1 /*
2 * Symmetric Power Basis - Bernstein Basis conversion routines
3 *
4 * Authors:
5 * Marco Cecchetti <mrcekets at gmail.com>
6 * Nathan Hurst <njh@mail.csse.monash.edu.au>
7 *
8 * Copyright 2007-2008 authors
9 *
10 * This library is free software; you can redistribute it and/or
11 * modify it either under the terms of the GNU Lesser General Public
12 * License version 2.1 as published by the Free Software Foundation
13 * (the "LGPL") or, at your option, under the terms of the Mozilla
14 * Public License Version 1.1 (the "MPL"). If you do not alter this
15 * notice, a recipient may use your version of this file under either
16 * the MPL or the LGPL.
17 *
18 * You should have received a copy of the LGPL along with this library
19 * in the file COPYING-LGPL-2.1; if not, write to the Free Software
20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
21 * You should have received a copy of the MPL along with this library
22 * in the file COPYING-MPL-1.1
23 *
24 * The contents of this file are subject to the Mozilla Public License
25 * Version 1.1 (the "License"); you may not use this file except in
26 * compliance with the License. You may obtain a copy of the License at
27 * http://www.mozilla.org/MPL/
28 *
29 * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
30 * OF ANY KIND, either express or implied. See the LGPL or the MPL for
31 * the specific language governing rights and limitations.
32 */
35 #include <2geom/sbasis-to-bezier.h>
36 #include <2geom/d2.h>
37 #include <2geom/choose.h>
38 #include <2geom/svg-path.h>
39 #include <2geom/exception.h>
41 #include <iostream>
46 namespace Geom
47 {
49 /*
50 * Symmetric Power Basis - Bernstein Basis conversion routines
51 *
52 * some remark about precision:
53 * interval [0,1], subdivisions: 10^3
54 * - bezier_to_sbasis : up to degree ~72 precision is at least 10^-5
55 * up to degree ~87 precision is at least 10^-3
56 * - sbasis_to_bezier : up to order ~63 precision is at least 10^-15
57 * precision is at least 10^-14 even beyond order 200
58 *
59 * interval [-1,1], subdivisions: 10^3
60 * - bezier_to_sbasis : up to degree ~21 precision is at least 10^-5
61 * up to degree ~24 precision is at least 10^-3
62 * - sbasis_to_bezier : up to order ~11 precision is at least 10^-5
63 * up to order ~13 precision is at least 10^-3
64 *
65 * interval [-10,10], subdivisions: 10^3
66 * - bezier_to_sbasis : up to degree ~7 precision is at least 10^-5
67 * up to degree ~8 precision is at least 10^-3
68 * - sbasis_to_bezier : up to order ~3 precision is at least 10^-5
69 * up to order ~4 precision is at least 10^-3
70 *
71 * references:
72 * this implementation is based on the following article:
73 * J.Sanchez-Reyes - The Symmetric Analogue of the Polynomial Power Basis
74 */
76 inline
77 double binomial(unsigned int n, unsigned int k)
78 {
79 return choose<double>(n, k);
80 }
82 inline
83 int sgn(unsigned int j, unsigned int k)
84 {
85 assert (j >= k);
86 // we are sure that j >= k
87 return ((j-k) & 1u) ? -1 : 1;
88 }
91 /** Changes the basis of p to be bernstein.
92 \param p the Symmetric basis polynomial
93 \returns the Bernstein basis polynomial
95 if the degree is even q is the order in the symmetrical power basis,
96 if the degree is odd q is the order + 1
97 n is always the polynomial degree, i. e. the Bezier order
98 */
99 void sbasis_to_bezier (Bezier & bz, SBasis const& sb, size_t sz)
100 {
101 size_t q, n;
102 bool even;
103 if (sz == 0)
104 {
105 q = sb.size();
106 if (sb[q-1][0] == sb[q-1][1])
107 {
108 even = true;
109 --q;
110 n = 2*q;
111 }
112 else
113 {
114 even = false;
115 n = 2*q-1;
116 }
117 }
118 else
119 {
120 q = (sz > sb.size()) ? sb.size() : sz;
121 n = 2*sz-1;
122 even = false;
123 }
124 bz.clear();
125 bz.resize(n+1);
126 double Tjk;
127 for (size_t k = 0; k < q; ++k)
128 {
129 for (size_t j = k; j < n-k; ++j) // j <= n-k-1
130 {
131 Tjk = binomial(n-2*k-1, j-k);
132 bz[j] += (Tjk * sb[k][0]);
133 bz[n-j] += (Tjk * sb[k][1]); // n-k <-> [k][1]
134 }
135 }
136 if (even)
137 {
138 bz[q] += sb[q][0];
139 }
140 // the resulting coefficients are with respect to the scaled Bernstein
141 // basis so we need to divide them by (n, j) binomial coefficient
142 for (size_t j = 1; j < n; ++j)
143 {
144 bz[j] /= binomial(n, j);
145 }
146 bz[0] = sb[0][0];
147 bz[n] = sb[0][1];
148 }
150 /** Changes the basis of p to be Bernstein.
151 \param p the D2 Symmetric basis polynomial
152 \returns the D2 Bernstein basis polynomial
154 if the degree is even q is the order in the symmetrical power basis,
155 if the degree is odd q is the order + 1
156 n is always the polynomial degree, i. e. the Bezier order
157 */
158 void sbasis_to_bezier (std::vector<Point> & bz, D2<SBasis> const& sb, size_t sz)
159 {
160 Bezier bzx, bzy;
161 sbasis_to_bezier(bzx, sb[X], sz);
162 sbasis_to_bezier(bzy, sb[Y], sz);
163 size_t n = (bzx.size() >= bzy.size()) ? bzx.size() : bzy.size();
165 bz.resize(n, Point(0,0));
166 for (size_t i = 0; i < bzx.size(); ++i)
167 {
168 bz[i][X] = bzx[i];
169 }
170 for (size_t i = 0; i < bzy.size(); ++i)
171 {
172 bz[i][Y] = bzy[i];
173 }
174 }
177 /** Changes the basis of p to be sbasis.
178 \param p the Bernstein basis polynomial
179 \returns the Symmetric basis polynomial
181 if the degree is even q is the order in the symmetrical power basis,
182 if the degree is odd q is the order + 1
183 n is always the polynomial degree, i. e. the Bezier order
184 */
185 void bezier_to_sbasis (SBasis & sb, Bezier const& bz)
186 {
187 size_t n = bz.order();
188 size_t q = (n+1) / 2;
189 size_t even = (n & 1u) ? 0 : 1;
190 sb.clear();
191 sb.resize(q + even, Linear(0, 0));
192 double Tjk;
193 for (size_t k = 0; k < q; ++k)
194 {
195 for (size_t j = k; j < q; ++j)
196 {
197 Tjk = sgn(j, k) * binomial(n-j-k, j-k) * binomial(n, k);
198 sb[j][0] += (Tjk * bz[k]);
199 sb[j][1] += (Tjk * bz[n-k]); // n-j <-> [j][1]
200 }
201 for (size_t j = k+1; j < q; ++j)
202 {
203 Tjk = sgn(j, k) * binomial(n-j-k-1, j-k-1) * binomial(n, k);
204 sb[j][0] += (Tjk * bz[n-k]);
205 sb[j][1] += (Tjk * bz[k]); // n-j <-> [j][1]
206 }
207 }
208 if (even)
209 {
210 for (size_t k = 0; k < q; ++k)
211 {
212 Tjk = sgn(q,k) * binomial(n, k);
213 sb[q][0] += (Tjk * (bz[k] + bz[n-k]));
214 }
215 sb[q][0] += (binomial(n, q) * bz[q]);
216 sb[q][1] = sb[q][0];
217 }
218 sb[0][0] = bz[0];
219 sb[0][1] = bz[n];
220 }
223 /** Changes the basis of d2 p to be sbasis.
224 \param p the d2 Bernstein basis polynomial
225 \returns the d2 Symmetric basis polynomial
227 if the degree is even q is the order in the symmetrical power basis,
228 if the degree is odd q is the order + 1
229 n is always the polynomial degree, i. e. the Bezier order
230 */
231 void bezier_to_sbasis (D2<SBasis> & sb, std::vector<Point> const& bz)
232 {
233 size_t n = bz.size() - 1;
234 size_t q = (n+1) / 2;
235 size_t even = (n & 1u) ? 0 : 1;
236 sb[X].clear();
237 sb[Y].clear();
238 sb[X].resize(q + even, Linear(0, 0));
239 sb[Y].resize(q + even, Linear(0, 0));
240 double Tjk;
241 for (size_t k = 0; k < q; ++k)
242 {
243 for (size_t j = k; j < q; ++j)
244 {
245 Tjk = sgn(j, k) * binomial(n-j-k, j-k) * binomial(n, k);
246 sb[X][j][0] += (Tjk * bz[k][X]);
247 sb[X][j][1] += (Tjk * bz[n-k][X]);
248 sb[Y][j][0] += (Tjk * bz[k][Y]);
249 sb[Y][j][1] += (Tjk * bz[n-k][Y]);
250 }
251 for (size_t j = k+1; j < q; ++j)
252 {
253 Tjk = sgn(j, k) * binomial(n-j-k-1, j-k-1) * binomial(n, k);
254 sb[X][j][0] += (Tjk * bz[n-k][X]);
255 sb[X][j][1] += (Tjk * bz[k][X]);
256 sb[Y][j][0] += (Tjk * bz[n-k][Y]);
257 sb[Y][j][1] += (Tjk * bz[k][Y]);
258 }
259 }
260 if (even)
261 {
262 for (size_t k = 0; k < q; ++k)
263 {
264 Tjk = sgn(q,k) * binomial(n, k);
265 sb[X][q][0] += (Tjk * (bz[k][X] + bz[n-k][X]));
266 sb[Y][q][0] += (Tjk * (bz[k][Y] + bz[n-k][Y]));
267 }
268 sb[X][q][0] += (binomial(n, q) * bz[q][X]);
269 sb[X][q][1] = sb[X][q][0];
270 sb[Y][q][0] += (binomial(n, q) * bz[q][Y]);
271 sb[Y][q][1] = sb[Y][q][0];
272 }
273 sb[X][0][0] = bz[0][X];
274 sb[X][0][1] = bz[n][X];
275 sb[Y][0][0] = bz[0][Y];
276 sb[Y][0][1] = bz[n][Y];
277 }
280 } // end namespace Geom
283 #if 0
284 /*
285 * This version works by inverting a reasonable upper bound on the error term after subdividing the
286 * curve at $a$. We keep biting off pieces until there is no more curve left.
287 *
288 * Derivation: The tail of the power series is $a_ks^k + a_{k+1}s^{k+1} + \ldots = e$. A
289 * subdivision at $a$ results in a tail error of $e*A^k, A = (1-a)a$. Let this be the desired
290 * tolerance tol $= e*A^k$ and invert getting $A = e^{1/k}$ and $a = 1/2 - \sqrt{1/4 - A}$
291 */
292 void
293 subpath_from_sbasis_incremental(Geom::OldPathSetBuilder &pb, D2<SBasis> B, double tol, bool initial) {
294 const unsigned k = 2; // cubic bezier
295 double te = B.tail_error(k);
296 assert(B[0].IS_FINITE());
297 assert(B[1].IS_FINITE());
299 //std::cout << "tol = " << tol << std::endl;
300 while(1) {
301 double A = std::sqrt(tol/te); // pow(te, 1./k)
302 double a = A;
303 if(A < 1) {
304 A = std::min(A, 0.25);
305 a = 0.5 - std::sqrt(0.25 - A); // quadratic formula
306 if(a > 1) a = 1; // clamp to the end of the segment
307 } else
308 a = 1;
309 assert(a > 0);
310 //std::cout << "te = " << te << std::endl;
311 //std::cout << "A = " << A << "; a=" << a << std::endl;
312 D2<SBasis> Bs = compose(B, Linear(0, a));
313 assert(Bs.tail_error(k));
314 std::vector<Geom::Point> bez = sbasis_to_bezier(Bs, 2);
315 reverse(bez.begin(), bez.end());
316 if (initial) {
317 pb.start_subpath(bez[0]);
318 initial = false;
319 }
320 pb.push_cubic(bez[1], bez[2], bez[3]);
322 // move to next piece of curve
323 if(a >= 1) break;
324 B = compose(B, Linear(a, 1));
325 te = B.tail_error(k);
326 }
327 }
329 #endif
331 namespace Geom{
333 /** Make a path from a d2 sbasis.
334 \param p the d2 Symmetric basis polynomial
335 \returns a Path
337 If only_cubicbeziers is true, the resulting path may only contain CubicBezier curves.
338 */
339 void build_from_sbasis(Geom::PathBuilder &pb, D2<SBasis> const &B, double tol, bool only_cubicbeziers) {
340 if (!B.isFinite()) {
341 THROW_EXCEPTION("assertion failed: B.isFinite()");
342 }
343 if(tail_error(B, 2) < tol || sbasis_size(B) == 2) { // nearly cubic enough
344 if( !only_cubicbeziers && (sbasis_size(B) <= 1) ) {
345 pb.lineTo(B.at1());
346 } else {
347 std::vector<Geom::Point> bez;
348 sbasis_to_bezier(bez, B, 2);
349 pb.curveTo(bez[1], bez[2], bez[3]);
350 }
351 } else {
352 build_from_sbasis(pb, compose(B, Linear(0, 0.5)), tol, only_cubicbeziers);
353 build_from_sbasis(pb, compose(B, Linear(0.5, 1)), tol, only_cubicbeziers);
354 }
355 }
357 /** Make a path from a d2 sbasis.
358 \param p the d2 Symmetric basis polynomial
359 \returns a Path
361 If only_cubicbeziers is true, the resulting path may only contain CubicBezier curves.
362 */
363 Path
364 path_from_sbasis(D2<SBasis> const &B, double tol, bool only_cubicbeziers) {
365 PathBuilder pb;
366 pb.moveTo(B.at0());
367 build_from_sbasis(pb, B, tol, only_cubicbeziers);
368 pb.finish();
369 return pb.peek().front();
370 }
372 /** Make a path from a d2 sbasis.
373 \param p the d2 Symmetric basis polynomial
374 \returns a Path
376 If only_cubicbeziers is true, the resulting path may only contain CubicBezier curves.
377 TODO: some of this logic should be lifted into svg-path
378 */
379 std::vector<Geom::Path>
380 path_from_piecewise(Geom::Piecewise<Geom::D2<Geom::SBasis> > const &B, double tol, bool only_cubicbeziers) {
381 Geom::PathBuilder pb;
382 if(B.size() == 0) return pb.peek();
383 Geom::Point start = B[0].at0();
384 pb.moveTo(start);
385 for(unsigned i = 0; ; i++) {
386 if(i+1 == B.size() || !are_near(B[i+1].at0(), B[i].at1(), tol)) {
387 //start of a new path
388 if(are_near(start, B[i].at1()) && sbasis_size(B[i]) <= 1) {
389 pb.closePath();
390 //last line seg already there (because of .closePath())
391 goto no_add;
392 }
393 build_from_sbasis(pb, B[i], tol, only_cubicbeziers);
394 if(are_near(start, B[i].at1())) {
395 //it's closed, the last closing segment was not a straight line so it needed to be added, but still make it closed here with degenerate straight line.
396 pb.closePath();
397 }
398 no_add:
399 if(i+1 >= B.size()) break;
400 start = B[i+1].at0();
401 pb.moveTo(start);
402 } else {
403 build_from_sbasis(pb, B[i], tol, only_cubicbeziers);
404 }
405 }
406 pb.finish();
407 return pb.peek();
408 }
410 }
412 /*
413 Local Variables:
414 mode:c++
415 c-file-style:"stroustrup"
416 c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
417 indent-tabs-mode:nil
418 fill-column:99
419 End:
420 */
421 // vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:encoding=utf-8:textwidth=99 :