assert( ( (x) > -SP_HUGE ) && ( (x) < SP_HUGE ) )
# define BEZIER_ASSERT(b) do { \
DOUBLE_ASSERT((b)[0][0]); DOUBLE_ASSERT((b)[0][1]); \
DOUBLE_ASSERT((b)[1][0]); DOUBLE_ASSERT((b)[1][1]); \
DOUBLE_ASSERT((b)[2][0]); DOUBLE_ASSERT((b)[2][1]); \
DOUBLE_ASSERT((b)[3][0]); DOUBLE_ASSERT((b)[3][1]); \
} 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.
*/
long
sp_bezier_fit_cubic(NR_Point *bezier, NR_Point const *data, long len, double error)
{
return sp_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.
*/
long
sp_bezier_fit_cubic_r(NR_Point bezier[], NR_Point const data[], long const len, double const error, unsigned const max_beziers)
{
g_return_val_if_fail(bezier != NULL, -1);
g_return_val_if_fail(data != NULL, -1);
g_return_val_if_fail(len > 0, -1);
g_return_val_if_fail(max_beziers < (1ul << (31 - 2 - 1 - 3)), -1);
NR_Point *uniqued_data = g_new(NR_Point, len);
unsigned uniqued_len = copy_without_nans_or_adjacent_duplicates(data, len, uniqued_data);
if ( uniqued_len < 2 ) {
g_free(uniqued_data);
return 0;
}
/* Call fit-cubic function with recursion. */
long const ret = sp_bezier_fit_cubic_full(bezier, NULL, uniqued_data, uniqued_len,
unconstrained_tangent, unconstrained_tangent,
error, max_beziers);
g_free(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(NR_Point const src[], unsigned src_len, NR_Point dest[])
{
unsigned si = 0;
for (;;) {
if ( si == src_len ) {
return 0;
}
if (!isNaN(src[si][0]) &&
!isNaN(src[si][1])) {
dest[0] = NR_Point(src[si]);
++si;
break;
}
}
unsigned di = 0;
for (; si < src_len; ++si) {
NR_Point const src_pt = NR_Point(src[si]);
if ( src_pt != dest[di]
&& !isNaN(src_pt[0])
&& !isNaN(src_pt[1])) {
dest[++di] = src_pt;
}
}
unsigned dest_len = di + 1;
g_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.
*/
long
sp_bezier_fit_cubic_full(NR_Point bezier[], int split_points[],
NR_Point const data[], long const len,
NR_Point const &tHat1, NR_Point const &tHat2,
double const error, unsigned const max_beziers)
{
int const maxIterations = 4; /* Max times to try iterating */
g_return_val_if_fail(bezier != NULL, -1);
g_return_val_if_fail(data != NULL, -1);
g_return_val_if_fail(len > 0, -1);
g_return_val_if_fail(max_beziers >= 1, -1);
g_return_val_if_fail(error >= 0.0, -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 = ( L2( data[len - 1]
- data[0] )
/ 3.0 );
if (isNaN(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 = 0; /* Point to split point set at. */
bool is_corner = false;
{
double *u = g_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.)
*/
g_free(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);
g_free(u);
return 1;
}
/* If error not too large, then try some reparameterization and iteration. */
if ( 0.0 <= maxErrorRatio && maxErrorRatio <= 3.0 ) {
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);
g_free(u);
return 1;
}
}
}
g_free(u);
is_corner = (maxErrorRatio < 0);
}
if (is_corner) {
g_assert(splitPoint < unsigned(len));
if (splitPoint == 0) {
if (is_zero(tHat1)) {
/* Got spike even with unconstrained initial tangent. */
++splitPoint;
} else {
return sp_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 sp_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;
NR_Point recTHat2, recTHat1;
if (is_corner) {
g_return_val_if_fail(0 < splitPoint && splitPoint < unsigned(len - 1), -1);
recTHat1 = recTHat2 = unconstrained_tangent;
} else {
/* Unit tangent vector at splitPoint. */
recTHat2 = sp_darray_center_tangent(data, splitPoint, len);
recTHat1 = -recTHat2;
}
long const nsegs1 = sp_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;
}
g_assert( nsegs1 != 0 );
if (split_points != NULL) {
split_points[nsegs1 - 1] = splitPoint;
}
unsigned const rec_max_beziers2 = max_beziers - nsegs1;
long const nsegs2 = sp_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(NR_Point bezier[],
NR_Point const data[], double const u[], unsigned const len,
NR_Point const &tHat1, NR_Point const &tHat2,
double const tolerance_sq)
{
bool const est1 = is_zero(tHat1);
bool const est2 = is_zero(tHat2);
NR_Point est_tHat1( est1
? sp_darray_left_tangent(data, len, tolerance_sq)
: tHat1 );
NR_Point est_tHat2( est2
? sp_darray_right_tangent(data, len, tolerance_sq)
: tHat2 );
estimate_lengths(bezier, data, u, len, est_tHat1, est_tHat2);
/* We find that sp_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(NR_Point bezier[],
NR_Point const data[], double const uPrime[], unsigned const len,
NR_Point const &tHat1, NR_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 */
NR_Point const a1 = b1 * tHat1;
NR_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]. */
NR_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 = ( L2( data[len - 1]
- data[0] )
/ 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(NR_Point const p) {
return dot(p, p);
}
static void
estimate_bi(NR_Point bezier[4], unsigned const ei,
NR_Point const data[], double const u[], unsigned const len)
{
g_return_if_fail(1 <= ei && ei <= 2);
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(NR_Point const d[],
unsigned const len,
double u[],
BezierCurve const bezCurve)
{
g_assert( 2 <= len );
unsigned const last = len - 1;
g_assert( bezCurve[0] == d[0] );
g_assert( bezCurve[3] == d[last] );
g_assert( u[0] == 0.0 );
g_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(BezierCurve const Q, NR_Point const &P, double const u)
{
g_assert( 0.0 <= u );
g_assert( u <= 1.0 );
/* Generate control vertices for Q'. */
NR_Point Q1[3];
for (unsigned i = 0; i < 3; i++) {
Q1[i] = 3.0 * ( Q[i+1] - Q[i] );
}
/* Generate control vertices for Q''. */
NR_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). */
NR_Point const Q_u = bezier_pt(3, Q, u);
NR_Point const Q1_u = bezier_pt(2, Q1, u);
NR_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)). */
NR_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 (!isFinite(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.
* \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].
*/
NR_Point
bezier_pt(unsigned const degree, NR_Point const V[], double const t)
{
/** Pascal's triangle. */