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spiro.c
1412 lines (1300 loc) · 42.3 KB
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spiro.c
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/*
libspiro - A sharable library of Spiro formula and functions.
This file is a fork from ppedit for use by the libspiro project.
Please see Changelog or git history for description of changes.
=============================================================
ppedit - A pattern plate editor for Spiro splines.
Copyright (C) 2007... Raph Levien (ppedit)
libspiro - A sharable library of Spiro formula and functions.
Copyright (C) 2007... George Williams (libspiro fork)
Copyright (C) 2013... Joe Da Silva (improvements plus 'a','h')
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 3
of the License, or (at your option) 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., 51 Franklin Street, Fifth Floor, Boston, MA
02110-1301, USA.
*/
/* C implementation of third-order polynomial spirals. */
#include <math.h>
#ifdef HAVE_FINITE
#include <float.h>
#endif
#include <stdlib.h>
#include <string.h>
#include "spiroentrypoints.h"
#include "bezctx_intf.h"
#include "spiro_intf.h"
#include "spiro.h"
#include "spiro-config.h"
#ifdef VERBOSE
#include <stdio.h>
#endif
typedef struct {
double a[11]; /* band-diagonal matrix */
double al[5]; /* lower part of band-diagonal decomposition */
} bandmat;
#ifndef M_PI
#define M_PI 3.14159265358979323846 /* pi */
#endif
#ifndef N_IS
/* int n = 4; */
#define N_IS 4
#endif
#ifndef ORDER
#define ORDER 12
#endif
/* Integrate polynomial spiral curve over range -.5 .. .5. */
static void
integrate_spiro(const double ks[4], double xy[2], int n)
{
#if 0
int n = 1024;
#endif
double th1 = ks[0];
double th2 = .5 * ks[1];
double th3 = (1./6) * ks[2];
double th4 = (1./24) * ks[3];
double x, y;
double ds = 1. / n;
double ds2 = ds * ds;
double ds3 = ds2 * ds;
double k0 = ks[0] * ds;
double k1 = ks[1] * ds;
double k2 = ks[2] * ds;
double k3 = ks[3] * ds;
int i;
double s = .5 * ds - .5;
x = 0;
y = 0;
for (i = 0; i < n; i++) {
#if ORDER > 2
double u, v;
double km0, km1, km2, km3;
if (n == 1) {
km0 = k0;
km1 = k1 * ds;
km2 = k2 * ds2;
} else {
km0 = (((1./6) * k3 * s + .5 * k2) * s + k1) * s + k0;
km1 = ((.5 * k3 * s + k2) * s + k1) * ds;
km2 = (k3 * s + k2) * ds2;
}
km3 = k3 * ds3;
#endif
{
#if ORDER == 4
double km0_2 = km0 * km0;
u = 24 - km0_2;
v = km1;
#endif
#if ORDER == 6
double km0_2 = km0 * km0;
double km0_4 = km0_2 * km0_2;
u = 24 - km0_2 + (km0_4 - 4 * km0 * km2 - 3 * km1 * km1) * (1./80);
v = km1 + (km3 - 6 * km0_2 * km1) * (1./80);
#endif
#if ORDER == 8
double t1_1 = km0;
double t1_2 = .5 * km1;
double t1_3 = (1./6) * km2;
double t1_4 = (1./24) * km3;
double t2_2 = t1_1 * t1_1;
double t2_3 = 2 * (t1_1 * t1_2);
double t2_4 = 2 * (t1_1 * t1_3) + t1_2 * t1_2;
double t2_5 = 2 * (t1_1 * t1_4 + t1_2 * t1_3);
double t2_6 = 2 * (t1_2 * t1_4) + t1_3 * t1_3;
double t3_4 = t2_2 * t1_2 + t2_3 * t1_1;
double t3_6 = t2_2 * t1_4 + t2_3 * t1_3 + t2_4 * t1_2 + t2_5 * t1_1;
double t4_4 = t2_2 * t2_2;
double t4_5 = 2 * (t2_2 * t2_3);
double t4_6 = 2 * (t2_2 * t2_4) + t2_3 * t2_3;
double t5_6 = t4_4 * t1_2 + t4_5 * t1_1;
double t6_6 = t4_4 * t2_2;
u = 1;
v = 0;
v += (1./12) * t1_2 + (1./80) * t1_4;
u -= (1./24) * t2_2 + (1./160) * t2_4 + (1./896) * t2_6;
v -= (1./480) * t3_4 + (1./2688) * t3_6;
u += (1./1920) * t4_4 + (1./10752) * t4_6;
v += (1./53760) * t5_6;
u -= (1./322560) * t6_6;
#endif
#if ORDER == 10
double t1_1 = km0;
double t1_2 = .5 * km1;
double t1_3 = (1./6) * km2;
double t1_4 = (1./24) * km3;
double t2_2 = t1_1 * t1_1;
double t2_3 = 2 * (t1_1 * t1_2);
double t2_4 = 2 * (t1_1 * t1_3) + t1_2 * t1_2;
double t2_5 = 2 * (t1_1 * t1_4 + t1_2 * t1_3);
double t2_6 = 2 * (t1_2 * t1_4) + t1_3 * t1_3;
double t2_7 = 2 * (t1_3 * t1_4);
double t2_8 = t1_4 * t1_4;
double t3_4 = t2_2 * t1_2 + t2_3 * t1_1;
double t3_6 = t2_2 * t1_4 + t2_3 * t1_3 + t2_4 * t1_2 + t2_5 * t1_1;
double t3_8 = t2_4 * t1_4 + t2_5 * t1_3 + t2_6 * t1_2 + t2_7 * t1_1;
double t4_4 = t2_2 * t2_2;
double t4_5 = 2 * (t2_2 * t2_3);
double t4_6 = 2 * (t2_2 * t2_4) + t2_3 * t2_3;
double t4_7 = 2 * (t2_2 * t2_5 + t2_3 * t2_4);
double t4_8 = 2 * (t2_2 * t2_6 + t2_3 * t2_5) + t2_4 * t2_4;
double t5_6 = t4_4 * t1_2 + t4_5 * t1_1;
double t5_8 = t4_4 * t1_4 + t4_5 * t1_3 + t4_6 * t1_2 + t4_7 * t1_1;
double t6_6 = t4_4 * t2_2;
double t6_7 = t4_4 * t2_3 + t4_5 * t2_2;
double t6_8 = t4_4 * t2_4 + t4_5 * t2_3 + t4_6 * t2_2;
double t7_8 = t6_6 * t1_2 + t6_7 * t1_1;
double t8_8 = t6_6 * t2_2;
u = 1;
v = 0;
v += (1./12) * t1_2 + (1./80) * t1_4;
u -= (1./24) * t2_2 + (1./160) * t2_4 + (1./896) * t2_6 + (1./4608) * t2_8;
v -= (1./480) * t3_4 + (1./2688) * t3_6 + (1./13824) * t3_8;
u += (1./1920) * t4_4 + (1./10752) * t4_6 + (1./55296) * t4_8;
v += (1./53760) * t5_6 + (1./276480) * t5_8;
u -= (1./322560) * t6_6 + (1./1.65888e+06) * t6_8;
v -= (1./1.16122e+07) * t7_8;
u += (1./9.28973e+07) * t8_8;
#endif
#if ORDER == 12
double t1_1 = km0;
double t1_2 = .5 * km1;
double t1_3 = (1./6) * km2;
double t1_4 = (1./24) * km3;
double t2_2 = t1_1 * t1_1;
double t2_3 = 2 * (t1_1 * t1_2);
double t2_4 = 2 * (t1_1 * t1_3) + t1_2 * t1_2;
double t2_5 = 2 * (t1_1 * t1_4 + t1_2 * t1_3);
double t2_6 = 2 * (t1_2 * t1_4) + t1_3 * t1_3;
double t2_7 = 2 * (t1_3 * t1_4);
double t2_8 = t1_4 * t1_4;
double t3_4 = t2_2 * t1_2 + t2_3 * t1_1;
double t3_6 = t2_2 * t1_4 + t2_3 * t1_3 + t2_4 * t1_2 + t2_5 * t1_1;
double t3_8 = t2_4 * t1_4 + t2_5 * t1_3 + t2_6 * t1_2 + t2_7 * t1_1;
double t3_10 = t2_6 * t1_4 + t2_7 * t1_3 + t2_8 * t1_2;
double t4_4 = t2_2 * t2_2;
double t4_5 = 2 * (t2_2 * t2_3);
double t4_6 = 2 * (t2_2 * t2_4) + t2_3 * t2_3;
double t4_7 = 2 * (t2_2 * t2_5 + t2_3 * t2_4);
double t4_8 = 2 * (t2_2 * t2_6 + t2_3 * t2_5) + t2_4 * t2_4;
double t4_9 = 2 * (t2_2 * t2_7 + t2_3 * t2_6 + t2_4 * t2_5);
double t4_10 = 2 * (t2_2 * t2_8 + t2_3 * t2_7 + t2_4 * t2_6) + t2_5 * t2_5;
double t5_6 = t4_4 * t1_2 + t4_5 * t1_1;
double t5_8 = t4_4 * t1_4 + t4_5 * t1_3 + t4_6 * t1_2 + t4_7 * t1_1;
double t5_10 = t4_6 * t1_4 + t4_7 * t1_3 + t4_8 * t1_2 + t4_9 * t1_1;
double t6_6 = t4_4 * t2_2;
double t6_7 = t4_4 * t2_3 + t4_5 * t2_2;
double t6_8 = t4_4 * t2_4 + t4_5 * t2_3 + t4_6 * t2_2;
double t6_9 = t4_4 * t2_5 + t4_5 * t2_4 + t4_6 * t2_3 + t4_7 * t2_2;
double t6_10 = t4_4 * t2_6 + t4_5 * t2_5 + t4_6 * t2_4 + t4_7 * t2_3 + t4_8 * t2_2;
double t7_8 = t6_6 * t1_2 + t6_7 * t1_1;
double t7_10 = t6_6 * t1_4 + t6_7 * t1_3 + t6_8 * t1_2 + t6_9 * t1_1;
double t8_8 = t6_6 * t2_2;
double t8_9 = t6_6 * t2_3 + t6_7 * t2_2;
double t8_10 = t6_6 * t2_4 + t6_7 * t2_3 + t6_8 * t2_2;
double t9_10 = t8_8 * t1_2 + t8_9 * t1_1;
double t10_10 = t8_8 * t2_2;
u = 1;
v = 0;
v += (1./12) * t1_2 + (1./80) * t1_4;
u -= (1./24) * t2_2 + (1./160) * t2_4 + (1./896) * t2_6 + (1./4608) * t2_8;
v -= (1./480) * t3_4 + (1./2688) * t3_6 + (1./13824) * t3_8 + (1./67584) * t3_10;
u += (1./1920) * t4_4 + (1./10752) * t4_6 + (1./55296) * t4_8 + (1./270336) * t4_10;
v += (1./53760) * t5_6 + (1./276480) * t5_8 + (1./1.35168e+06) * t5_10;
u -= (1./322560) * t6_6 + (1./1.65888e+06) * t6_8 + (1./8.11008e+06) * t6_10;
v -= (1./1.16122e+07) * t7_8 + (1./5.67706e+07) * t7_10;
u += (1./9.28973e+07) * t8_8 + (1./4.54164e+08) * t8_10;
v += (1./4.08748e+09) * t9_10;
u -= (1./4.08748e+10) * t10_10;
#endif
#if ORDER == 14
double t1_1 = km0;
double t1_2 = .5 * km1;
double t1_3 = (1./6) * km2;
double t1_4 = (1./24) * km3;
double t2_2 = t1_1 * t1_1;
double t2_3 = 2 * (t1_1 * t1_2);
double t2_4 = 2 * (t1_1 * t1_3) + t1_2 * t1_2;
double t2_5 = 2 * (t1_1 * t1_4 + t1_2 * t1_3);
double t2_6 = 2 * (t1_2 * t1_4) + t1_3 * t1_3;
double t2_7 = 2 * (t1_3 * t1_4);
double t2_8 = t1_4 * t1_4;
double t3_4 = t2_2 * t1_2 + t2_3 * t1_1;
double t3_6 = t2_2 * t1_4 + t2_3 * t1_3 + t2_4 * t1_2 + t2_5 * t1_1;
double t3_8 = t2_4 * t1_4 + t2_5 * t1_3 + t2_6 * t1_2 + t2_7 * t1_1;
double t3_10 = t2_6 * t1_4 + t2_7 * t1_3 + t2_8 * t1_2;
double t3_12 = t2_8 * t1_4;
double t4_4 = t2_2 * t2_2;
double t4_5 = 2 * (t2_2 * t2_3);
double t4_6 = 2 * (t2_2 * t2_4) + t2_3 * t2_3;
double t4_7 = 2 * (t2_2 * t2_5 + t2_3 * t2_4);
double t4_8 = 2 * (t2_2 * t2_6 + t2_3 * t2_5) + t2_4 * t2_4;
double t4_9 = 2 * (t2_2 * t2_7 + t2_3 * t2_6 + t2_4 * t2_5);
double t4_10 = 2 * (t2_2 * t2_8 + t2_3 * t2_7 + t2_4 * t2_6) + t2_5 * t2_5;
double t4_11 = 2 * (t2_3 * t2_8 + t2_4 * t2_7 + t2_5 * t2_6);
double t4_12 = 2 * (t2_4 * t2_8 + t2_5 * t2_7) + t2_6 * t2_6;
double t5_6 = t4_4 * t1_2 + t4_5 * t1_1;
double t5_8 = t4_4 * t1_4 + t4_5 * t1_3 + t4_6 * t1_2 + t4_7 * t1_1;
double t5_10 = t4_6 * t1_4 + t4_7 * t1_3 + t4_8 * t1_2 + t4_9 * t1_1;
double t5_12 = t4_8 * t1_4 + t4_9 * t1_3 + t4_10 * t1_2 + t4_11 * t1_1;
double t6_6 = t4_4 * t2_2;
double t6_7 = t4_4 * t2_3 + t4_5 * t2_2;
double t6_8 = t4_4 * t2_4 + t4_5 * t2_3 + t4_6 * t2_2;
double t6_9 = t4_4 * t2_5 + t4_5 * t2_4 + t4_6 * t2_3 + t4_7 * t2_2;
double t6_10 = t4_4 * t2_6 + t4_5 * t2_5 + t4_6 * t2_4 + t4_7 * t2_3 + t4_8 * t2_2;
double t6_11 = t4_4 * t2_7 + t4_5 * t2_6 + t4_6 * t2_5 + t4_7 * t2_4 + t4_8 * t2_3 + t4_9 * t2_2;
double t6_12 = t4_4 * t2_8 + t4_5 * t2_7 + t4_6 * t2_6 + t4_7 * t2_5 + t4_8 * t2_4 + t4_9 * t2_3 + t4_10 * t2_2;
double t7_8 = t6_6 * t1_2 + t6_7 * t1_1;
double t7_10 = t6_6 * t1_4 + t6_7 * t1_3 + t6_8 * t1_2 + t6_9 * t1_1;
double t7_12 = t6_8 * t1_4 + t6_9 * t1_3 + t6_10 * t1_2 + t6_11 * t1_1;
double t8_8 = t6_6 * t2_2;
double t8_9 = t6_6 * t2_3 + t6_7 * t2_2;
double t8_10 = t6_6 * t2_4 + t6_7 * t2_3 + t6_8 * t2_2;
double t8_11 = t6_6 * t2_5 + t6_7 * t2_4 + t6_8 * t2_3 + t6_9 * t2_2;
double t8_12 = t6_6 * t2_6 + t6_7 * t2_5 + t6_8 * t2_4 + t6_9 * t2_3 + t6_10 * t2_2;
double t9_10 = t8_8 * t1_2 + t8_9 * t1_1;
double t9_12 = t8_8 * t1_4 + t8_9 * t1_3 + t8_10 * t1_2 + t8_11 * t1_1;
double t10_10 = t8_8 * t2_2;
double t10_11 = t8_8 * t2_3 + t8_9 * t2_2;
double t10_12 = t8_8 * t2_4 + t8_9 * t2_3 + t8_10 * t2_2;
double t11_12 = t10_10 * t1_2 + t10_11 * t1_1;
double t12_12 = t10_10 * t2_2;
u = 1;
v = 0;
v += (1./12) * t1_2 + (1./80) * t1_4;
u -= (1./24) * t2_2 + (1./160) * t2_4 + (1./896) * t2_6 + (1./4608) * t2_8;
v -= (1./480) * t3_4 + (1./2688) * t3_6 + (1./13824) * t3_8 + (1./67584) * t3_10 + (1./319488) * t3_12;
u += (1./1920) * t4_4 + (1./10752) * t4_6 + (1./55296) * t4_8 + (1./270336) * t4_10 + (1./1.27795e+06) * t4_12;
v += (1./53760) * t5_6 + (1./276480) * t5_8 + (1./1.35168e+06) * t5_10 + (1./6.38976e+06) * t5_12;
u -= (1./322560) * t6_6 + (1./1.65888e+06) * t6_8 + (1./8.11008e+06) * t6_10 + (1./3.83386e+07) * t6_12;
v -= (1./1.16122e+07) * t7_8 + (1./5.67706e+07) * t7_10 + (1./2.6837e+08) * t7_12;
u += (1./9.28973e+07) * t8_8 + (1./4.54164e+08) * t8_10 + (1./2.14696e+09) * t8_12;
v += (1./4.08748e+09) * t9_10 + (1./1.93226e+10) * t9_12;
u -= (1./4.08748e+10) * t10_10 + (1./1.93226e+11) * t10_12;
v -= (1./2.12549e+12) * t11_12;
u += (1./2.55059e+13) * t12_12;
#endif
#if ORDER == 16
double t1_1 = km0;
double t1_2 = .5 * km1;
double t1_3 = (1./6) * km2;
double t1_4 = (1./24) * km3;
double t2_2 = t1_1 * t1_1;
double t2_3 = 2 * (t1_1 * t1_2);
double t2_4 = 2 * (t1_1 * t1_3) + t1_2 * t1_2;
double t2_5 = 2 * (t1_1 * t1_4 + t1_2 * t1_3);
double t2_6 = 2 * (t1_2 * t1_4) + t1_3 * t1_3;
double t2_7 = 2 * (t1_3 * t1_4);
double t2_8 = t1_4 * t1_4;
double t3_4 = t2_2 * t1_2 + t2_3 * t1_1;
double t3_6 = t2_2 * t1_4 + t2_3 * t1_3 + t2_4 * t1_2 + t2_5 * t1_1;
double t3_8 = t2_4 * t1_4 + t2_5 * t1_3 + t2_6 * t1_2 + t2_7 * t1_1;
double t3_10 = t2_6 * t1_4 + t2_7 * t1_3 + t2_8 * t1_2;
double t3_12 = t2_8 * t1_4;
double t4_4 = t2_2 * t2_2;
double t4_5 = 2 * (t2_2 * t2_3);
double t4_6 = 2 * (t2_2 * t2_4) + t2_3 * t2_3;
double t4_7 = 2 * (t2_2 * t2_5 + t2_3 * t2_4);
double t4_8 = 2 * (t2_2 * t2_6 + t2_3 * t2_5) + t2_4 * t2_4;
double t4_9 = 2 * (t2_2 * t2_7 + t2_3 * t2_6 + t2_4 * t2_5);
double t4_10 = 2 * (t2_2 * t2_8 + t2_3 * t2_7 + t2_4 * t2_6) + t2_5 * t2_5;
double t4_11 = 2 * (t2_3 * t2_8 + t2_4 * t2_7 + t2_5 * t2_6);
double t4_12 = 2 * (t2_4 * t2_8 + t2_5 * t2_7) + t2_6 * t2_6;
double t4_13 = 2 * (t2_5 * t2_8 + t2_6 * t2_7);
double t4_14 = 2 * (t2_6 * t2_8) + t2_7 * t2_7;
double t5_6 = t4_4 * t1_2 + t4_5 * t1_1;
double t5_8 = t4_4 * t1_4 + t4_5 * t1_3 + t4_6 * t1_2 + t4_7 * t1_1;
double t5_10 = t4_6 * t1_4 + t4_7 * t1_3 + t4_8 * t1_2 + t4_9 * t1_1;
double t5_12 = t4_8 * t1_4 + t4_9 * t1_3 + t4_10 * t1_2 + t4_11 * t1_1;
double t5_14 = t4_10 * t1_4 + t4_11 * t1_3 + t4_12 * t1_2 + t4_13 * t1_1;
double t6_6 = t4_4 * t2_2;
double t6_7 = t4_4 * t2_3 + t4_5 * t2_2;
double t6_8 = t4_4 * t2_4 + t4_5 * t2_3 + t4_6 * t2_2;
double t6_9 = t4_4 * t2_5 + t4_5 * t2_4 + t4_6 * t2_3 + t4_7 * t2_2;
double t6_10 = t4_4 * t2_6 + t4_5 * t2_5 + t4_6 * t2_4 + t4_7 * t2_3 + t4_8 * t2_2;
double t6_11 = t4_4 * t2_7 + t4_5 * t2_6 + t4_6 * t2_5 + t4_7 * t2_4 + t4_8 * t2_3 + t4_9 * t2_2;
double t6_12 = t4_4 * t2_8 + t4_5 * t2_7 + t4_6 * t2_6 + t4_7 * t2_5 + t4_8 * t2_4 + t4_9 * t2_3 + t4_10 * t2_2;
double t6_13 = t4_5 * t2_8 + t4_6 * t2_7 + t4_7 * t2_6 + t4_8 * t2_5 + t4_9 * t2_4 + t4_10 * t2_3 + t4_11 * t2_2;
double t6_14 = t4_6 * t2_8 + t4_7 * t2_7 + t4_8 * t2_6 + t4_9 * t2_5 + t4_10 * t2_4 + t4_11 * t2_3 + t4_12 * t2_2;
double t7_8 = t6_6 * t1_2 + t6_7 * t1_1;
double t7_10 = t6_6 * t1_4 + t6_7 * t1_3 + t6_8 * t1_2 + t6_9 * t1_1;
double t7_12 = t6_8 * t1_4 + t6_9 * t1_3 + t6_10 * t1_2 + t6_11 * t1_1;
double t7_14 = t6_10 * t1_4 + t6_11 * t1_3 + t6_12 * t1_2 + t6_13 * t1_1;
double t8_8 = t6_6 * t2_2;
double t8_9 = t6_6 * t2_3 + t6_7 * t2_2;
double t8_10 = t6_6 * t2_4 + t6_7 * t2_3 + t6_8 * t2_2;
double t8_11 = t6_6 * t2_5 + t6_7 * t2_4 + t6_8 * t2_3 + t6_9 * t2_2;
double t8_12 = t6_6 * t2_6 + t6_7 * t2_5 + t6_8 * t2_4 + t6_9 * t2_3 + t6_10 * t2_2;
double t8_13 = t6_6 * t2_7 + t6_7 * t2_6 + t6_8 * t2_5 + t6_9 * t2_4 + t6_10 * t2_3 + t6_11 * t2_2;
double t8_14 = t6_6 * t2_8 + t6_7 * t2_7 + t6_8 * t2_6 + t6_9 * t2_5 + t6_10 * t2_4 + t6_11 * t2_3 + t6_12 * t2_2;
double t9_10 = t8_8 * t1_2 + t8_9 * t1_1;
double t9_12 = t8_8 * t1_4 + t8_9 * t1_3 + t8_10 * t1_2 + t8_11 * t1_1;
double t9_14 = t8_10 * t1_4 + t8_11 * t1_3 + t8_12 * t1_2 + t8_13 * t1_1;
double t10_10 = t8_8 * t2_2;
double t10_11 = t8_8 * t2_3 + t8_9 * t2_2;
double t10_12 = t8_8 * t2_4 + t8_9 * t2_3 + t8_10 * t2_2;
double t10_13 = t8_8 * t2_5 + t8_9 * t2_4 + t8_10 * t2_3 + t8_11 * t2_2;
double t10_14 = t8_8 * t2_6 + t8_9 * t2_5 + t8_10 * t2_4 + t8_11 * t2_3 + t8_12 * t2_2;
double t11_12 = t10_10 * t1_2 + t10_11 * t1_1;
double t11_14 = t10_10 * t1_4 + t10_11 * t1_3 + t10_12 * t1_2 + t10_13 * t1_1;
double t12_12 = t10_10 * t2_2;
double t12_13 = t10_10 * t2_3 + t10_11 * t2_2;
double t12_14 = t10_10 * t2_4 + t10_11 * t2_3 + t10_12 * t2_2;
double t13_14 = t12_12 * t1_2 + t12_13 * t1_1;
double t14_14 = t12_12 * t2_2;
u = 1;
u -= 1./24 * t2_2 + 1./160 * t2_4 + 1./896 * t2_6 + 1./4608 * t2_8;
u += 1./1920 * t4_4 + 1./10752 * t4_6 + 1./55296 * t4_8 + 1./270336 * t4_10 + 1./1277952 * t4_12 + 1./5898240 * t4_14;
u -= 1./322560 * t6_6 + 1./1658880 * t6_8 + 1./8110080 * t6_10 + 1./38338560 * t6_12 + 1./176947200 * t6_14;
u += 1./92897280 * t8_8 + 1./454164480 * t8_10 + 4.6577500191e-10 * t8_12 + 1.0091791708e-10 * t8_14;
u -= 2.4464949595e-11 * t10_10 + 5.1752777990e-12 * t10_12 + 1.1213101898e-12 * t10_14;
u += 3.9206649992e-14 * t12_12 + 8.4947741650e-15 * t12_14;
u -= 4.6674583324e-17 * t14_14;
v = 0;
v += 1./12 * t1_2 + 1./80 * t1_4;
v -= 1./480 * t3_4 + 1./2688 * t3_6 + 1./13824 * t3_8 + 1./67584 * t3_10 + 1./319488 * t3_12;
v += 1./53760 * t5_6 + 1./276480 * t5_8 + 1./1351680 * t5_10 + 1./6389760 * t5_12 + 1./29491200 * t5_14;
v -= 1./11612160 * t7_8 + 1./56770560 * t7_10 + 1./268369920 * t7_12 + 8.0734333664e-10 * t7_14;
v += 2.4464949595e-10 * t9_10 + 5.1752777990e-11 * t9_12 + 1.1213101898e-11 * t9_14;
v -= 4.7047979991e-13 * t11_12 + 1.0193728998e-13 * t11_14;
v += 6.5344416654e-16 * t13_14;
#endif
}
if (n == 1) {
#if ORDER == 2
x = 1;
y = 0;
#else
x = u;
y = v;
#endif
} else {
double th = (((th4 * s + th3) * s + th2) * s + th1) * s;
double cth = cos(th);
double sth = sin(th);
#if ORDER == 2
x += cth;
y += sth;
#else
x += cth * u - sth * v;
y += cth * v + sth * u;
#endif
s += ds;
}
}
#if ORDER == 4 || ORDER == 6
xy[0] = x * (1./24 * ds);
xy[1] = y * (1./24 * ds);
#else
xy[0] = x * ds;
xy[1] = y * ds;
#endif
}
static void
set_dm_to_1(double *dm)
{
/* assume spiro within -0.5 to +0.5 */
/* values loaded: scale, xoff, yoff */
dm[0] = 1.; dm[1] = dm[2] = 0.;
}
static void
set_di_to_x1y1(double *di, double *dm, double x1, double y1)
{
/* assume IEEE 754 rounding errors. */
di[3] = di[4] = x1;
di[6] = di[7] = y1;
di[3] -= di[1]; di[4] += di[1];
di[6] -= di[1]; di[7] += di[1];
di[2] = x1 * dm[0] + dm[1];
di[5] = y1 * dm[0] + dm[2];
}
static double
compute_ends(const double ks[4], double ends[2][4], double seg_ch)
{
double xy[2];
double ch, th;
double l, l2, l3;
double th_even, th_odd;
double k0_even, k0_odd;
double k1_even, k1_odd;
double k2_even, k2_odd;
integrate_spiro(ks, xy, N_IS);
ch = hypot(xy[0], xy[1]);
th = atan2(xy[1], xy[0]);
l = ch / seg_ch;
th_even = .5 * ks[0] + (1./48) * ks[2];
th_odd = .125 * ks[1] + (1./384) * ks[3] - th;
ends[0][0] = th_even - th_odd;
ends[1][0] = th_even + th_odd;
k0_even = l * (ks[0] + .125 * ks[2]);
k0_odd = l * (.5 * ks[1] + (1./48) * ks[3]);
ends[0][1] = k0_even - k0_odd;
ends[1][1] = k0_even + k0_odd;
l2 = l * l;
k1_even = l2 * (ks[1] + .125 * ks[3]);
k1_odd = l2 * .5 * ks[2];
ends[0][2] = k1_even - k1_odd;
ends[1][2] = k1_even + k1_odd;
l3 = l2 * l;
k2_even = l3 * ks[2];
k2_odd = l3 * .5 * ks[3];
ends[0][3] = k2_even - k2_odd;
ends[1][3] = k2_even + k2_odd;
return l;
}
static void
compute_pderivs(const spiro_seg *s, double ends[2][4], double derivs[4][2][4],
int jinc)
{
double recip_d = 2e6;
double delta = 1./ recip_d;
double try_ks[4];
double try_ends[2][4];
int i, j, k;
compute_ends(s->ks, ends, s->seg_ch);
for (i = 0; i < jinc; i++) {
for (j = 0; j < 4; j++)
try_ks[j] = s->ks[j];
try_ks[i] += delta;
compute_ends(try_ks, try_ends, s->seg_ch);
for (k = 0; k < 2; k++)
for (j = 0; j < 4; j++)
derivs[j][k][i] = recip_d * (try_ends[k][j] - ends[k][j]);
}
}
static double
mod_2pi(double th)
{
double u = th / (2 * M_PI);
return 2 * M_PI * (u - floor(u + 0.5));
}
static spiro_seg *
setup_path0(const spiro_cp *src, double *dm, int n)
{
int i, ilast, n_seg, z;
double dx, dy;
double xmin, xmax, ymin, ymax;
spiro_seg *r;
z = -1;
if (src[n - 1].ty == 'z') z = --n;
if (src[0].ty == ']' || src[n - 1].ty == '[') { /* pair */
#ifdef VERBOSE
fprintf(stderr, "ERROR: LibSpiro: cannot use cp type ']' as start, or '[' as end.\n");
#endif
return 0;
}
if (src[0].ty == 'h' || src[n - 1].ty == 'a') { /* pair */
#ifdef VERBOSE
fprintf(stderr, "ERROR: LibSpiro: cannot use cp type 'h' as start, or 'a' as end.\n");
#endif
return 0;
}
#ifdef CHECK_INPUT_FINITENESS
/* Verify that input values are within realistic limits */
for (i = 0; i < n; i++) {
if (IS_FINITE(src[i].x)==0 || IS_FINITE(src[i].y)==0) {
#ifdef VERBOSE
fprintf(stderr, "ERROR: LibSpiro: #%d={'%c',%g,%g} is not finite.\n", \
i, src[i].ty, src[i].x, src[i].y);
#endif
return 0;
}
}
#endif
n_seg = src[0].ty == '{' ? n - 1 : n;
i = (int)((unsigned int)(n_seg + 1) * sizeof(spiro_seg));
if (i <= 0 || (r=(spiro_seg *)malloc((unsigned int)(i))) == NULL) return 0;
if (dm[0] < 0.9) {
/* for math to be scalable fit it within -0.5..+0.5 */
xmin = xmax = src[0].x; ymin = ymax = src[0].y;
for (i = 0; i < n_seg; i++) {
if (src[i].ty != 'z' && src[i].ty != 'h') {
if (src[i].x < xmin) xmin = src[i].x; else
if (src[i].x > xmax) xmax = src[i].x;
if (src[i].y < ymin) ymin = src[i].y; else
if (src[i].y > ymax) ymax = src[i].y;
}
}
dm[1] /* xoff */ = (xmin + xmax) / 2; xmax -= xmin;
dm[2] /* yoff */ = (ymin + ymax) / 2; ymax -= ymin;
dm[0] /* scale */ = fabs((fabs(xmax) >= fabs(ymax)) ? xmax : ymax);
dm[0] /* scale */ /= 500.; /* ~ backward compatible */
}
#ifdef VERBOSE
printf("scale=%g, x_offset=%g, y_offset=%g, n=%d, n_seg=%d\n", dm[0], dm[1], dm[2], n, n_seg);
#endif
for (i = 0; i < n_seg; i++) {
/* gigo test: error if src[i].ty isn't a known type */
if (src[i].ty == 'a') {
if (src[i + 1].ty == 'h' || (i == n_seg-1 && src[i + 1].ty == '}')) {
if (src[i].x == src[i + 1].x && src[i].y == src[i + 1].y)
goto setup_path_error1;
} else
/* did not find 'ah' (or 'a}' as last pair) */
goto setup_path_error1;
} else if (src[i].ty == 'h') {
if (src[i - 1].ty == 'a' || (i == 1 || src[0].ty == '{')) {
if (src[i - 1].x == src[i].x && src[i - 1].y == src[i].y)
goto setup_path_error1;
} else
/* didn't find 'ah' (or '{h' as first pair) */
goto setup_path_error1;
}
r[i].ty = src[i].ty;
r[i].x = (src[i].x - dm[1]) / dm[0];
r[i].y = (src[i].y - dm[2]) / dm[0];
r[i].ks[0] = 0.;
r[i].ks[1] = 0.;
r[i].ks[2] = 0.;
r[i].ks[3] = 0.;
}
r[n_seg].x = (src[n_seg % n].x - dm[1]) / dm[0];
r[n_seg].y = (src[n_seg % n].y - dm[2]) / dm[0];
r[n_seg].ty = src[n_seg % n].ty;
for (i = 0; i < n_seg; i++) {
if (r[i].ty == 'h' || (i == n_seg-1 && i > 0 && r[i].ty == '}' && r[i - 1].ty == 'a')) {
/* behave like a disconnected pair of '[' & ']' */
/* point 'a' holds vector to old 'h' and now we */
/* change x,y here to be the same as point 'a'. */
/* curve fitting is based on vectors and angles */
/* but final curves will be based on x,y points */
r[i].x = r[i - 1].x;
r[i].y = r[i - 1].y;
}
dx = r[i + 1].x - r[i].x;
dy = r[i + 1].y - r[i].y;
#ifndef CHECK_INPUT_FINITENESS
r[i].seg_ch = hypot(dx, dy);
#else
if (IS_FINITE(dx)==0 || IS_FINITE(dy)==0 || \
IS_FINITE((r[i].seg_ch = hypot(dx, dy)))==0) {
#ifdef VERBOSE
fprintf(stderr, "ERROR: LibSpiro: #%d={'%c',%g,%g} hypot error.\n", \
i, src[i].ty, src[i].x, src[i].y);
#endif
goto setup_path_error0;
}
#endif
r[i].seg_th = atan2(dy, dx);
}
ilast = n_seg - 1;
for (i = 0; i < n_seg; i++) {
if (r[i].ty == '{' || r[i].ty == '}' || r[i].ty == 'v')
r[i].bend_th = 0.;
else
r[i].bend_th = mod_2pi(r[i].seg_th - r[ilast].seg_th);
ilast = i;
#ifdef VERBOSE
printf("input #%d={'%c',%g=>%g,%g=>%g}, hypot=%g, atan2=%g, bend_th=%g\n", \
i, src[i].ty, src[i].x, r[i].x * dm[0] + dm[1], \
src[i].y, r[i].y * dm[0] + dm[2], r[i].seg_ch * dm[0], \
r[i].seg_th, r[i].bend_th);
#endif
}
#ifdef VERBOSE
if (n_seg < n)
printf("input #%d={'%c',%g=>%g,%g=>%g}\n", i, src[i].ty, \
src[i].x, r[i].x * dm[0] + dm[1], src[i].y, r[i].y * dm[0] + dm[2]);
#endif
if (z >= 0) r[z].ty = 'z'; /* wrong n, maintain z. */
return r;
setup_path_error1:
#ifdef VERBOSE
fprintf(stderr, "ERROR: LibSpiro: #%d={'%c',%g,%g} found unpaired anchor+handle 'ah'.\n", \
i, src[i].ty, src[i].x, src[i].y);
#endif
#ifdef CHECK_INPUT_FINITENESS
setup_path_error0:
#endif
free(r);
return 0;
}
/* deprecated / backwards compatibility / not scalable */
static spiro_seg *
setup_path(const spiro_cp *src, int n)
{
double dm[6];
set_dm_to_1(dm);
return setup_path0(src, dm, n);
}
static void
bandec11(bandmat *m, int *perm, int n)
{
int i, j, k, l, pivot;
double pivot_val, pivot_scale, tmp, x;
/* pack top triangle to the left. */
for (i = 0; i < 5; i++) {
for (j = 0; j < i + 6; j++)
m[i].a[j] = m[i].a[j + 5 - i];
for (; j < 11; j++)
m[i].a[j] = 0.;
}
l = 5;
for (k = 0; k < n; k++) {
pivot = k;
pivot_val = m[k].a[0];
l = l < n ? l + 1 : n;
for (j = k + 1; j < l; j++)
if (fabs(m[j].a[0]) > fabs(pivot_val)) {
pivot_val = m[j].a[0];
pivot = j;
}
perm[k] = pivot;
if (pivot != k) {
for (j = 0; j < 11; j++) {
tmp = m[k].a[j];
m[k].a[j] = m[pivot].a[j];
m[pivot].a[j] = tmp;
}
}
if (fabs(pivot_val) < 1e-12) pivot_val = 1e-12;
pivot_scale = 1. / pivot_val;
for (i = k + 1; i < l; i++) {
x = m[i].a[0] * pivot_scale;
m[k].al[i - k - 1] = x;
for (j = 1; j < 11; j++)
m[i].a[j - 1] = m[i].a[j] - x * m[k].a[j];
m[i].a[10] = 0.;
}
}
}
static void
banbks11(const bandmat *m, const int *perm, double *v, int n)
{
int i, k, l;
double tmp, x;
/* forward substitution */
l = 5;
for (k = 0; k < n; k++) {
i = perm[k];
if (i != k) {
tmp = v[k];
v[k] = v[i];
v[i] = tmp;
}
if (l < n) l++;
for (i = k + 1; i < l; i++)
v[i] -= m[k].al[i - k - 1] * v[k];
}
/* back substitution */
l = 1;
for (i = n - 1; i >= 0; i--) {
x = v[i];
for (k = 1; k < l; k++)
x -= m[i].a[k] * v[k + i];
v[i] = x / m[i].a[0];
if (l < 11) l++;
}
}
static int compute_jinc(char ty0, char ty1)
{
if (ty0 == 'o' || ty1 == 'o' || \
ty0 == ']' || ty1 == '[' || \
ty0 == 'h' || ty1 == 'a')
return 4;
else if (ty0 == 'c' && ty1 == 'c')
return 2;
else if ((ty1 == 'c' && (ty0 == '{' || ty0 == 'v' || ty0 == '[' || ty0 == 'a')) ||
(ty0 == 'c' && (ty1 == '}' || ty1 == 'v' || ty1 == ']' || ty1 == 'h')))
return 1;
else
return 0;
}
static int count_vec(const spiro_seg *s, int *jinca, int nseg)
{
int i, n;
n = 0;
for (i = 0; i < nseg; i++)
n += (jinca[i] = compute_jinc(s[i].ty, s[i + 1].ty));
return n;
}
static void
add_mat_line(bandmat *m, double *v,double derivs[4],
double x, double y, int j, int jj, int jinc, int nmat)
{
int joff, k;
if (jj >= 0) {
jj %= nmat;
joff = (j + 5 - jj + nmat) % nmat;
if (nmat < 6) {
joff = j + 5 - jj;
} else if (nmat == 6) {
joff = 2 + (j + 3 - jj + nmat) % nmat;
}
#ifdef VERBOSE
printf("add_mat_line j=%d jj=%d jinc=%d nmat=%d joff=%d\n", j, jj, jinc, nmat, joff);
#endif
v[jj] += x;
for (k = 0; k < jinc; k++)
m[jj].a[joff + k] += y * derivs[k];
}
}
static double
spiro_iter(spiro_seg *s, bandmat *m, int *perm, double *v, int *jinca, int n, int cyclic, int nmat)
{
unsigned int l;
int i, j, jthl, jthr, jk0l, jk0r, jk1l, jk1r, jk2l, jk2r, jinc, jj, k, n_invert;
char ty0, ty1;
double dk, norm, th;
double ends[2][4];
double derivs[4][2][4];
for (i = 0; i < nmat; i++) {
v[i] = 0.;
for (j = 0; j < 11; j++)
m[i].a[j] = 0.;
for (j = 0; j < 5; j++)
m[i].al[j] = 0.;
}
i = j = jj = 0;
if (s[0].ty == 'o')
jj = nmat - 2;
else if (s[0].ty == 'c')
jj = nmat - 1;
else if (s[0].ty == '[' || s[0].ty == 'a') {
if (cyclic) {
/* start at v, c or o */
for (i = 0; i < n; i++) {
switch (s[i].ty) {
case 'o':
--jj;
case 'c':
--jj;
case 'v':
jj = (jj + nmat) % nmat;
j %= nmat;
goto spiro_iter_1;
break;
default:
jj += jinca[i];
j += jinca[i];
}
}
i = j = jj = 0;
spiro_iter_1: ;
}
}
for (k = 0; k < n; i++, k++) {
i %= n;
ty0 = s[i].ty;
ty1 = s[i + 1].ty;
jinc = jinca[i];
th = s[i].bend_th;
jthl = jk0l = jk1l = jk2l = -1;
jthr = jk0r = jk1r = jk2r = -1;
compute_pderivs(&s[i], ends, derivs, jinc);
/* constraints crossing left */
if (ty0 == 'o' || ty0 == 'c' || ty0 == '[' || ty0 == ']' || \
ty0 == 'a' || ty0 == 'h') {
jthl = jj++;
jthl %= nmat;
jj %= nmat;
jk0l = jj++;
if (ty0 == 'o') {
jj %= nmat;
jk1l = jj++;
jk2l = jj++;
}
}
if (jinc == 4) {
/* constraints on left */
if (ty0 == 'c' || ty0 == 'v' || ty0 == '[' || \
ty0 == 'a' || ty0 == '{') {
if (ty0 != 'c')
jk1l = jj++;
jk2l = jj++;
}
/* constraints on right */
if (ty1 == 'c' || ty1 == 'v' || ty1 == ']' || \
ty1 == 'h' || ty1 == '}') {
if (ty1 != 'c')
jk1r = jj++;
jk2r = jj++;
}
}
/* constraints crossing right */
if (ty1 == 'o' || ty1 == 'c' || ty1 == '[' || ty1 == ']' || \
ty1 == 'a' || ty1 == 'h') {
jj %= nmat;
jthr = jj;
jk0r = (jj + 1) % nmat;
if (ty1 == 'o') {
jk1r = (jj + 2) % nmat;
jk2r = (jj + 3) % nmat;
}
}
add_mat_line(m, v, derivs[0][0], th - ends[0][0], 1, j, jthl, jinc, nmat);
add_mat_line(m, v, derivs[1][0], ends[0][1], -1, j, jk0l, jinc, nmat);
add_mat_line(m, v, derivs[2][0], ends[0][2], -1, j, jk1l, jinc, nmat);
add_mat_line(m, v, derivs[3][0], ends[0][3], -1, j, jk2l, jinc, nmat);
add_mat_line(m, v, derivs[0][1], -ends[1][0], 1, j, jthr, jinc, nmat);
add_mat_line(m, v, derivs[1][1], -ends[1][1], 1, j, jk0r, jinc, nmat);
add_mat_line(m, v, derivs[2][1], -ends[1][2], 1, j, jk1r, jinc, nmat);
add_mat_line(m, v, derivs[3][1], -ends[1][3], 1, j, jk2r, jinc, nmat);
if (jthl >= 0)
v[jthl] = mod_2pi(v[jthl]);
if (jthr >= 0)
v[jthr] = mod_2pi(v[jthr]);
j += jinc;
j %= nmat;
}
if (cyclic) {
l = sizeof(bandmat) * (unsigned int)(nmat);
memcpy(m + nmat, m, l);
memcpy(m + 2 * nmat, m, l);
l = sizeof(double) * (unsigned int)(nmat);
memcpy(v + nmat, v, l);
memcpy(v + 2 * nmat, v, l);
n_invert = 3 * nmat;
j = nmat;
#ifdef VERBOSE
printf("cyclic\n");
#endif
} else {
n_invert = nmat;
j = 0;
}
#ifdef VERBOSE
for (i = 0; i < n; i++) {
for (k = 0; k < 11; k++)
printf(" %2.4f", m[i].a[k]);
printf(": %2.4f\n", v[i]);
}
printf("---\n");
#endif
bandec11(m, perm, n_invert);
banbks11(m, perm, v, n_invert);
norm = 0.;
for (i = 0; i < n; i++) {
jinc = jinca[i];
for (k = 0; k < jinc; k++) {
dk = v[j++];
#ifdef VERBOSE
printf("s[%d].ks[%d] += %f\n", i, k, dk);
#endif
s[i].ks[k] += dk;
norm += dk * dk;
/* Break if calculations are headed for failure */
if (IS_FINITE(s[i].ks[k]) == 0) return s[i].ks[k];
}
s[i].ks[0] = 2.0 * mod_2pi(s[i].ks[0]/2.0);
}
return norm;
}
static int
solve_spiro(spiro_seg *s, int n)
{
int i, converged, cyclic, nmat, n_alloc, nseg, z;
bandmat *m;
double *v;
int *perm, *jinca;
double norm;
i = converged = 0; /* not solved (yet) */
z = -1;
if (s[0].ty == '{')
nseg = n - 1;
else {
if (s[n - 1].ty == 'z') {
z = --n;
s[z].ty = s[0].ty;
}
nseg = n;
}
if (nseg <= 1) {
converged = 1; /* means no convergence problems */
goto solve_spiroerr0;
}
if ((jinca = (int *)malloc((unsigned int)(sizeof(int) * nseg))) == NULL) {
#ifdef VERBOSE
fprintf(stderr, "ERROR: LibSpiro: failed to alloc memory.\n");
#endif
goto solve_spiroerr0;
}
nmat = count_vec(s, jinca, nseg);
if (nmat == 0) {