public Sphere(Transform objectToWorld, bool reverseOrientation,
float radius, float z0, float z1, float phiMax)
: base(objectToWorld, reverseOrientation)
{
_radius = radius;
_zMin = MathUtility.Clamp(Math.Min(z0, z1), -radius, radius);
_zMax = MathUtility.Clamp(Math.Max(z0, z1), -radius, radius);
_thetaMin = MathUtility.Acos(MathUtility.Clamp(_zMin / radius, -1.0f, 1.0f));
_thetaMax = MathUtility.Acos(MathUtility.Clamp(_zMax / radius, -1.0f, 1.0f));
_phiMax = MathUtility.ToRadians(MathUtility.Clamp(phiMax, 0.0f, 360.0f));
}
public static Quaternion Slerp(float t, Quaternion q1, Quaternion q2)
{
float cosTheta = Dot(q1, q2);
if (cosTheta > .9995f)
{
return(Normalize((1.0f - t) * q1 + t * q2));
}
float theta = MathUtility.Acos(MathUtility.Clamp(cosTheta, -1.0f, 1.0f));
float thetap = theta * t;
Quaternion qperp = Normalize(q2 - q1 * cosTheta);
return(q1 * MathUtility.Cos(thetap) + qperp * MathUtility.Sin(thetap));
}
public override bool TryIntersect(Ray r, out float tHit, out float rayEpsilon, out DifferentialGeometry dg)
{
tHit = float.NegativeInfinity;
rayEpsilon = 0.0f;
dg = null;
float phi;
Point phit;
float thit;
if (!DoIntersection(r, out phi, out phit, out thit))
{
return(false);
}
// Find parametric representation of sphere hit
float u = phi / _phiMax;
float theta = MathUtility.Acos(MathUtility.Clamp(phit.Z / _radius, -1.0f, 1.0f));
float v = (theta - _thetaMin) / (_thetaMax - _thetaMin);
// Compute sphere $\dpdu$ and $\dpdv$
float zradius = MathUtility.Sqrt(phit.X * phit.X + phit.Y * phit.Y);
float invzradius = 1.0f / zradius;
float cosphi = phit.X * invzradius;
float sinphi = phit.Y * invzradius;
var dpdu = new Vector(-_phiMax * phit.Y, _phiMax * phit.X, 0);
var dpdv = (_thetaMax - _thetaMin) *
new Vector(phit.Z * cosphi, phit.Z * sinphi, -_radius * MathUtility.Sin(theta));
// Compute sphere $\dndu$ and $\dndv$
Vector d2Pduu = -_phiMax * _phiMax * new Vector(phit.X, phit.Y, 0);
Vector d2Pduv = (_thetaMax - _thetaMin) * phit.Z * _phiMax * new Vector(-sinphi, cosphi, 0.0f);
Vector d2Pdvv = -(_thetaMax - _thetaMin) * (_thetaMax - _thetaMin) * new Vector(phit.X, phit.Y, phit.Z);
// Compute coefficients for fundamental forms
float E = Vector.Dot(dpdu, dpdu);
float F = Vector.Dot(dpdu, dpdv);
float G = Vector.Dot(dpdv, dpdv);
Vector N = Vector.Normalize(Vector.Cross(dpdu, dpdv));
float e = Vector.Dot(N, d2Pduu);
float f = Vector.Dot(N, d2Pduv);
float g = Vector.Dot(N, d2Pdvv);
// Compute $\dndu$ and $\dndv$ from fundamental form coefficients
float invEGF2 = 1.0f / (E * G - F * F);
var dndu = (Normal)((f * F - e * G) * invEGF2 * dpdu + (e * F - f * E) * invEGF2 * dpdv);
var dndv = (Normal)((g * F - f * G) * invEGF2 * dpdu + (f * F - g * E) * invEGF2 * dpdv);
// Initialize _DifferentialGeometry_ from parametric information
var o2w = ObjectToWorld;
dg = new DifferentialGeometry(
o2w.TransformPoint(ref phit),
o2w.TransformVector(ref dpdu),
o2w.TransformVector(ref dpdv),
o2w.TransformNormal(ref dndu),
o2w.TransformNormal(ref dndv),
u, v, this);
// Update _tHit_ for quadric intersection
tHit = thit;
// Compute _rayEpsilon_ for quadric intersection
rayEpsilon = 5e-4f * tHit;
return(true);
}
public static float SphericalTheta(Vector v)
{
return(MathUtility.Acos(MathUtility.Clamp(v.Z, -1.0f, 1.0f)));
}