libcruft-util/polynomial.cpp
Danny Robson f6056153e3 rename root namespace from util to cruft
This places, at long last, the core library code into the same namespace
as the extended library code.
2018-08-05 14:42:02 +10:00

150 lines
4.4 KiB
C++

/*
* This Source Code Form is subject to the terms of the Mozilla Public
* License, v. 2.0. If a copy of the MPL was not distributed with this
* file, You can obtain one at http://mozilla.org/MPL/2.0/.
*
* Copyright 2015 Danny Robson <danny@nerdcruft.net>
*/
#include "polynomial.hpp"
#include "maths.hpp"
#include <limits>
#include <cmath>
static const size_t NEWTON_ITERATIONS = 1u;
//-----------------------------------------------------------------------------
namespace cruft::polynomial {
template <>
std::array<float,1>
roots (std::array<float,2> coeff)
{
const float a = coeff[0];
const float b = coeff[1];
if (almost_zero (a))
return { std::numeric_limits<float>::quiet_NaN () };
return { -b / a };
}
}
//-----------------------------------------------------------------------------
namespace cruft::polynomial {
template <>
std::array<float,2>
roots (std::array<float,3> coeff)
{
const float a = coeff[0];
const float b = coeff[1];
const float c = coeff[2];
if (almost_zero (a)) {
auto s = roots<1> ({b, c});
return { s[0], std::numeric_limits<float>::quiet_NaN () };
}
auto descriminator = std::sqrt (pow (b,2) - 4 * a * c);
return {
(-b - descriminator) / (2 * a),
(-b + descriminator) / (2 * a)
};
}
}
//-----------------------------------------------------------------------------
// From graphics gems: http://tog.acm.org/resources/GraphicsGems/gemsiv/vec_mat/ray/solver.c
namespace cruft::polynomial {
template <>
std::array<float,3>
roots (std::array<float,4> coeffs)
{
const float _a = coeffs[0];
const float _b = coeffs[1];
const float _c = coeffs[2];
const float _d = coeffs[3];
// Take care of degenerate quadratic cases. We can also pass off if 'd'
// is zero, but the benefit isn't clear given we have to merge results
// at the end anyway.
if (almost_zero (_a)) {
auto s = roots<2> ({_b, _c, _d});
return {s[0], s[1], std::numeric_limits<float>::quiet_NaN () };
}
std::array<float,3> s;
// Normalise to x^3 + ax^2 + bx + c = 0
const float a = _b / _a;
const float b = _c / _a;
const float c = _d / _a;
// Substituate x = y - a / 3 to eliminate the quadric. Now: x^3 + px + q = 0
const float p = (-a * a / 3.f + b) / 3.f;
const float q = (2 * a * a * a / 27.f - a * b /3.f + c) / 2.f;
// Polynomial descriminant
const float D = q * q + p * p * p;
// Solve using Cardano's method
if (almost_zero (D))
{
if (almost_zero (q)) {
s[0] = 0.f;
s[1] = std::numeric_limits<float>::quiet_NaN ();
s[2] = std::numeric_limits<float>::quiet_NaN ();
} else {
const float u = std::cbrt (-q);
s[0] = 2 * u;
s[1] = -u;
s[2] = std::numeric_limits<float>::quiet_NaN ();
}
} else if (D < 0) {
const float phi = std::acos (-q / std::sqrt (-p * p * p)) / 3.f;
const float t = 2 * std::sqrt (-p);
s[0] = t * std::cos (phi);
s[1] = -t * std::cos (phi + pi<float> / 3.f);
s[2] = -t * std::cos (phi - pi<float> / 3.f);
} else {
float u = std::cbrt (std::sqrt (D) + abs (q));
if (q > 0.f)
s[0] = -u + p / u;
else
s[0] = u - p / u;
s[1] = std::numeric_limits<float>::quiet_NaN ();
s[2] = std::numeric_limits<float>::quiet_NaN ();
}
// Resubstitute a / 3 from above
const float sub = a / 3.f;
for (auto &i: s)
i -= sub;
// Run some iterations of Newtons method to make the results slightly
// more accurate, they're a little loose straight out of the bat.
const float da = 3 * _a;
const float db = 2 * _b;
const float dc = 1 * _c;
for (auto &i: s) {
for (size_t j = 0; j < NEWTON_ITERATIONS; ++j) {
float deriv = da * i * i + db * i + dc;
if (almost_zero (deriv))
continue;
i = i - eval (coeffs, i) / deriv;
}
}
return s;
}
}