DLLEXPORT int solveCubic(double c3, double c2, double c1, double c0,
double & s0, double & s1, double & s2)
{
int i, num;
double sub,
A, B, C,
sq_A, p, q,
cb_p, D;
if (isZero(c3))
return solveQuadric(c2, c1, c0, s0, s1);
// normalize the equation:x ^ 3 + Ax ^ 2 + Bx + C = 0
A = c2 / c3;
B = c1 / c3;
C = c0 / c3;
// substitute x = y - A / 3 to eliminate the quadric term: x^3 + px + q = 0
sq_A = A * A;
p = 1.0/3.0 * (-1.0/3.0 * sq_A + B);
q = 1.0/2.0 * (2.0/27.0 * A *sq_A - 1.0/3.0 * A * B + C);
// use Cardano's formula
cb_p = p * p * p;
D = q * q + cb_p;
if (isZero(D))
{
if (isZero(q))
{
// one triple solution
s0 = 0.0;
num = 1;
}
else
{
// one single and one double solution
double u = cbrt(-q);
s0 = 2.0 * u;
s1 = - u;
num = 2;
}
}
else
if (D < 0.0)
{
// casus irreductibilis: three real solutions
double phi = 1.0/3.0 * acos(-q / sqrt(-cb_p));
double t = 2.0 * sqrt(-p);
s0 = t * cos(phi);
s1 = -t * cos(phi + M_PI / 3.0);
s2 = -t * cos(phi - M_PI / 3.0);
num = 3;
}
else
{
// one real solution
double sqrt_D = sqrt(D);
double u = cbrt(sqrt_D + fabs(q));
if (q > 0.0)
s0 = - u + p / u ;
else
s0 = u - p / u;
num = 1;
}
// resubstitute
sub = 1.0 / 3.0 * A;
s0 -= sub;
s1 -= sub;
s2 -= sub;
return num;
}
void Photon::choosePointInRect(Float& x, Float& y, const int rectNum, const Float randX, const Float randY)
{
Rect& rect = m_chunk->m_rects[rectNum];
Float b1 = m_knotValues[rect.tl];
Float b2 = m_knotValues[rect.tr] - b1;
Float b3 = m_knotValues[rect.bl] - b1;
Float b4 = b1 - m_knotValues[rect.tr] - m_knotValues[rect.bl] + m_knotValues[rect.br];
int roots;
{
Float x1, x2;
roots = solveQuadric(b2 + 0.5*b4, 0.5*(b1 + 0.5*b3), -randX*(b2+0.5*b4+0.5*b1+0.25*b3), x1, x2);
if (roots == 1)
x = x1;
else if (roots == 2) {
if (x1 >= 0. && x1 < 1.) {
x = x1;
}
else if (x2 >= 0. && x2 < 1.) {
x = x2;
}
else {
x = 0.5;
fprintf(stderr, "x out of range, %f\t%f\n", x1, x2);
}
}
}
{
Float y1, y2;
roots = solveQuadric(0.5*(b3 + b4*x), b1 + b2*x, -randY*(b1+b2*x + 0.5*(b3+b4*x)), y1, y2);
if (roots == 1)
y = y1;
else if (roots == 2) {
if (y1 >= 0. && y1 < 1.) {
y = y1;
}
else if (y2 >= 0. && y2 < 1.) {
y = y2;
}
else {
y = 0.5;
fprintf(stderr, "y out of range, %f\t%f\n", y1, y2);
}
}
}
x *= rect.width;
y *= rect.height;
x += m_chunk->m_knots[rect.tl].x;
y += m_chunk->m_knots[rect.tl].y;
}
int PolynomialSolver::solveQuartic(float c[5], float s[4])
{
float coeffs[4],
z, u, v, sub,
A, B, C, D,
sq_A, p, q, r;
int i, num;
// normalize the equation:x ^ 4 + Ax ^ 3 + Bx ^ 2 + Cx + D = 0
A = c[3] / c[4];
B = c[2] / c[4];
C = c[1] / c[4];
D = c[0] / c[4];
// subsitute x = y - A / 4 to eliminate the cubic term: x^4 + px^2 + qx + r = 0
sq_A = A * A;
p = -3.0f / 8.0f * sq_A + B;
q = 1.0f / 8.0f * sq_A * A - 1.0f / 2.0f * A * B + C;
r = -3.0f / 256.0f * sq_A * sq_A + 1.0f / 16.0f * sq_A * B - 1.0f / 4.0f * A * C + D;
if(isZero(r))
{
// no absolute term:y(y ^ 3 + py + q) = 0
coeffs[0] = q;
coeffs[1] = p;
coeffs[2] = 0.0;
coeffs[3] = 1.0;
num = solveCubic(coeffs, s);
s[num++] = 0;
}
else
{
// solve the resolvent cubic...
coeffs[0] = 1.0f / 2.0f * r * p - 1.0f / 8.0f * q * q;
coeffs[1] = -r;
coeffs[2] = -1.0f / 2.0f * p;
coeffs[3] = 1.0f;
(void) solveCubic(coeffs, s);
// ...and take the one real solution...
z = s[0];
// ...to build two quadratic equations
u = z * z - r;
v = 2.0f * z - p;
if(isZero(u))
u = 0.0;
else if(u > 0.0f)
u = std::sqrt(u);
else
return 0;
if(isZero(v))
v = 0;
else if(v > 0.0f)
v = std::sqrt(v);
else
return 0;
coeffs[0] = z - u;
coeffs[1] = q < 0 ? -v : v;
coeffs[2] = 1.0f;
num = solveQuadric(coeffs, s);
coeffs[0] = z + u;
coeffs[1] = q < 0 ? v : -v;
coeffs[2] = 1.0f;
num += solveQuadric(coeffs, s + num);
}
// resubstitute
sub = 1.0f / 4 * A;
for(i = 0; i < num; i++)
s[i] -= sub;
return num;
}
int PolySolver::solveCubic(FCL_REAL c[4], FCL_REAL s[3])
{
int i, num;
FCL_REAL sub, A, B, C, sq_A, p, q, cb_p, D;
const FCL_REAL ONE_OVER_THREE = 1 / 3.0;
const FCL_REAL PI = 3.14159265358979323846;
// make sure we have a d2 equation
if(isZero(c[3]))
return solveQuadric(c, s);
// normalize the equation:x ^ 3 + Ax ^ 2 + Bx + C = 0
A = c[2] / c[3];
B = c[1] / c[3];
C = c[0] / c[3];
// substitute x = y - A / 3 to eliminate the quadratic term: x^3 + px + q = 0
sq_A = A * A;
p = (-ONE_OVER_THREE * sq_A + B) * ONE_OVER_THREE;
q = 0.5 * (2.0 / 27.0 * A * sq_A - ONE_OVER_THREE * A * B + C);
// use Cardano's formula
cb_p = p * p * p;
D = q * q + cb_p;
if(isZero(D))
{
if(isZero(q))
{
// one triple solution
s[0] = 0.0;
num = 1;
}
else
{
// one single and one FCL_REAL solution
FCL_REAL u = cbrt(-q);
s[0] = 2.0 * u;
s[1] = -u;
num = 2;
}
}
else
{
if(D < 0.0)
{
// three real solutions
FCL_REAL phi = ONE_OVER_THREE * acos(-q / sqrt(-cb_p));
FCL_REAL t = 2.0 * sqrt(-p);
s[0] = t * cos(phi);
s[1] = -t * cos(phi + PI / 3.0);
s[2] = -t * cos(phi - PI / 3.0);
num = 3;
}
else
{
// one real solution
FCL_REAL sqrt_D = sqrt(D);
FCL_REAL u = cbrt(sqrt_D + fabs(q));
if(q > 0.0)
s[0] = - u + p / u ;
else
s[0] = u - p / u;
num = 1;
}
}
// re-substitute
sub = ONE_OVER_THREE * A;
for(i = 0; i < num; i++)
s[i] -= sub;
return num;
}
int solveQuartic(float c[5], float s[4])
{
float coeffs[4];
float z, u, v, sub;
float A, B, C, D;
float sq_A, p, q, r;
int i, num;
/* normal form: x^4 + Ax^3 + Bx^2 + Cx + D = 0 */
A = c[ 3 ] / c[ 4 ];
B = c[ 2 ] / c[ 4 ];
C = c[ 1 ] / c[ 4 ];
D = c[ 0 ] / c[ 4 ];
/* substitute x = y - A/4 to eliminate cubic term:
x^4 + px^2 + qx + r = 0 */
sq_A = A * A;
p = - 3.0/8 * sq_A + B;
q = 1.0/8 * sq_A * A - 1.0/2 * A * B + C;
r = - 3.0/256*sq_A*sq_A + 1.0/16*sq_A*B - 1.0/4*A*C + D;
if (IS_ZERO(r))
{
/* no absolute term: y(y^3 + py + q) = 0 */
coeffs[ 0 ] = q;
coeffs[ 1 ] = p;
coeffs[ 2 ] = 0;
coeffs[ 3 ] = 1;
num = solveCubic(coeffs, s);
s[ num++ ] = 0;
}
else {
/* solve the resolvent cubic ... */
coeffs[ 0 ] = 1.0/2 * r * p - 1.0/8 * q * q;
coeffs[ 1 ] = - r;
coeffs[ 2 ] = - 1.0/2 * p;
coeffs[ 3 ] = 1;
(void) solveCubic(coeffs, s);
/* ... and take the one real solution ... */
z = s[ 0 ];
/* ... to build two quadric equations */
u = z * z - r;
v = 2 * z - p;
if (IS_ZERO(u))
u = 0;
else if (u > 0)
u = sqrt(u);
else
return 0;
if (IS_ZERO(v))
v = 0;
else if (v > 0)
v = sqrt(v);
else
return 0;
coeffs[ 0 ] = z - u;
coeffs[ 1 ] = q < 0 ? -v : v;
coeffs[ 2 ] = 1;
num = solveQuadric(coeffs, s);
coeffs[ 0 ]= z + u;
coeffs[ 1 ] = q < 0 ? v : -v;
coeffs[ 2 ] = 1;
num += solveQuadric(coeffs, s + num);
}
/* resubstitute */
sub = 1.0/4 * A;
for (i = 0; i < num; ++i)
s[ i ] -= sub;
return num;
}
