/// <summary>
/// Solves the equation \f$(q(x))^2 = b_0 + b_1 x + b_2 x^2 + b_3 x^3 + b_4 x^4\f$, returning all values of
/// \f$x\f$ for which the equation is true. \f$q(x)\f$ is the quadratic spline. The _parameters z0 and c
/// can be used to substitute x, such that \f$x = z0 + c t\f$. This is useful for raytracing.
/// </summary>
public override IEnumerable <double> SolveRaytrace(QuarticFunction surfaceFunction, double z0 = 0.0, double c = 1.0)
{
// Solve the polynomial equation for each segment:
for (int i = 1; i < Points.Count; i++)
{
double x1 = Points.Key[i - 1];
double x2 = Points.Key[i];
double y1 = Points.Value[i - 1];
double y2 = Points.Value[i];
// Calculate and return the interpolated value:
double dx = x2 - x1;
double div = 1.0 / dx;
double dy = y2 - y1;
double a = -_parameters[i] * dx + dy;
// Write in the form of a0 + a1 z + a2 z^2:
double a0 = -a * x1 * x1 * div * div - (a + dy) * x1 * div + y1;
double a1 = 2.0 * a * x1 * div * div + (a + dy) * div;
double a2 = -a * div * div;
// Substitute z = z0 + c t:
double A0 = a0 + a1 * z0 + a2 * z0 * z0;
double A1 = (a1 + 2.0 * a2 * z0) * c;
double A2 = a2 * c * c;
// Find the quartic polynomial to solve:
double p0 = surfaceFunction.a0 - A0 * A0;
double p1 = surfaceFunction.a1 - 2.0 * A0 * A1;
double p2 = surfaceFunction.a2 - (2.0 * A0 * A2 + A1 * A1);
double p3 = surfaceFunction.a3 - 2.0 * A1 * A2;
double p4 = surfaceFunction.a4 - A2 * A2;
// Solve the quartic polynomial:
IEnumerable <double> intersections = QuarticFunction.Solve(p0, p1, p2, p3, p4);
// Only return the value if it is sampled within the segment that we are currently considering,
// otherwise the value we got is invalid:
foreach (var j in intersections)
{
if (((z0 + c * j) > x1) && ((z0 + c * j) <= x2))
{
yield return(j);
}
}
}
}
/// <summary>
/// Solves the equation \f$(q(x))^2 = b_0 + b_1 x + b_2 x^2 + b_3 x^3 + b_4 x^4\f$, returning all values of
/// \f$x\f$ for which the equation is true. \f$q(x)\f$ is the quadratic spline. The parameters z0 and c
/// can be used to substitute x, such that \f$x = z0 + c t\f$. This is useful for raytracing.
/// </summary>
public override IEnumerable <float> SolveRaytrace(QuarticFunction surfaceFunction, float z0 = 0.0f, float c = 1.0f)
{
// Solve the polynomial equation for each segment:
for (int i = 1; i < Points.Count; i++)
{
Real x1 = Points.Key[i - 1];
Real x2 = Points.Key[i];
Real y1 = Points.Value[i - 1];
Real y2 = Points.Value[i];
// Calculate and return the interpolated value:
Real dx = x2 - x1;
Real div = 1.0 / dx;
Real dy = y2 - y1;
Real a = -parameters[i] * dx + dy;
// Write in the form of a0 + a1 z + a2 z^2:
Real a0 = -a * x1 * x1 * div * div - (a + dy) * x1 * div + y1;
Real a1 = 2.0 * a * x1 * div * div + (a + dy) * div;
Real a2 = -a * div * div;
// Substitute z = z0 + c t:
Real A0 = a0 + a1 * z0 + a2 * z0 * z0;
Real A1 = (a1 + 2.0 * a2 * z0) * c;
Real A2 = a2 * c * c;
// Find the quartic polynomial to solve:
Real p0 = surfaceFunction.a0 - A0 * A0;
Real p1 = surfaceFunction.a1 - 2.0 * A0 * A1;
Real p2 = surfaceFunction.a2 - (2.0 * A0 * A2 + A1 * A1);
Real p3 = surfaceFunction.a3 - 2.0 * A1 * A2;
Real p4 = surfaceFunction.a4 - A2 * A2;
// Solve the quartic polynomial:
IEnumerable <float> intersections = QuarticFunction.Solve((float)p0, (float)p1, (float)p2, (float)p3, (float)p4);
// Only return the value if it is sampled within the segment that we are currently considering,
// otherwise the value we got is invalid:
foreach (var j in intersections)
{
if (((z0 + c * j) > x1) && ((z0 + c * j) <= x2))
{
yield return(j);
}
}
}
}
public IEnumerable <double> Roots()
{
return(QuarticFunction.Solve(A0, A1, A2, A3, A4));
}
public IEnumerable <double> Roots()
{
return(QuarticFunction.Solve(a0, a1, a2, a3, a4));
}