242 lines
5.6 KiB
C++
Executable File
242 lines
5.6 KiB
C++
Executable File
//-----------------------------------------------------------------------------
|
|
// Torque Game Engine
|
|
// Copyright (C) GarageGames.com, Inc.
|
|
//-----------------------------------------------------------------------------
|
|
|
|
#include "platform/platform.h"
|
|
#include "math/mMathFn.h"
|
|
|
|
//--------------------------------------------------------------------------
|
|
#define EQN_EPSILON (1e-8)
|
|
|
|
static inline void swap(F32 & a, F32 & b)
|
|
{
|
|
F32 t = b;
|
|
b = a;
|
|
a = t;
|
|
}
|
|
|
|
static inline bool mIsZero(F32 val)
|
|
{
|
|
return((val > -EQN_EPSILON) && (val < EQN_EPSILON));
|
|
}
|
|
|
|
static inline F32 mCbrt(F32 val)
|
|
{
|
|
if(val < 0.f)
|
|
return(-mPow(-val, F32(1.f/3.f)));
|
|
else
|
|
return(mPow(val, F32(1.f/3.f)));
|
|
}
|
|
|
|
static inline U32 mSolveLinear(F32 a, F32 b, F32 * x)
|
|
{
|
|
if(mIsZero(a))
|
|
return(0);
|
|
|
|
x[0] = -b/a;
|
|
return(1);
|
|
}
|
|
|
|
static U32 mSolveQuadratic_c(F32 a, F32 b, F32 c, F32 * x)
|
|
{
|
|
// really linear?
|
|
if(mIsZero(a))
|
|
return(mSolveLinear(b, c, x));
|
|
|
|
// get the descriminant: (b^2 - 4ac)
|
|
F32 desc = (b * b) - (4.f * a * c);
|
|
|
|
// solutions:
|
|
// desc < 0: two imaginary solutions
|
|
// desc > 0: two real solutions (b +- sqrt(desc)) / 2a
|
|
// desc = 0: one real solution (b / 2a)
|
|
if(mIsZero(desc))
|
|
{
|
|
x[0] = b / (2.f * a);
|
|
return(1);
|
|
}
|
|
else if(desc > 0.f)
|
|
{
|
|
F32 sqrdesc = mSqrt(desc);
|
|
F32 den = (2.f * a);
|
|
x[0] = (b + sqrdesc) / den;
|
|
x[1] = (b - sqrdesc) / den;
|
|
|
|
if(x[1] < x[0])
|
|
swap(x[0], x[1]);
|
|
|
|
return(2);
|
|
}
|
|
else
|
|
return(0);
|
|
}
|
|
|
|
//--------------------------------------------------------------------------
|
|
// from Graphics Gems I: pp 738-742
|
|
U32 mSolveCubic_c(F32 a, F32 b, F32 c, F32 d, F32 * x)
|
|
{
|
|
if(mIsZero(a))
|
|
return(mSolveQuadratic(b, c, d, x));
|
|
|
|
// normal form: x^3 + Ax^2 + BX + C = 0
|
|
F32 A = b / a;
|
|
F32 B = c / a;
|
|
F32 C = d / a;
|
|
|
|
// substitute x = y - A/3 to eliminate quadric term and depress
|
|
// the cubic equation to (x^3 + px + q = 0)
|
|
F32 A2 = A * A;
|
|
F32 A3 = A2 * A;
|
|
|
|
F32 p = (1.f/3.f) * (((-1.f/3.f) * A2) + B);
|
|
F32 q = (1.f/2.f) * (((2.f/27.f) * A3) - ((1.f/3.f) * A * B) + C);
|
|
|
|
// use Cardano's fomula to solve the depressed cubic
|
|
F32 p3 = p * p * p;
|
|
F32 q2 = q * q;
|
|
|
|
F32 D = q2 + p3;
|
|
|
|
U32 num = 0;
|
|
|
|
if(mIsZero(D)) // 1 or 2 solutions
|
|
{
|
|
if(mIsZero(q)) // 1 triple solution
|
|
{
|
|
x[0] = 0.f;
|
|
num = 1;
|
|
}
|
|
else // 1 single and 1 double
|
|
{
|
|
F32 u = mCbrt(-q);
|
|
x[0] = 2.f * u;
|
|
x[1] = -u;
|
|
num = 2;
|
|
}
|
|
}
|
|
else if(D < 0.f) // 3 solutions: casus irreducibilis
|
|
{
|
|
F32 phi = (1.f/3.f) * mAcos(-q / mSqrt(-p3));
|
|
F32 t = 2.f * mSqrt(-p);
|
|
|
|
x[0] = t * mCos(phi);
|
|
x[1] = -t * mCos(phi + (M_PI / 3.f));
|
|
x[2] = -t * mCos(phi - (M_PI / 3.f));
|
|
num = 3;
|
|
}
|
|
else // 1 solution
|
|
{
|
|
F32 sqrtD = mSqrt(D);
|
|
F32 u = mCbrt(sqrtD - q);
|
|
F32 v = -mCbrt(sqrtD + q);
|
|
|
|
x[0] = u + v;
|
|
num = 1;
|
|
}
|
|
|
|
// resubstitute
|
|
F32 sub = (1.f/3.f) * A;
|
|
for(U32 i = 0; i < num; i++)
|
|
x[i] -= sub;
|
|
|
|
// sort the roots
|
|
for(S32 j = 0; j < (num - 1); j++)
|
|
for(S32 k = j + 1; k < num; k++)
|
|
if(x[k] < x[j])
|
|
swap(x[k], x[j]);
|
|
|
|
return(num);
|
|
}
|
|
|
|
//--------------------------------------------------------------------------
|
|
// from Graphics Gems I: pp 738-742
|
|
U32 mSolveQuartic_c(F32 a, F32 b, F32 c, F32 d, F32 e, F32 * x)
|
|
{
|
|
if(mIsZero(a))
|
|
return(mSolveCubic(b, c, d, e, x));
|
|
|
|
// normal form: x^4 + ax^3 + bx^2 + cx + d = 0
|
|
F32 A = b / a;
|
|
F32 B = c / a;
|
|
F32 C = d / a;
|
|
F32 D = e / a;
|
|
|
|
// substitue x = y - A/4 to eliminate cubic term:
|
|
// x^4 + px^2 + qx + r = 0
|
|
F32 A2 = A * A;
|
|
F32 A3 = A2 * A;
|
|
F32 A4 = A2 * A2;
|
|
|
|
F32 p = ((-3.f/8.f) * A2) + B;
|
|
F32 q = ((1.f/8.f) * A3) - ((1.f/2.f) * A * B) + C;
|
|
F32 r = ((-3.f/256.f) * A4) + ((1.f/16.f) * A2 * B) - ((1.f/4.f) * A * C) + D;
|
|
|
|
U32 num = 0;
|
|
if(mIsZero(r)) // no absolute term: y(y^3 + py + q) = 0
|
|
{
|
|
num = mSolveCubic(1.f, 0.f, p, q, x);
|
|
x[num++] = 0.f;
|
|
}
|
|
else
|
|
{
|
|
// solve the resolvent cubic
|
|
F32 q2 = q * q;
|
|
|
|
a = 1.f;
|
|
b = (-1.f/2.f) * p;
|
|
c = -r;
|
|
d = ((1.f/2.f) * r * p) - ((1.f/8.f) * q2);
|
|
|
|
mSolveCubic(a, b, c, d, x);
|
|
|
|
F32 z = x[0];
|
|
|
|
// build 2 quadratic equations from the one solution
|
|
F32 u = (z * z) - r;
|
|
F32 v = (2.f * z) - p;
|
|
|
|
if(mIsZero(u))
|
|
u = 0.f;
|
|
else if(u > 0.f)
|
|
u = mSqrt(u);
|
|
else
|
|
return(0);
|
|
|
|
if(mIsZero(v))
|
|
v = 0.f;
|
|
else if(v > 0.f)
|
|
v = mSqrt(v);
|
|
else
|
|
return(0);
|
|
|
|
// solve the two quadratics
|
|
a = 1.f;
|
|
b = v;
|
|
c = z - u;
|
|
num = mSolveQuadratic(a, b, c, x);
|
|
|
|
a = 1.f;
|
|
b = -v;
|
|
c = z + u;
|
|
num += mSolveQuadratic(a, b, c, x + num);
|
|
}
|
|
|
|
// resubstitute
|
|
F32 sub = (1.f/4.f) * A;
|
|
for(U32 i = 0; i < num; i++)
|
|
x[i] -= sub;
|
|
|
|
// sort the roots
|
|
for(S32 j = 0; j < (num - 1); j++)
|
|
for(S32 k = j + 1; k < num; k++)
|
|
if(x[k] < x[j])
|
|
swap(x[k], x[j]);
|
|
|
|
return(num);
|
|
}
|
|
|
|
U32 (*mSolveQuadratic)( F32 a, F32 b, F32 c, F32* x ) = mSolveQuadratic_c;
|
|
U32 (*mSolveCubic)( F32 a, F32 b, F32 c, F32 d, F32* x ) = mSolveCubic_c;
|
|
U32 (*mSolveQuartic)( F32 a, F32 b, F32 c, F32 d, F32 e, F32* x ) = mSolveQuartic_c;
|