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+// Begin License:
+// Copyright (C) 2006-2011 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 the GNU General Public
+// License version 2.0 as published by the Free Software Foundation
+// and appearing in the file LICENSE.GPL2 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) {}
+ };
+
+ 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;
+ }
+ }
+
+ 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);
+ }
+ }
+
+ 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();
+ }
+
+ 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;
+ }
+
+ 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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