forked from bartvdbraak/blender
356 lines
13 KiB
C++
356 lines
13 KiB
C++
|
/*
|
||
|
|
* Copyright (c) 2005 Erwin Coumans http://www.erwincoumans.com
|
||
|
|
*
|
||
|
|
* Permission to use, copy, modify, distribute and sell this software
|
||
|
|
* and its documentation for any purpose is hereby granted without fee,
|
||
|
|
* provided that the above copyright notice appear in all copies.
|
||
|
|
* Erwin Coumans makes no representations about the suitability
|
||
|
|
* of this software for any purpose.
|
||
|
|
* It is provided "as is" without express or implied warranty.
|
||
|
|
*/
|
||
|
|
|
||
|
|
#include "BU_AlgebraicPolynomialSolver.h"
|
||
|
|
#include <math.h>
|
||
|
|
#include <SimdMinMax.h>
|
||
|
|
|
||
|
|
int BU_AlgebraicPolynomialSolver::Solve2Quadratic(SimdScalar p, SimdScalar q)
|
||
|
|
{
|
||
|
|
|
||
|
|
SimdScalar basic_h_local;
|
||
|
|
SimdScalar basic_h_local_delta;
|
||
|
|
|
||
|
|
basic_h_local = p * 0.5f;
|
||
|
|
basic_h_local_delta = basic_h_local * basic_h_local - q;
|
||
|
|
if (basic_h_local_delta > 0.0f) {
|
||
|
|
basic_h_local_delta = sqrtf(basic_h_local_delta);
|
||
|
|
m_roots[0] = - basic_h_local + basic_h_local_delta;
|
||
|
|
m_roots[1] = - basic_h_local - basic_h_local_delta;
|
||
|
|
return 2;
|
||
|
|
}
|
||
|
|
else if (SimdGreaterEqual(basic_h_local_delta, SIMD_EPSILON)) {
|
||
|
|
m_roots[0] = - basic_h_local;
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
int BU_AlgebraicPolynomialSolver::Solve2QuadraticFull(SimdScalar a,SimdScalar b, SimdScalar c)
|
||
|
|
{
|
||
|
|
SimdScalar radical = b * b - 4.0f * a * c;
|
||
|
|
if(radical >= 0.f)
|
||
|
|
{
|
||
|
|
SimdScalar sqrtRadical = sqrtf(radical);
|
||
|
|
SimdScalar idenom = 1.0f/(2.0f * a);
|
||
|
|
m_roots[0]=(-b + sqrtRadical) * idenom;
|
||
|
|
m_roots[1]=(-b - sqrtRadical) * idenom;
|
||
|
|
return 2;
|
||
|
|
}
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
#define cubic_rt(x) \
|
||
|
|
((x) > 0.0f ? powf((SimdScalar)(x), 0.333333333333333333333333f) : \
|
||
|
|
((x) < 0.0f ? -powf((SimdScalar)-(x), 0.333333333333333333333333f) : 0.0f))
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
/* */
|
||
|
|
/* this function solves the following cubic equation: */
|
||
|
|
/* */
|
||
|
|
/* 3 2 */
|
||
|
|
/* lead * x + a * x + b * x + c = 0. */
|
||
|
|
/* */
|
||
|
|
/* it returns the number of different roots found, and stores the roots in */
|
||
|
|
/* roots[0,2]. it returns -1 for a degenerate equation 0 = 0. */
|
||
|
|
/* */
|
||
|
|
int BU_AlgebraicPolynomialSolver::Solve3Cubic(SimdScalar lead, SimdScalar a, SimdScalar b, SimdScalar c)
|
||
|
|
{
|
||
|
|
SimdScalar p, q, r;
|
||
|
|
SimdScalar delta, u, phi;
|
||
|
|
SimdScalar dummy;
|
||
|
|
|
||
|
|
if (lead != 1.0) {
|
||
|
|
/* */
|
||
|
|
/* transform into normal form: x^3 + a x^2 + b x + c = 0 */
|
||
|
|
/* */
|
||
|
|
if (SimdEqual(lead, SIMD_EPSILON)) {
|
||
|
|
/* */
|
||
|
|
/* we have a x^2 + b x + c = 0 */
|
||
|
|
/* */
|
||
|
|
if (SimdEqual(a, SIMD_EPSILON)) {
|
||
|
|
/* */
|
||
|
|
/* we have b x + c = 0 */
|
||
|
|
/* */
|
||
|
|
if (SimdEqual(b, SIMD_EPSILON)) {
|
||
|
|
if (SimdEqual(c, SIMD_EPSILON)) {
|
||
|
|
return -1;
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
m_roots[0] = -c / b;
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
p = c / a;
|
||
|
|
q = b / a;
|
||
|
|
return Solve2QuadraticFull(a,b,c);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
a = a / lead;
|
||
|
|
b = b / lead;
|
||
|
|
c = c / lead;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* */
|
||
|
|
/* we substitute x = y - a / 3 in order to eliminate the quadric term. */
|
||
|
|
/* we get x^3 + p x + q = 0 */
|
||
|
|
/* */
|
||
|
|
a /= 3.0f;
|
||
|
|
u = a * a;
|
||
|
|
p = b / 3.0f - u;
|
||
|
|
q = a * (2.0f * u - b) + c;
|
||
|
|
|
||
|
|
/* */
|
||
|
|
/* now use Cardano's formula */
|
||
|
|
/* */
|
||
|
|
if (SimdEqual(p, SIMD_EPSILON)) {
|
||
|
|
if (SimdEqual(q, SIMD_EPSILON)) {
|
||
|
|
/* */
|
||
|
|
/* one triple root */
|
||
|
|
/* */
|
||
|
|
m_roots[0] = -a;
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
/* */
|
||
|
|
/* one real and two complex roots */
|
||
|
|
/* */
|
||
|
|
m_roots[0] = cubic_rt(-q) - a;
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
q /= 2.0f;
|
||
|
|
delta = p * p * p + q * q;
|
||
|
|
if (delta > 0.0f) {
|
||
|
|
/* */
|
||
|
|
/* one real and two complex roots. note that v = -p / u. */
|
||
|
|
/* */
|
||
|
|
u = -q + sqrtf(delta);
|
||
|
|
u = cubic_rt(u);
|
||
|
|
m_roots[0] = u - p / u - a;
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
else if (delta < 0.0) {
|
||
|
|
/* */
|
||
|
|
/* Casus irreducibilis: we have three real roots */
|
||
|
|
/* */
|
||
|
|
r = sqrtf(-p);
|
||
|
|
p *= -r;
|
||
|
|
r *= 2.0;
|
||
|
|
phi = acosf(-q / p) / 3.0f;
|
||
|
|
dummy = SIMD_2_PI / 3.0f;
|
||
|
|
m_roots[0] = r * cosf(phi) - a;
|
||
|
|
m_roots[1] = r * cosf(phi + dummy) - a;
|
||
|
|
m_roots[2] = r * cosf(phi - dummy) - a;
|
||
|
|
return 3;
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
/* */
|
||
|
|
/* one single and one SimdScalar root */
|
||
|
|
/* */
|
||
|
|
r = cubic_rt(-q);
|
||
|
|
m_roots[0] = 2.0f * r - a;
|
||
|
|
m_roots[1] = -r - a;
|
||
|
|
return 2;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
/* */
|
||
|
|
/* this function solves the following quartic equation: */
|
||
|
|
/* */
|
||
|
|
/* 4 3 2 */
|
||
|
|
/* lead * x + a * x + b * x + c * x + d = 0. */
|
||
|
|
/* */
|
||
|
|
/* it returns the number of different roots found, and stores the roots in */
|
||
|
|
/* roots[0,3]. it returns -1 for a degenerate equation 0 = 0. */
|
||
|
|
/* */
|
||
|
|
int BU_AlgebraicPolynomialSolver::Solve4Quartic(SimdScalar lead, SimdScalar a, SimdScalar b, SimdScalar c, SimdScalar d)
|
||
|
|
{
|
||
|
|
SimdScalar p, q ,r;
|
||
|
|
SimdScalar u, v, w;
|
||
|
|
int i, num_roots, num_tmp;
|
||
|
|
//SimdScalar tmp[2];
|
||
|
|
|
||
|
|
if (lead != 1.0) {
|
||
|
|
/* */
|
||
|
|
/* transform into normal form: x^4 + a x^3 + b x^2 + c x + d = 0 */
|
||
|
|
/* */
|
||
|
|
if (SimdEqual(lead, SIMD_EPSILON)) {
|
||
|
|
/* */
|
||
|
|
/* we have a x^3 + b x^2 + c x + d = 0 */
|
||
|
|
/* */
|
||
|
|
if (SimdEqual(a, SIMD_EPSILON)) {
|
||
|
|
/* */
|
||
|
|
/* we have b x^2 + c x + d = 0 */
|
||
|
|
/* */
|
||
|
|
if (SimdEqual(b, SIMD_EPSILON)) {
|
||
|
|
/* */
|
||
|
|
/* we have c x + d = 0 */
|
||
|
|
/* */
|
||
|
|
if (SimdEqual(c, SIMD_EPSILON)) {
|
||
|
|
if (SimdEqual(d, SIMD_EPSILON)) {
|
||
|
|
return -1;
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
m_roots[0] = -d / c;
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
p = c / b;
|
||
|
|
q = d / b;
|
||
|
|
return Solve2QuadraticFull(b,c,d);
|
||
|
|
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
return Solve3Cubic(1.0, b / a, c / a, d / a);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
a = a / lead;
|
||
|
|
b = b / lead;
|
||
|
|
c = c / lead;
|
||
|
|
d = d / lead;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* */
|
||
|
|
/* we substitute x = y - a / 4 in order to eliminate the cubic term. */
|
||
|
|
/* we get: y^4 + p y^2 + q y + r = 0. */
|
||
|
|
/* */
|
||
|
|
a /= 4.0f;
|
||
|
|
p = b - 6.0f * a * a;
|
||
|
|
q = a * (8.0f * a * a - 2.0f * b) + c;
|
||
|
|
r = a * (a * (b - 3.f * a * a) - c) + d;
|
||
|
|
if (SimdEqual(q, SIMD_EPSILON)) {
|
||
|
|
/* */
|
||
|
|
/* biquadratic equation: y^4 + p y^2 + r = 0. */
|
||
|
|
/* */
|
||
|
|
num_roots = Solve2Quadratic(p, r);
|
||
|
|
if (num_roots > 0) {
|
||
|
|
if (m_roots[0] > 0.0f) {
|
||
|
|
if (num_roots > 1) {
|
||
|
|
if ((m_roots[1] > 0.0f) && (m_roots[1] != m_roots[0])) {
|
||
|
|
u = sqrtf(m_roots[1]);
|
||
|
|
m_roots[2] = u - a;
|
||
|
|
m_roots[3] = -u - a;
|
||
|
|
u = sqrtf(m_roots[0]);
|
||
|
|
m_roots[0] = u - a;
|
||
|
|
m_roots[1] = -u - a;
|
||
|
|
return 4;
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
u = sqrtf(m_roots[0]);
|
||
|
|
m_roots[0] = u - a;
|
||
|
|
m_roots[1] = -u - a;
|
||
|
|
return 2;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
u = sqrtf(m_roots[0]);
|
||
|
|
m_roots[0] = u - a;
|
||
|
|
m_roots[1] = -u - a;
|
||
|
|
return 2;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
else if (SimdEqual(r, SIMD_EPSILON)) {
|
||
|
|
/* */
|
||
|
|
/* no absolute term: y (y^3 + p y + q) = 0. */
|
||
|
|
/* */
|
||
|
|
num_roots = Solve3Cubic(1.0, 0.0, p, q);
|
||
|
|
for (i = 0; i < num_roots; ++i) m_roots[i] -= a;
|
||
|
|
if (num_roots != -1) {
|
||
|
|
m_roots[num_roots] = -a;
|
||
|
|
++num_roots;
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
m_roots[0] = -a;
|
||
|
|
num_roots = 1;;
|
||
|
|
}
|
||
|
|
return num_roots;
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
/* */
|
||
|
|
/* we solve the resolvent cubic equation */
|
||
|
|
/* */
|
||
|
|
num_roots = Solve3Cubic(1.0f, -0.5f * p, -r, 0.5f * r * p - 0.125f * q * q);
|
||
|
|
if (num_roots == -1) {
|
||
|
|
num_roots = 1;
|
||
|
|
m_roots[0] = 0.0f;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* */
|
||
|
|
/* build two quadric equations */
|
||
|
|
/* */
|
||
|
|
w = m_roots[0];
|
||
|
|
u = w * w - r;
|
||
|
|
v = 2.0f * w - p;
|
||
|
|
|
||
|
|
if (SimdEqual(u, SIMD_EPSILON))
|
||
|
|
u = 0.0;
|
||
|
|
else if (u > 0.0f)
|
||
|
|
u = sqrtf(u);
|
||
|
|
else
|
||
|
|
return 0;
|
||
|
|
|
||
|
|
if (SimdEqual(v, SIMD_EPSILON))
|
||
|
|
v = 0.0;
|
||
|
|
else if (v > 0.0f)
|
||
|
|
v = sqrtf(v);
|
||
|
|
else
|
||
|
|
return 0;
|
||
|
|
|
||
|
|
if (q < 0.0f) v = -v;
|
||
|
|
w -= u;
|
||
|
|
num_roots=Solve2Quadratic(v, w);
|
||
|
|
for (i = 0; i < num_roots; ++i)
|
||
|
|
{
|
||
|
|
m_roots[i] -= a;
|
||
|
|
}
|
||
|
|
w += 2.0f *u;
|
||
|
|
SimdScalar tmp[2];
|
||
|
|
tmp[0] = m_roots[0];
|
||
|
|
tmp[1] = m_roots[1];
|
||
|
|
|
||
|
|
num_tmp = Solve2Quadratic(-v, w);
|
||
|
|
for (i = 0; i < num_tmp; ++i)
|
||
|
|
{
|
||
|
|
m_roots[i + num_roots] = tmp[i] - a;
|
||
|
|
m_roots[i]=tmp[i];
|
||
|
|
}
|
||
|
|
|
||
|
|
return (num_tmp + num_roots);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|