/**
* Compute dz/dx and dz/dy at x,y
*/
static Vector2 compute_derivative(Asphere surface, Vector2 xy)
{
/*
* Let s^2 = x^2 + y^2
* and,
* z = f(s) = c*s^2/(1 + (1 - c^2*k*s^2)^(1/2)) + A_4*s^4 + A_6*s^6 + A_8*s^8
* + A_10*s^10 + A_12*s^12 + A_14*s^14
*
* Then,
* dz/dx = dz/ds * ds/dx
*
* Now,
* dz/ds = c*s/(1 - c^2*k*s^2)^(1/2) + 4*A_4*s^3 + 6*A_6*s^5 + 8*A_8*s^7 +
* 10*A_10*s^9 + 12*A_12*s^11 + 14*A_14*s^13 and, ds/dx = x/s
*
* using
* E = dz/ds * 1/s = c/(1 - c^2*k*s^2)^(1/2) + 4*A_4*s^2 + 6*A_6*s^4 +
* 8*A_8*s^6 + 10*A_10*s^8 + 12*A_12*s^10 + 14*A_14*s^12 dz/dx = x*E and
* dz/dy = y*E
*/
double s = Math.Sqrt(xy.x() * xy.x() + xy.y() * xy.y());
double E = compute_derivative(surface, s) / s;
return(new Vector2(xy.x() * E, xy.y() * E));
}
/**
* Compute 4*A_4*s^2 + 6*A_6*s^4 + 8*A_8*s^6 + 10*A_10*s^8 + 12*A_12*s^10 +
* 14*A_14*s^12 s2 = x^2 + y^2 Used in dz/dy and dz/dx calculations
*/
static double deform_dz_dxy(Asphere S, double s2)
{
double s4 = s2 * s2;
double s6 = s4 * s2;
double s8 = s6 * s2;
double s10 = s8 * s2;
double s12 = s10 * s2;
return(4 * S._A4 * s2 + 6 * S._A6 * s4 + 8 * S._A8 * s6
+ 10 * S._A10 * s8 + 12 * S._A12 * s10 + 14 * S._A14 * s12);
}
/**
* Compute z at x,y
*/
static double compute_Z(Asphere surface, Vector2 xy)
{
/* Our formula is:
* z = f(s) = c*s^2/(1 + (1 - c^2*k*s^2)^(1/2)) + A_4*s^4 + A_6*s^6 + A_8*s^8
* + A_10*s^10 + + A_12*s^12 + A_14*s^14
*
* where s = (x^2 + y^2)^(1/2)
*/
double s2 = xy.x() * xy.x() + xy.y() * xy.y();
return(compute_Z(surface, s2));
}
/**
* Compute A_4*s^4 + A_6*s^6 + A_8*s^8 + A_10*s^10 + A_12*s^12 + A_14*s^14
* s2 = x^2 + y^2
*/
static double deform_sagitta(Asphere S, double s2)
{
double s4 = s2 * s2;
double s6 = s4 * s2;
double s8 = s6 * s2;
double s10 = s8 * s2;
double s12 = s10 * s2;
double s14 = s12 * s2;
return(S._A4 * s4 + S._A6 * s6 + S._A8 * s8 + S._A10 * s10
+ S._A12 * s12 + S._A14 * s14);
}
/**
* Compute 4*A_4*s^3 + 6*A_6*s^5 + 8*A_8*s^7 + 10*A_10*s^9 + 12*A_12*s^11 +
* 14*A_14*s^13 s2 = x^2 + y^2 Used in dz/ds calculation
*/
static double deform_dz_ds(Asphere S, double s)
{
double s2 = s * s;
double s3 = s2 * s;
double s5 = s3 * s2;
double s7 = s5 * s2;
double s9 = s7 * s2;
double s11 = s9 * s2;
double s13 = s11 * s2;
return(4 * S._A4 * s3 + 6 * S._A6 * s5 + 8 * S._A8 * s7
+ 10 * S._A10 * s9 + 12 * S._A12 * s11 + 14 * S._A14 * s13);
}
/**
* Compute dz/ds. For the equation see next function below.
*/
static double compute_derivative(Asphere surface, double s)
{
double s2 = s * s;
double c = surface._c; /* curvature = 1/radius */
double c2 = c * c;
double K = surface._k;
double l = Math.Sqrt(1 - s2 * K * c2);
if (l == 0.0)
{
// division by zero, really an error
return(0);
}
return((c * s) / l + deform_dz_ds(surface, s));
}
/**
* Compute z at s^2, where s^2 = x^2 + y^2
*/
static double compute_Z(Asphere surface, double s2)
{
/* Our formula is:
* z = f(s) = c*s^2/(1 + (1 - c^2*k*s^2)^(1/2)) + A_4*s^4 + A_6*s^6 + A_8*s^8
* + A_10*s^10 + + A_12*s^12 + A_14*s^14
*
* where s = (x^2 + y^2)^(1/2)
*/
double c = surface._c; /* curvature = 1/radius */
double c2 = c * c;
double K = surface._k;
double l = Math.Sqrt(1 - s2 * K * c2);
double temp = 1 + l;
if (temp == 0.0)
{
// division by zero, really an error
return(0);
}
return(c * s2 / temp + deform_sagitta(surface, s2));
}
static Vector3 compute_normal(Asphere S, Vector3 point)
{
/* General ray tracing procedure - Spencer and Murty */
/* See eq 18, 19 */
/* Also same as p632 Feder - but z axis swapped with x */
double s_2 = point.y() * point.y() + point.x() * point.x();
double temp = Math.Sqrt(1.0 - S._c * S._c * s_2 * S._k);
if (temp == 0.0)
{
return(Vector3.vector3_001);
}
if (Double.IsNaN(temp))
{
return(null);
}
double E = S._c / temp + deform_dz_dxy(S, s_2); // eq 19
double y1 = -point.y() * E; // eq 18
double x1 = -point.x() * E; // eq 18
double z1 = 1.0; // eq 18
// Following is from Goptical - tbc
return(new Vector3(x1, y1, z1).normalize());
}
/* computes intersection using Feder's equations - code is taken from
* https://github.com/dibyendumajumdar/ray.
* Note that Feder's paper uses x-axis rather than z-axis as the
* optical axis, so below we switch from x to z.
*/
public static Vector3Pair compute_intersection(Vector3 origin, Vector3 direction, Asphere S)
{
/* direction (X,Y,Z) is the vector along the ray to the surface */
/* origin (x,y,z) is the vector form of the vertex of the surface */
/*
* NOTE: variable 't' was
* used by Feder as the vertex separation between previous
* surface and this surface. In the new scheme, in which rays
* are transformed to the coordinate system of the surface
* before tracing, 't' is _zero_. It is still present in this
* code to enable comparison with the Feder paper; the
* optimizing compiler will eliminate it from the
* expressions. */
/* Feder paper equation (1) */
double t = 0;
double e = (t * origin.z()) - origin.dot(direction);
/* Feder paper equation (2) */
double M_1x = origin.z() + e * direction.z() - t;
/* Feder paper equation (3) */
double M_1_2 = origin.dot(origin) - (e * e) + (t * t) - (2.0 * t * origin.z());
double r_1_2 = 1.0 / (S._c * S._c);
if (M_1_2 > r_1_2)
{
M_1_2 = r_1_2; /* SPECIAL RULE! 96-01-22 */
}
/* Feder paper equation (4) */
double xi_1 = Math.Sqrt((direction.z() * direction.z())
- S._c * (S._c * M_1_2 - 2.0 * M_1x));
if (Double.IsNaN(xi_1))
{ /* NaN! reject this ray! */
Console.Write("Nan value\n");
return(null);
}
/* Feder paper equation (5) */
double L = e + (S._c * M_1_2 - 2.0 * M_1x) / (direction.z() + xi_1);
/* Get intercept with new (spherical) surface: */
double[] delta_length = new double[3];
for (int k = 0; k < 3; k++)
{
delta_length[k] = -origin.v(k);
}
Vector3 result = origin.plus(direction.times(L));
result = result.z(result.z() - t);
Vector3 N = Vector3.vector3_0;
/* Now (result) has x1, y1, z1 */
/*
* The ray has been traced to the osculating sphere with
* curvature c1. Now we will iterate to get the intercept with
* the nearby aspheric surface. Suppose the (rotationally
* symmetric) aspheric is given by $$x = f(y,z)$$, and is a
* function of $y^2 + z^2$ only. For a spherical surface, one
* has $$x = r - (r^2 - s^2)^{1\over2}$$, where $s^2 = y^2 +
* z^2$. For a general surface one may add deformation terms
* to this expression and obtain $$x = c s^2 / (1 + (1 - c^2
* s^2)^{1\over2})) + (A_2 s^2 + A_4 s^4 + ...) = f$$. The
* equation is expressed in this form in order to avoid
* indeterminacy as c approaches zero, and in order to
* represent surfaces that are nearly spherical. Near-spheres
* cannot be handled well by a power series alone, especially
* in the neighborhood of $s = 1 / c$.
*
* In this implementation we include a term for the numerical
* eccentricity so that we can trace any pure conic section
* without using the $A_i$ terms.
*/
const int TOLMAX = 10;
double tolerance = 1e-15;
int j = 0;
double delta = 0.0;
do
{
/* Get square of radius of intercept: */
/* Feder equation s^2 = x^2 + y^2, section E */
double s_2 = result.y() * result.y() + result.x() * result.x();
/*
* Get the point on aspheric which is at the same radius as
* the intercept of the ray. Then compute a tangent plane to
* the aspheric at this point and find where it intersects
* the ray. This point will lie very close to the aspheric
* surface. The first step is to compute the z-coordinate
* on the aspheric surface using $\overline{z}_0 = f(x_0,
* y_0)$.
*/
/* (1 - k*c^2*s^2)^(1/2) - part of equation (12) */
double temp = Math.Sqrt(1.0 - S._c * S._c * s_2 * S._k);
if (Double.IsNaN(temp) || (1.0 + temp) == 0.0)
{
Console.Write("Nan or zero divide value\n");
return(null);
}
/* Feder equation (12) */
/* But using c*s^2/[1 + (1 - k*c^2*s^2)^(1/2)] + aspheric A_2*s^2 +
* A_4*s^4 + ... */
double x_bar_0 = (S._c * s_2) / (1.0 + temp) + deform_sagitta(S, s_2);
delta = Math.Abs(result.z() - x_bar_0);
/* Get the direction numbers for the normal to the
* aspheric: */
/* Feder equation (13), l */
double z1 = temp;
temp = S._c + N.z() * deform_dz_dxy(S, s_2);
/* Feder equation (14), m */
double y1 = -result.y() * temp;
/* Feder equation (15), n */
double x1 = -result.x() * temp;
N = new Vector3(x1, y1, z1);
/* Get the distance from aspheric point to ray intercept */
double G_0 = N.z() * (x_bar_0 - result.z()) / (direction.dot(N));
/* and compute new estimate of intercept point: */
result = result.plus(direction.times(G_0));
}while ((delta > tolerance) && (++j < TOLMAX));
if (j >= TOLMAX)
{
Console.Write(string.Format("rayTrace: delta=%g, reached %d iterations!?!\n", delta, j));
return(null);
}
return(new Vector3Pair(result, N));
}