bool Disk::Intersect(const Ray &r, Float *tHit,
SurfaceInteraction *isect) const {
// Transform _Ray_ to object space
Vector3f oErr, dErr;
Ray ray = (*WorldToObject)(r, &oErr, &dErr);
// Compute plane intersection for disk
// Reject disk intersections for rays parallel to the disk's plane
if (ray.d.z == 0) return false;
Float tShapeHit = (height - ray.o.z) / ray.d.z;
if (tShapeHit <= 0 || tShapeHit >= ray.tMax) return false;
// See if hit point is inside disk radii and $\phimax$
Point3f pHit = ray(tShapeHit);
Float dist2 = pHit.x * pHit.x + pHit.y * pHit.y;
if (dist2 > radius * radius || dist2 < innerRadius * innerRadius)
return false;
// Refine disk intersection point
pHit.z = height;
// Test disk $\phi$ value against $\phimax$
Float phi = std::atan2(pHit.y, pHit.x);
if (phi < 0) phi += 2 * Pi;
if (phi > phiMax) return false;
// Find parametric representation of disk hit
Float u = phi / phiMax;
Float rHit = std::sqrt(dist2);
Float oneMinusV = ((rHit - innerRadius) / (radius - innerRadius));
Float v = 1 - oneMinusV;
Vector3f dpdu(-phiMax * pHit.y, phiMax * pHit.x, 0.);
Vector3f dpdv =
Vector3f(pHit.x, pHit.y, 0.) * (innerRadius - radius) / rHit;
Normal3f dndu(0, 0, 0), dndv(0, 0, 0);
// Compute error bounds for disk intersection
Vector3f pError(0., 0., 0.);
// Initialize _SurfaceInteraction_ from parametric information
*isect = (*ObjectToWorld)(SurfaceInteraction(pHit, pError, Point2f(u, v),
-ray.d, dpdu, dpdv, dndu, dndv,
ray.time, this));
// Update _tHit_ for quadric intersection
*tHit = (Float)tShapeHit;
return true;
}
bool Cylinder::Intersect(const Ray &r, Float *tHit, SurfaceInteraction *isect,
bool testAlphaTexture) const {
Float phi;
Point3f pHit;
// Transform _Ray_ to object space
Vector3f oErr, dErr;
Ray ray = (*WorldToObject)(r, &oErr, &dErr);
// Compute quadratic cylinder coefficients
// Initialize _EFloat_ ray coordinate values
EFloat ox(ray.o.x, oErr.x), oy(ray.o.y, oErr.y), oz(ray.o.z, oErr.z);
EFloat dx(ray.d.x, dErr.x), dy(ray.d.y, dErr.y), dz(ray.d.z, dErr.z);
EFloat a = dx * dx + dy * dy;
EFloat b = 2 * (dx * ox + dy * oy);
EFloat c = ox * ox + oy * oy - EFloat(radius) * EFloat(radius);
// Solve quadratic equation for _t_ values
EFloat t0, t1;
if (!Quadratic(a, b, c, &t0, &t1)) return false;
// Check quadric shape _t0_ and _t1_ for nearest intersection
if (t0.UpperBound() > ray.tMax || t1.LowerBound() <= 0) return false;
EFloat tShapeHit = t0;
if (tShapeHit.LowerBound() <= 0) {
tShapeHit = t1;
if (tShapeHit.UpperBound() > ray.tMax) return false;
}
// Compute cylinder hit point and $\phi$
pHit = ray((Float)tShapeHit);
// Refine cylinder intersection point
Float hitRad = std::sqrt(pHit.x * pHit.x + pHit.y * pHit.y);
pHit.x *= radius / hitRad;
pHit.y *= radius / hitRad;
phi = std::atan2(pHit.y, pHit.x);
if (phi < 0) phi += 2 * Pi;
// Test cylinder intersection against clipping parameters
if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) {
if (tShapeHit == t1) return false;
tShapeHit = t1;
if (t1.UpperBound() > ray.tMax) return false;
// Compute cylinder hit point and $\phi$
pHit = ray((Float)tShapeHit);
// Refine cylinder intersection point
Float hitRad = std::sqrt(pHit.x * pHit.x + pHit.y * pHit.y);
pHit.x *= radius / hitRad;
pHit.y *= radius / hitRad;
phi = std::atan2(pHit.y, pHit.x);
if (phi < 0) phi += 2 * Pi;
if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) return false;
}
// Find parametric representation of cylinder hit
Float u = phi / phiMax;
Float v = (pHit.z - zMin) / (zMax - zMin);
// Compute cylinder $\dpdu$ and $\dpdv$
Vector3f dpdu(-phiMax * pHit.y, phiMax * pHit.x, 0);
Vector3f dpdv(0, 0, zMax - zMin);
// Compute cylinder $\dndu$ and $\dndv$
Vector3f d2Pduu = -phiMax * phiMax * Vector3f(pHit.x, pHit.y, 0);
Vector3f d2Pduv(0, 0, 0), d2Pdvv(0, 0, 0);
// Compute coefficients for fundamental forms
Float E = Dot(dpdu, dpdu);
Float F = Dot(dpdu, dpdv);
Float G = Dot(dpdv, dpdv);
Vector3f N = Normalize(Cross(dpdu, dpdv));
Float e = Dot(N, d2Pduu);
Float f = Dot(N, d2Pduv);
Float g = Dot(N, d2Pdvv);
// Compute $\dndu$ and $\dndv$ from fundamental form coefficients
Float invEGF2 = 1 / (E * G - F * F);
Normal3f dndu = Normal3f((f * F - e * G) * invEGF2 * dpdu +
(e * F - f * E) * invEGF2 * dpdv);
Normal3f dndv = Normal3f((g * F - f * G) * invEGF2 * dpdu +
(f * F - g * E) * invEGF2 * dpdv);
// Compute error bounds for cylinder intersection
Vector3f pError = gamma(3) * Abs(Vector3f(pHit.x, pHit.y, 0));
// Initialize _SurfaceInteraction_ from parametric information
*isect = (*ObjectToWorld)(SurfaceInteraction(pHit, pError, Point2f(u, v),
-ray.d, dpdu, dpdv, dndu, dndv,
ray.time, this));
// Update _tHit_ for quadric intersection
*tHit = (Float)tShapeHit;
return true;
}
bool Curve::recursiveIntersect(const Ray &ray, Float *tHit,
SurfaceInteraction *isect, const Point3f cp[4],
const Transform &rayToObject, Float u0, Float u1,
int depth) const {
// Try to cull curve segment versus ray
// Compute bounding box of curve segment, _curveBounds_
Bounds3f curveBounds =
Union(Bounds3f(cp[0], cp[1]), Bounds3f(cp[2], cp[3]));
Float maxWidth = std::max(Lerp(u0, common->width[0], common->width[1]),
Lerp(u1, common->width[0], common->width[1]));
curveBounds = Expand(curveBounds, 0.5 * maxWidth);
// Compute bounding box of ray, _rayBounds_
Float rayLength = ray.d.Length();
Float zMax = rayLength * ray.tMax;
Bounds3f rayBounds(Point3f(0, 0, 0), Point3f(0, 0, zMax));
if (Overlaps(curveBounds, rayBounds) == false) return false;
if (depth > 0) {
// Split curve segment into sub-segments and test for intersection
Float uMid = 0.5f * (u0 + u1);
Point3f cpSplit[7];
SubdivideBezier(cp, cpSplit);
return (recursiveIntersect(ray, tHit, isect, &cpSplit[0], rayToObject,
u0, uMid, depth - 1) ||
recursiveIntersect(ray, tHit, isect, &cpSplit[3], rayToObject,
uMid, u1, depth - 1));
} else {
// Intersect ray with curve segment
// Test ray against segment endpoint boundaries
Vector2f segmentDirection = Point2f(cp[3]) - Point2f(cp[0]);
// Test sample point against tangent perpendicular at curve start
Float edge =
(cp[1].y - cp[0].y) * -cp[0].y + cp[0].x * (cp[0].x - cp[1].x);
if (edge < 0) return false;
// Test sample point against tangent perpendicular at curve end
// TODO: update to match the starting test.
Vector2f endTangent = Point2f(cp[2]) - Point2f(cp[3]);
if (Dot(segmentDirection, endTangent) < 0) endTangent = -endTangent;
if (Dot(endTangent, Vector2f(cp[3])) < 0) return false;
// Compute line $w$ that gives minimum distance to sample point
Float denom = Dot(segmentDirection, segmentDirection);
if (denom == 0) return false;
Float w = Dot(-Vector2f(cp[0]), segmentDirection) / denom;
// Compute $u$ coordinate of curve intersection point and _hitWidth_
Float u = Clamp(Lerp(w, u0, u1), u0, u1);
Float hitWidth = Lerp(u, common->width[0], common->width[1]);
Normal3f nHit;
if (common->type == CurveType::Ribbon) {
// Scale _hitWidth_ based on ribbon orientation
Float sin0 = std::sin((1 - u) * common->normalAngle) *
common->invSinNormalAngle;
Float sin1 =
std::sin(u * common->normalAngle) * common->invSinNormalAngle;
nHit = sin0 * common->n[0] + sin1 * common->n[1];
hitWidth *= AbsDot(nHit, -ray.d / rayLength);
}
// Test intersection point against curve width
Vector3f dpcdw;
Point3f pc = EvalBezier(cp, Clamp(w, 0, 1), &dpcdw);
Float ptCurveDist2 = pc.x * pc.x + pc.y * pc.y;
if (ptCurveDist2 > hitWidth * hitWidth * .25) return false;
if (pc.z < 0 || pc.z > zMax) return false;
// Compute $v$ coordinate of curve intersection point
Float ptCurveDist = std::sqrt(ptCurveDist2);
Float edgeFunc = dpcdw.x * -pc.y + pc.x * dpcdw.y;
Float v = (edgeFunc > 0.) ? 0.5f + ptCurveDist / hitWidth
: 0.5f - ptCurveDist / hitWidth;
// Compute hit _t_ and partial derivatives for curve intersection
if (tHit != nullptr) {
// FIXME: this tHit isn't quite right for ribbons...
*tHit = pc.z / rayLength;
// Compute error bounds for curve intersection
Vector3f pError(2 * hitWidth, 2 * hitWidth, 2 * hitWidth);
// Compute $\dpdu$ and $\dpdv$ for curve intersection
Vector3f dpdu, dpdv;
EvalBezier(common->cpObj, u, &dpdu);
if (common->type == CurveType::Ribbon)
dpdv = Normalize(Cross(nHit, dpdu)) * hitWidth;
else {
// Compute curve $\dpdv$ for flat and cylinder curves
Vector3f dpduPlane = (Inverse(rayToObject))(dpdu);
Vector3f dpdvPlane =
Normalize(Vector3f(-dpduPlane.y, dpduPlane.x, 0)) *
hitWidth;
if (common->type == CurveType::Cylinder) {
// Rotate _dpdvPlane_ to give cylindrical appearance
Float theta = Lerp(v, -90., 90.);
Transform rot = Rotate(-theta, dpduPlane);
dpdvPlane = rot(dpdvPlane);
}
dpdv = rayToObject(dpdvPlane);
}
*isect = (*ObjectToWorld)(SurfaceInteraction(
ray(pc.z), pError, Point2f(u, v), -ray.d, dpdu, dpdv,
Normal3f(0, 0, 0), Normal3f(0, 0, 0), ray.time, this));
}
++nHits;
return true;
}
}
bool Hyperboloid::Intersect(const Ray &r, Float *tHit,
SurfaceInteraction *isect) const {
Float phi, v;
Point3f pHit;
// Transform _Ray_ to object space
Vector3f oErr, dErr;
Ray ray = (*WorldToObject)(r, &oErr, &dErr);
// Compute quadratic hyperboloid coefficients
// Initialize _EFloat_ ray coordinate values
EFloat ox(ray.o.x, oErr.x), oy(ray.o.y, oErr.y), oz(ray.o.z, oErr.z);
EFloat dx(ray.d.x, dErr.x), dy(ray.d.y, dErr.y), dz(ray.d.z, dErr.z);
EFloat a = ah * dx * dx + ah * dy * dy - ch * dz * dz;
EFloat b = 2.f * (ah * dx * ox + ah * dy * oy - ch * dz * oz);
EFloat c = ah * ox * ox + ah * oy * oy - ch * oz * oz - 1.f;
// Solve quadratic equation for _t_ values
EFloat t0, t1;
if (!Quadratic(a, b, c, &t0, &t1)) return false;
// Check quadric shape _t0_ and _t1_ for nearest intersection
if (t0.UpperBound() > ray.tMax || t1.LowerBound() <= 0) return false;
EFloat tShapeHit = t0;
if (t0.LowerBound() <= 0) {
tShapeHit = t1;
if (tShapeHit.UpperBound() > ray.tMax) return false;
}
// Compute hyperboloid inverse mapping
pHit = ray((Float)tShapeHit);
v = (pHit.z - p1.z) / (p2.z - p1.z);
Point3f pr = (1 - v) * p1 + v * p2;
phi = std::atan2(pr.x * pHit.y - pHit.x * pr.y,
pHit.x * pr.x + pHit.y * pr.y);
if (phi < 0) phi += 2 * Pi;
// Test hyperboloid intersection against clipping parameters
if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) {
if (tShapeHit == t1) return false;
tShapeHit = t1;
if (t1.UpperBound() > ray.tMax) return false;
// Compute hyperboloid inverse mapping
pHit = ray((Float)tShapeHit);
v = (pHit.z - p1.z) / (p2.z - p1.z);
Point3f pr = (1 - v) * p1 + v * p2;
phi = std::atan2(pr.x * pHit.y - pHit.x * pr.y,
pHit.x * pr.x + pHit.y * pr.y);
if (phi < 0) phi += 2 * Pi;
if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) return false;
}
// Compute parametric representation of hyperboloid hit
Float u = phi / phiMax;
// Compute hyperboloid $\dpdu$ and $\dpdv$
Float cosPhi = std::cos(phi), sinPhi = std::sin(phi);
Vector3f dpdu(-phiMax * pHit.y, phiMax * pHit.x, 0.);
Vector3f dpdv((p2.x - p1.x) * cosPhi - (p2.y - p1.y) * sinPhi,
(p2.x - p1.x) * sinPhi + (p2.y - p1.y) * cosPhi, p2.z - p1.z);
// Compute hyperboloid $\dndu$ and $\dndv$
Vector3f d2Pduu = -phiMax * phiMax * Vector3f(pHit.x, pHit.y, 0);
Vector3f d2Pduv = phiMax * Vector3f(-dpdv.y, dpdv.x, 0.);
Vector3f d2Pdvv(0, 0, 0);
// Compute coefficients for fundamental forms
Float E = Dot(dpdu, dpdu);
Float F = Dot(dpdu, dpdv);
Float G = Dot(dpdv, dpdv);
Vector3f N = Normalize(Cross(dpdu, dpdv));
Float e = Dot(N, d2Pduu);
Float f = Dot(N, d2Pduv);
Float g = Dot(N, d2Pdvv);
// Compute $\dndu$ and $\dndv$ from fundamental form coefficients
Float invEGF2 = 1 / (E * G - F * F);
Normal3f dndu = Normal3f((f * F - e * G) * invEGF2 * dpdu +
(e * F - f * E) * invEGF2 * dpdv);
Normal3f dndv = Normal3f((g * F - f * G) * invEGF2 * dpdu +
(f * F - g * E) * invEGF2 * dpdv);
// Compute error bounds for hyperboloid intersection
// Compute error bounds for intersection computed with ray equation
EFloat px = ox + tShapeHit * dx;
EFloat py = oy + tShapeHit * dy;
EFloat pz = oz + tShapeHit * dz;
Vector3f pError = Vector3f(px.GetAbsoluteError(), py.GetAbsoluteError(),
pz.GetAbsoluteError());
// Initialize _SurfaceInteraction_ from parametric information
*isect = (*ObjectToWorld)(SurfaceInteraction(pHit, pError, Point2f(u, v),
-ray.d, dpdu, dpdv, dndu, dndv,
ray.time, this));
*tHit = (Float)tShapeHit;
return true;
}
bool Paraboloid::Intersect(const Ray &r, Float *tHit,
SurfaceInteraction *isect) const {
Float phi;
Point3f pHit;
// Transform _Ray_ to object space
Vector3f oErr, dErr;
Ray ray = (*WorldToObject)(r, &oErr, &dErr);
// Compute quadratic paraboloid coefficients
// Initialize _EFloat_ ray coordinate values
EFloat ox(ray.o.x, oErr.x), oy(ray.o.y, oErr.y), oz(ray.o.z, oErr.z);
EFloat dx(ray.d.x, dErr.x), dy(ray.d.y, dErr.y), dz(ray.d.z, dErr.z);
EFloat k = EFloat(zMax) / (EFloat(radius) * EFloat(radius));
EFloat a = k * (dx * dx + dy * dy);
EFloat b = 2.f * k * (dx * ox + dy * oy) - dz;
EFloat c = k * (ox * ox + oy * oy) - oz;
// Solve quadratic equation for _t_ values
EFloat t0, t1;
if (!Quadratic(a, b, c, &t0, &t1)) return false;
// Check quadric shape _t0_ and _t1_ for nearest intersection
if (t0.UpperBound() > ray.tMax || t1.LowerBound() <= 0) return false;
EFloat tShapeHit = t0;
if (tShapeHit.LowerBound() <= 0) {
tShapeHit = t1;
if (tShapeHit.UpperBound() > ray.tMax) return false;
}
// Compute paraboloid inverse mapping
pHit = ray((Float)tShapeHit);
phi = std::atan2(pHit.y, pHit.x);
if (phi < 0.) phi += 2 * Pi;
// Test paraboloid intersection against clipping parameters
if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) {
if (tShapeHit == t1) return false;
tShapeHit = t1;
if (t1.UpperBound() > ray.tMax) return false;
// Compute paraboloid inverse mapping
pHit = ray((Float)tShapeHit);
phi = std::atan2(pHit.y, pHit.x);
if (phi < 0.) phi += 2 * Pi;
if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) return false;
}
// Find parametric representation of paraboloid hit
Float u = phi / phiMax;
Float v = (pHit.z - zMin) / (zMax - zMin);
// Compute paraboloid $\dpdu$ and $\dpdv$
Vector3f dpdu(-phiMax * pHit.y, phiMax * pHit.x, 0.);
Vector3f dpdv = (zMax - zMin) *
Vector3f(pHit.x / (2 * pHit.z), pHit.y / (2 * pHit.z), 1.);
// Compute paraboloid $\dndu$ and $\dndv$
Vector3f d2Pduu = -phiMax * phiMax * Vector3f(pHit.x, pHit.y, 0);
Vector3f d2Pduv =
(zMax - zMin) * phiMax *
Vector3f(-pHit.y / (2 * pHit.z), pHit.x / (2 * pHit.z), 0);
Vector3f d2Pdvv = -(zMax - zMin) * (zMax - zMin) *
Vector3f(pHit.x / (4 * pHit.z * pHit.z),
pHit.y / (4 * pHit.z * pHit.z), 0.);
// Compute coefficients for fundamental forms
Float E = Dot(dpdu, dpdu);
Float F = Dot(dpdu, dpdv);
Float G = Dot(dpdv, dpdv);
Vector3f N = Normalize(Cross(dpdu, dpdv));
Float e = Dot(N, d2Pduu);
Float f = Dot(N, d2Pduv);
Float g = Dot(N, d2Pdvv);
// Compute $\dndu$ and $\dndv$ from fundamental form coefficients
Float invEGF2 = 1 / (E * G - F * F);
Normal3f dndu = Normal3f((f * F - e * G) * invEGF2 * dpdu +
(e * F - f * E) * invEGF2 * dpdv);
Normal3f dndv = Normal3f((g * F - f * G) * invEGF2 * dpdu +
(f * F - g * E) * invEGF2 * dpdv);
// Compute error bounds for paraboloid intersection
// Compute error bounds for intersection computed with ray equation
EFloat px = ox + tShapeHit * dx;
EFloat py = oy + tShapeHit * dy;
EFloat pz = oz + tShapeHit * dz;
Vector3f pError = Vector3f(px.GetAbsoluteError(), py.GetAbsoluteError(),
pz.GetAbsoluteError());
// Initialize _SurfaceInteraction_ from parametric information
*isect = (*ObjectToWorld)(SurfaceInteraction(pHit, pError, Point2f(u, v),
-ray.d, dpdu, dpdv, dndu, dndv,
ray.time, this));
*tHit = (Float)tShapeHit;
return true;
}
bool Sphere::Intersect(const Ray &r, Float *tHit, SurfaceInteraction *isect,
bool testAlphaTexture) const {
Float phi;
Point3f pHit;
// Transform _Ray_ to object space
Vector3f oErr, dErr;
Ray ray = (*WorldToObject)(r, &oErr, &dErr);
// Compute quadratic sphere coefficients
// Initialize _EFloat_ ray coordinate values
EFloat ox(ray.o.x, oErr.x), oy(ray.o.y, oErr.y), oz(ray.o.z, oErr.z);
EFloat dx(ray.d.x, dErr.x), dy(ray.d.y, dErr.y), dz(ray.d.z, dErr.z);
EFloat a = dx * dx + dy * dy + dz * dz;
EFloat b = 2 * (dx * ox + dy * oy + dz * oz);
EFloat c = ox * ox + oy * oy + oz * oz - EFloat(radius) * EFloat(radius);
// Solve quadratic equation for _t_ values
EFloat t0, t1;
if (!Quadratic(a, b, c, &t0, &t1)) return false;
// Check quadric shape _t0_ and _t1_ for nearest intersection
if (t0.UpperBound() > ray.tMax || t1.LowerBound() <= 0) return false;
EFloat tShapeHit = t0;
if (tShapeHit.LowerBound() <= 0) {
tShapeHit = t1;
if (tShapeHit.UpperBound() > ray.tMax) return false;
}
// Compute sphere hit position and $\phi$
pHit = ray((Float)tShapeHit);
// Refine sphere intersection point
pHit *= radius / Distance(pHit, Point3f(0, 0, 0));
if (pHit.x == 0 && pHit.y == 0) pHit.x = 1e-5f * radius;
phi = std::atan2(pHit.y, pHit.x);
if (phi < 0) phi += 2 * Pi;
// Test sphere intersection against clipping parameters
if ((zMin > -radius && pHit.z < zMin) || (zMax < radius && pHit.z > zMax) ||
phi > phiMax) {
if (tShapeHit == t1) return false;
if (t1.UpperBound() > ray.tMax) return false;
tShapeHit = t1;
// Compute sphere hit position and $\phi$
pHit = ray((Float)tShapeHit);
// Refine sphere intersection point
pHit *= radius / Distance(pHit, Point3f(0, 0, 0));
if (pHit.x == 0 && pHit.y == 0) pHit.x = 1e-5f * radius;
phi = std::atan2(pHit.y, pHit.x);
if (phi < 0) phi += 2 * Pi;
if ((zMin > -radius && pHit.z < zMin) ||
(zMax < radius && pHit.z > zMax) || phi > phiMax)
return false;
}
// Find parametric representation of sphere hit
Float u = phi / phiMax;
Float theta = std::acos(Clamp(pHit.z / radius, -1, 1));
Float v = (theta - thetaMin) / (thetaMax - thetaMin);
// Compute sphere $\dpdu$ and $\dpdv$
Float zRadius = std::sqrt(pHit.x * pHit.x + pHit.y * pHit.y);
Float invZRadius = 1 / zRadius;
Float cosPhi = pHit.x * invZRadius;
Float sinPhi = pHit.y * invZRadius;
Vector3f dpdu(-phiMax * pHit.y, phiMax * pHit.x, 0);
Vector3f dpdv =
(thetaMax - thetaMin) *
Vector3f(pHit.z * cosPhi, pHit.z * sinPhi, -radius * std::sin(theta));
// Compute sphere $\dndu$ and $\dndv$
Vector3f d2Pduu = -phiMax * phiMax * Vector3f(pHit.x, pHit.y, 0);
Vector3f d2Pduv =
(thetaMax - thetaMin) * pHit.z * phiMax * Vector3f(-sinPhi, cosPhi, 0.);
Vector3f d2Pdvv = -(thetaMax - thetaMin) * (thetaMax - thetaMin) *
Vector3f(pHit.x, pHit.y, pHit.z);
// Compute coefficients for fundamental forms
Float E = Dot(dpdu, dpdu);
Float F = Dot(dpdu, dpdv);
Float G = Dot(dpdv, dpdv);
Vector3f N = Normalize(Cross(dpdu, dpdv));
Float e = Dot(N, d2Pduu);
Float f = Dot(N, d2Pduv);
Float g = Dot(N, d2Pdvv);
// Compute $\dndu$ and $\dndv$ from fundamental form coefficients
Float invEGF2 = 1 / (E * G - F * F);
Normal3f dndu = Normal3f((f * F - e * G) * invEGF2 * dpdu +
(e * F - f * E) * invEGF2 * dpdv);
Normal3f dndv = Normal3f((g * F - f * G) * invEGF2 * dpdu +
(f * F - g * E) * invEGF2 * dpdv);
// Compute error bounds for sphere intersection
Vector3f pError = gamma(5) * Abs((Vector3f)pHit);
// Initialize _SurfaceInteraction_ from parametric information
*isect = (*ObjectToWorld)(SurfaceInteraction(pHit, pError, Point2f(u, v),
-ray.d, dpdu, dpdv, dndu, dndv,
ray.time, this));
// Update _tHit_ for quadric intersection
*tHit = (Float)tShapeHit;
return true;
}
bool Triangle::Intersect(const Ray &ray, Float *tHit, SurfaceInteraction *isect,
bool testAlphaTexture) const {
ProfilePhase p(Prof::TriIntersect);
++nTests;
// Get triangle vertices in _p0_, _p1_, and _p2_
const Point3f &p0 = mesh->p[v[0]];
const Point3f &p1 = mesh->p[v[1]];
const Point3f &p2 = mesh->p[v[2]];
// Perform ray--triangle intersection test
// Transform triangle vertices to ray coordinate space
// Translate vertices based on ray origin
Point3f p0t = p0 - Vector3f(ray.o);
Point3f p1t = p1 - Vector3f(ray.o);
Point3f p2t = p2 - Vector3f(ray.o);
// Permute components of triangle vertices and ray direction
int kz = MaxDimension(Abs(ray.d));
int kx = kz + 1;
if (kx == 3) kx = 0;
int ky = kx + 1;
if (ky == 3) ky = 0;
Vector3f d = Permute(ray.d, kx, ky, kz);
p0t = Permute(p0t, kx, ky, kz);
p1t = Permute(p1t, kx, ky, kz);
p2t = Permute(p2t, kx, ky, kz);
// Apply shear transformation to translated vertex positions
Float Sx = -d.x / d.z;
Float Sy = -d.y / d.z;
Float Sz = 1.f / d.z;
p0t.x += Sx * p0t.z;
p0t.y += Sy * p0t.z;
p1t.x += Sx * p1t.z;
p1t.y += Sy * p1t.z;
p2t.x += Sx * p2t.z;
p2t.y += Sy * p2t.z;
// Compute edge function coefficients _e0_, _e1_, and _e2_
Float e0 = p1t.x * p2t.y - p1t.y * p2t.x;
Float e1 = p2t.x * p0t.y - p2t.y * p0t.x;
Float e2 = p0t.x * p1t.y - p0t.y * p1t.x;
// Fall back to double precision test at triangle edges
if (sizeof(Float) == sizeof(float) &&
(e0 == 0.0f || e1 == 0.0f || e2 == 0.0f)) {
double p2txp1ty = (double)p2t.x * (double)p1t.y;
double p2typ1tx = (double)p2t.y * (double)p1t.x;
e0 = (float)(p2typ1tx - p2txp1ty);
double p0txp2ty = (double)p0t.x * (double)p2t.y;
double p0typ2tx = (double)p0t.y * (double)p2t.x;
e1 = (float)(p0typ2tx - p0txp2ty);
double p1txp0ty = (double)p1t.x * (double)p0t.y;
double p1typ0tx = (double)p1t.y * (double)p0t.x;
e2 = (float)(p1typ0tx - p1txp0ty);
}
// Perform triangle edge and determinant tests
if ((e0 < 0 || e1 < 0 || e2 < 0) && (e0 > 0 || e1 > 0 || e2 > 0))
return false;
Float det = e0 + e1 + e2;
if (det == 0) return false;
// Compute scaled hit distance to triangle and test against ray $t$ range
p0t.z *= Sz;
p1t.z *= Sz;
p2t.z *= Sz;
Float tScaled = e0 * p0t.z + e1 * p1t.z + e2 * p2t.z;
if (det < 0 && (tScaled >= 0 || tScaled < ray.tMax * det))
return false;
else if (det > 0 && (tScaled <= 0 || tScaled > ray.tMax * det))
return false;
// Compute barycentric coordinates and $t$ value for triangle intersection
Float invDet = 1 / det;
Float b0 = e0 * invDet;
Float b1 = e1 * invDet;
Float b2 = e2 * invDet;
Float t = tScaled * invDet;
// Ensure that computed triangle $t$ is conservatively greater than zero
// Compute $\delta_z$ term for triangle $t$ error bounds
Float maxZt = MaxComponent(Abs(Vector3f(p0t.z, p1t.z, p2t.z)));
Float deltaZ = gamma(3) * maxZt;
// Compute $\delta_x$ and $\delta_y$ terms for triangle $t$ error bounds
Float maxXt = MaxComponent(Abs(Vector3f(p0t.x, p1t.x, p2t.x)));
Float maxYt = MaxComponent(Abs(Vector3f(p0t.y, p1t.y, p2t.y)));
Float deltaX = gamma(5) * (maxXt + maxZt);
Float deltaY = gamma(5) * (maxYt + maxZt);
// Compute $\delta_e$ term for triangle $t$ error bounds
Float deltaE =
2 * (gamma(2) * maxXt * maxYt + deltaY * maxXt + deltaX * maxYt);
// Compute $\delta_t$ term for triangle $t$ error bounds and check _t_
Float maxE = MaxComponent(Abs(Vector3f(e0, e1, e2)));
Float deltaT = 3 *
(gamma(3) * maxE * maxZt + deltaE * maxZt + deltaZ * maxE) *
std::abs(invDet);
if (t <= deltaT) return false;
// Compute triangle partial derivatives
Vector3f dpdu, dpdv;
Point2f uv[3];
GetUVs(uv);
// Compute deltas for triangle partial derivatives
Vector2f duv02 = uv[0] - uv[2], duv12 = uv[1] - uv[2];
Vector3f dp02 = p0 - p2, dp12 = p1 - p2;
Float determinant = duv02[0] * duv12[1] - duv02[1] * duv12[0];
if (determinant == 0) {
// Handle zero determinant for triangle partial derivative matrix
CoordinateSystem(Normalize(Cross(p2 - p0, p1 - p0)), &dpdu, &dpdv);
} else {
Float invdet = 1 / determinant;
dpdu = (duv12[1] * dp02 - duv02[1] * dp12) * invdet;
dpdv = (-duv12[0] * dp02 + duv02[0] * dp12) * invdet;
}
// Compute error bounds for triangle intersection
Float xAbsSum =
(std::abs(b0 * p0.x) + std::abs(b1 * p1.x) + std::abs(b2 * p2.x));
Float yAbsSum =
(std::abs(b0 * p0.y) + std::abs(b1 * p1.y) + std::abs(b2 * p2.y));
Float zAbsSum =
(std::abs(b0 * p0.z) + std::abs(b1 * p1.z) + std::abs(b2 * p2.z));
Vector3f pError = gamma(7) * Vector3f(xAbsSum, yAbsSum, zAbsSum);
// Interpolate $(u,v)$ parametric coordinates and hit point
Point3f pHit = b0 * p0 + b1 * p1 + b2 * p2;
Point2f uvHit = b0 * uv[0] + b1 * uv[1] + b2 * uv[2];
// Test intersection against alpha texture, if present
if (testAlphaTexture && mesh->alphaMask) {
SurfaceInteraction isectLocal(pHit, Vector3f(0, 0, 0), uvHit, -ray.d,
dpdu, dpdv, Normal3f(0, 0, 0),
Normal3f(0, 0, 0), ray.time, this);
if (mesh->alphaMask->Evaluate(isectLocal) == 0) return false;
}
// Fill in _SurfaceInteraction_ from triangle hit
*isect = SurfaceInteraction(pHit, pError, uvHit, -ray.d, dpdu, dpdv,
Normal3f(0, 0, 0), Normal3f(0, 0, 0), ray.time,
this);
// Override surface normal in _isect_ for triangle
isect->n = isect->shading.n = Normal3f(Normalize(Cross(dp02, dp12)));
if (mesh->n || mesh->s) {
// Initialize _Triangle_ shading geometry
// Compute shading normal _ns_ for triangle
Normal3f ns;
if (mesh->n) {
ns = (b0 * mesh->n[v[0]] + b1 * mesh->n[v[1]] +
b2 * mesh->n[v[2]]);
if (ns.LengthSquared() > 0)
ns = Normalize(ns);
else
ns = isect->n;
} else
ns = isect->n;
// Compute shading tangent _ss_ for triangle
Vector3f ss;
if (mesh->s) {
ss = (b0 * mesh->s[v[0]] + b1 * mesh->s[v[1]] +
b2 * mesh->s[v[2]]);
if (ss.LengthSquared() > 0)
ss = Normalize(ss);
else
ss = Normalize(isect->dpdu);
}
else
ss = Normalize(isect->dpdu);
// Compute shading bitangent _ts_ for triangle and adjust _ss_
Vector3f ts = Cross(ss, ns);
if (ts.LengthSquared() > 0.f) {
ts = Normalize(ts);
ss = Cross(ts, ns);
} else
CoordinateSystem((Vector3f)ns, &ss, &ts);
// Compute $\dndu$ and $\dndv$ for triangle shading geometry
Normal3f dndu, dndv;
if (mesh->n) {
// Compute deltas for triangle partial derivatives of normal
Vector2f duv02 = uv[0] - uv[2];
Vector2f duv12 = uv[1] - uv[2];
Normal3f dn1 = mesh->n[v[0]] - mesh->n[v[2]];
Normal3f dn2 = mesh->n[v[1]] - mesh->n[v[2]];
Float determinant = duv02[0] * duv12[1] - duv02[1] * duv12[0];
if (determinant == 0)
dndu = dndv = Normal3f(0, 0, 0);
else {
Float invDet = 1 / determinant;
dndu = (duv12[1] * dn1 - duv02[1] * dn2) * invDet;
dndv = (-duv12[0] * dn1 + duv02[0] * dn2) * invDet;
}
} else
dndu = dndv = Normal3f(0, 0, 0);
isect->SetShadingGeometry(ss, ts, dndu, dndv, true);
}
// Ensure correct orientation of the geometric normal
if (mesh->n)
isect->n = Faceforward(isect->n, isect->shading.n);
else if (reverseOrientation ^ transformSwapsHandedness)
isect->n = isect->shading.n = -isect->n;
*tHit = t;
++nHits;
return true;
}
