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-// Begin License:
-// Copyright (C) 2006-2014 Tobias Sargeant (tobias.sargeant@gmail.com).
-// All rights reserved.
-//
-// This file is part of the Carve CSG Library (http://carve-csg.com/)
-//
-// This file may be used under the terms of either the GNU General
-// Public License version 2 or 3 (at your option) as published by the
-// Free Software Foundation and appearing in the files LICENSE.GPL2
-// and LICENSE.GPL3 included in the packaging of this file.
-//
-// This file is provided "AS IS" with NO WARRANTY OF ANY KIND,
-// INCLUDING THE WARRANTIES OF DESIGN, MERCHANTABILITY AND FITNESS FOR
-// A PARTICULAR PURPOSE.
-// End:
-
-
-#if defined(HAVE_CONFIG_H)
-# include <carve_config.h>
-#endif
-
-#include <carve/math.hpp>
-#include <carve/matrix.hpp>
-
-#include <iostream>
-#include <limits>
-
-#include <stdio.h>
-
-#define M_2PI_3 2.0943951023931953
-#define M_SQRT_3_4 0.8660254037844386
-#define EPS std::numeric_limits<double>::epsilon()
-
-namespace carve {
- namespace math {
-
- struct Root {
- double root;
- int multiplicity;
-
- Root(double r) : root(r), multiplicity(1) {}
- Root(double r, int m) : root(r), multiplicity(m) {}
- };
-
- namespace {
-#if 0
- void cplx_sqrt(double re, double im,
- double &re_1, double &im_1,
- double &re_2, double &im_2) {
- if (re == 0.0 && im == 0.0) {
- re_1 = re_2 = re;
- im_1 = im_2 = im;
- } else {
- double d = sqrt(re * re + im * im);
- re_1 = sqrt((d + re) / 2.0);
- re_2 = re_1;
- im_1 = fabs(sqrt((d - re) / 2.0));
- im_2 = -im_1;
- }
- }
-#endif
-
-#if 0
- void cplx_cbrt(double re, double im,
- double &re_1, double &im_1,
- double &re_2, double &im_2,
- double &re_3, double &im_3) {
- if (re == 0.0 && im == 0.0) {
- re_1 = re_2 = re_3 = re;
- im_1 = im_2 = im_3 = im;
- } else {
- double r = cbrt(sqrt(re * re + im * im));
- double t = atan2(im, re) / 3.0;
- re_1 = r * cos(t);
- im_1 = r * sin(t);
- re_2 = r * cos(t + M_TWOPI / 3.0);
- im_2 = r * sin(t + M_TWOPI / 3.0);
- re_3 = r * cos(t + M_TWOPI * 2.0 / 3.0);
- im_3 = r * sin(t + M_TWOPI * 2.0 / 3.0);
- }
- }
-#endif
-
- void add_root(std::vector<Root> &roots, double root) {
- for (size_t i = 0; i < roots.size(); ++i) {
- if (roots[i].root == root) {
- roots[i].multiplicity++;
- return;
- }
- }
- roots.push_back(Root(root));
- }
-
- void linear_roots(double c1, double c0, std::vector<Root> &roots) {
- roots.push_back(Root(c0 / c1));
- }
-
- void quadratic_roots(double c2, double c1, double c0, std::vector<Root> &roots) {
- if (fabs(c2) < EPS) {
- linear_roots(c1, c0, roots);
- return;
- }
-
- double p = 0.5 * c1 / c2;
- double dis = p * p - c0 / c2;
-
- if (dis > 0.0) {
- dis = sqrt(dis);
- if (-p - dis != -p + dis) {
- roots.push_back(Root(-p - dis));
- roots.push_back(Root(-p + dis));
- } else {
- roots.push_back(Root(-p, 2));
- }
- }
- }
-
- void cubic_roots(double c3, double c2, double c1, double c0, std::vector<Root> &roots) {
- int n_sol = 0;
- double _r[3];
-
- if (fabs(c3) < EPS) {
- quadratic_roots(c2, c1, c0, roots);
- return;
- }
-
- if (fabs(c0) < EPS) {
- quadratic_roots(c3, c2, c1, roots);
- add_root(roots, 0.0);
- return;
- }
-
- double xN = -c2 / (3.0 * c3);
- double yN = c0 + xN * (c1 + xN * (c2 + c3 * xN));
-
- double delta_sq = (c2 * c2 - 3.0 * c3 * c1) / (9.0 * c3 * c3);
- double h_sq = 4.0 / 9.0 * (c2 * c2 - 3.0 * c3 * c1) * (delta_sq * delta_sq);
- double dis = yN * yN - h_sq;
-
- if (dis > EPS) {
- // One real root, two complex roots.
-
- double dis_sqrt = sqrt(dis);
- double r_p = yN - dis_sqrt;
- double r_q = yN + dis_sqrt;
- double p = cbrt(fabs(r_p)/(2.0 * c3));
- double q = cbrt(fabs(r_q)/(2.0 * c3));
-
- if (r_p > 0.0) p = -p;
- if (r_q > 0.0) q = -q;
-
- _r[0] = xN + p + q;
- n_sol = 1;
-
- double re = xN - p * .5 - q * .5;
- double im = p * M_SQRT_3_4 - q * M_SQRT_3_4;
-
- // root 2: xN + p * exp(M_2PI_3.i) + q * exp(-M_2PI_3.i);
- // root 3: complex conjugate of root 2
-
- if (im < EPS) {
- _r[1] = _r[2] = re;
- n_sol += 2;
- }
- } else if (dis < -EPS) {
- // Three distinct real roots.
- double theta = acos(-yN / sqrt(h_sq)) / 3.0;
- double delta = sqrt(c2 * c2 - 3.0 * c3 * c1) / (3.0 * c3);
-
- _r[0] = xN + (2.0 * delta) * cos(theta);
- _r[1] = xN + (2.0 * delta) * cos(M_2PI_3 - theta);
- _r[2] = xN + (2.0 * delta) * cos(M_2PI_3 + theta);
- n_sol = 3;
- } else {
- // Three real roots (two or three equal).
- double r = yN / (2.0 * c3);
- double delta = cbrt(r);
-
- _r[0] = xN + delta;
- _r[1] = xN + delta;
- _r[2] = xN - 2.0 * delta;
- n_sol = 3;
- }
-
- for (int i=0; i < n_sol; i++) {
- add_root(roots, _r[i]);
- }
- }
- }
-
- static void U(const Matrix3 &m,
- double l,
- double u[6],
- double &u_max,
- int &u_argmax) {
- u[0] = (m._22 - l) * (m._33 - l) - m._23 * m._23;
- u[1] = m._13 * m._23 - m._12 * (m._33 - l);
- u[2] = m._12 * m._23 - m._13 * (m._22 - l);
- u[3] = (m._11 - l) * (m._33 - l) - m._13 * m._13;
- u[4] = m._12 * m._13 - m._23 * (m._11 - l);
- u[5] = (m._11 - l) * (m._22 - l) - m._12 * m._12;
-
- u_max = -1.0;
- u_argmax = -1;
-
- for (int i = 0; i < 6; ++i) {
- if (u_max < fabs(u[i])) { u_max = fabs(u[i]); u_argmax = i; }
- }
- }
-
- static void eig1(const Matrix3 &m, double l, carve::geom::vector<3> &e) {
- double u[6];
- double u_max;
- int u_argmax;
-
- U(m, l, u, u_max, u_argmax);
-
- switch(u_argmax) {
- case 0:
- e.x = u[0]; e.y = u[1]; e.z = u[2]; break;
- case 1: case 3:
- e.x = u[1]; e.y = u[3]; e.z = u[4]; break;
- case 2: case 4: case 5:
- e.x = u[2]; e.y = u[4]; e.z = u[5]; break;
- }
- e.normalize();
- }
-
- static void eig2(const Matrix3 &m, double l, carve::geom::vector<3> &e1, carve::geom::vector<3> &e2) {
- double u[6];
- double u_max;
- int u_argmax;
-
- U(m, l, u, u_max, u_argmax);
-
- switch(u_argmax) {
- case 0: case 1:
- e1.x = -m._12; e1.y = m._11; e1.z = 0.0;
- e2.x = -m._13 * m._11; e2.y = -m._13 * m._12; e2.z = m._11 * m._11 + m._12 * m._12;
- break;
- case 2:
- e1.x = m._12; e1.y = 0.0; e1.z = -m._11;
- e2.x = -m._12 * m._11; e2.y = m._11 * m._11 + m._13 * m._13; e2.z = -m._12 * m._13;
- break;
- case 3: case 4:
- e1.x = 0.0; e1.y = -m._23; e1.z = -m._22;
- e2.x = m._22 * m._22 + m._23 * m._23; e2.y = -m._12 * m._22; e2.z = -m._12 * m._23;
- break;
- case 5:
- e1.x = 0.0; e1.y = -m._33; e1.z = m._23;
- e2.x = m._23 * m._23 + m._33 * m._33; e2.y = -m._13 * m._23; e2.z = -m._13 * m._33;
- }
- e1.normalize();
- e2.normalize();
- }
-
-#if 0
- static void eig3(const Matrix3 &m,
- double l,
- carve::geom::vector<3> &e1,
- carve::geom::vector<3> &e2,
- carve::geom::vector<3> &e3) {
- e1.x = 1.0; e1.y = 0.0; e1.z = 0.0;
- e2.x = 0.0; e2.y = 1.0; e2.z = 0.0;
- e3.x = 0.0; e3.y = 0.0; e3.z = 1.0;
- }
-#endif
-
- void eigSolveSymmetric(const Matrix3 &m,
- double &l1, carve::geom::vector<3> &e1,
- double &l2, carve::geom::vector<3> &e2,
- double &l3, carve::geom::vector<3> &e3) {
- double c0 =
- m._11 * m._22 * m._33 +
- 2.0 * m._12 * m._13 * m._23 -
- m._11 * m._23 * m._23 -
- m._22 * m._13 * m._13 -
- m._33 * m._12 * m._12;
- double c1 =
- m._11 * m._22 -
- m._12 * m._12 +
- m._11 * m._33 -
- m._13 * m._13 +
- m._22 * m._33 -
- m._23 * m._23;
- double c2 =
- m._11 +
- m._22 +
- m._33;
-
- double a = (3.0 * c1 - c2 * c2) / 3.0;
- double b = (-2.0 * c2 * c2 * c2 + 9.0 * c1 * c2 - 27.0 * c0) / 27.0;
-
- double Q = b * b / 4.0 + a * a * a / 27.0;
-
- if (fabs(Q) < 1e-16) {
- l1 = m._11; e1.x = 1.0; e1.y = 0.0; e1.z = 0.0;
- l2 = m._22; e2.x = 0.0; e2.y = 1.0; e2.z = 0.0;
- l3 = m._33; e3.x = 0.0; e3.y = 0.0; e3.z = 1.0;
- } else if (Q > 0) {
- l1 = l2 = c2 / 3.0 + cbrt(b / 2.0);
- l3 = c2 / 3.0 - 2.0 * cbrt(b / 2.0);
-
- eig2(m, l1, e1, e2);
- eig1(m, l3, e3);
- } else if (Q < 0) {
- double t = atan2(sqrt(-Q), -b / 2.0);
- double cos_t3 = cos(t / 3.0);
- double sin_t3 = sin(t / 3.0);
- double r = cbrt(sqrt(b * b / 4.0 - Q));
-
- l1 = c2 / 3.0 + 2 * r * cos_t3;
- l2 = c2 / 3.0 - r * (cos_t3 + M_SQRT_3 * sin_t3);
- l3 = c2 / 3.0 - r * (cos_t3 - M_SQRT_3 * sin_t3);
-
- eig1(m, l1, e1);
- eig1(m, l2, e2);
- eig1(m, l3, e3);
- }
- }
-
- void eigSolve(const Matrix3 &m, double &l1, double &l2, double &l3) {
- double c3, c2, c1, c0;
- std::vector<Root> roots;
-
- c3 = -1.0;
- c2 = m._11 + m._22 + m._33;
- c1 =
- -(m._22 * m._33 + m._11 * m._22 + m._11 * m._33)
- +(m._23 * m._32 + m._13 * m._31 + m._12 * m._21);
- c0 =
- +(m._11 * m._22 - m._12 * m._21) * m._33
- -(m._11 * m._23 - m._13 * m._21) * m._32
- +(m._12 * m._23 - m._13 * m._22) * m._31;
-
- cubic_roots(c3, c2, c1, c0, roots);
-
- for (size_t i = 0; i < roots.size(); i++) {
- Matrix3 M(m);
- M._11 -= roots[i].root;
- M._22 -= roots[i].root;
- M._33 -= roots[i].root;
- // solve M.v = 0
- }
-
- std::cerr << "n_roots=" << roots.size() << std::endl;
- for (size_t i = 0; i < roots.size(); i++) {
- fprintf(stderr, " %.24f(%d)", roots[i].root, roots[i].multiplicity);
- }
- std::cerr << std::endl;
- }
-
- }
-}
-
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