#ifndef UNITY_GEOMETRICTOOLS_INCLUDED
#define UNITY_GEOMETRICTOOLS_INCLUDED
//-----------------------------------------------------------------------------
// Transform functions
//-----------------------------------------------------------------------------
// Rotate around a pivot point and an axis
float3 Rotate(float3 pivot, float3 position, float3 rotationAxis, float angle)
{
rotationAxis = normalize(rotationAxis);
float3 cpa = pivot + rotationAxis * dot(rotationAxis, position - pivot);
return cpa + ((position - cpa) * cos(angle) + cross(rotationAxis, (position - cpa)) * sin(angle));
}
float3x3 RotationFromAxisAngle(float3 A, float sinAngle, float cosAngle)
{
float c = cosAngle;
float s = sinAngle;
return float3x3(A.x * A.x * (1 - c) + c, A.x * A.y * (1 - c) - A.z * s, A.x * A.z * (1 - c) + A.y * s,
A.x * A.y * (1 - c) + A.z * s, A.y * A.y * (1 - c) + c, A.y * A.z * (1 - c) - A.x * s,
A.x * A.z * (1 - c) - A.y * s, A.y * A.z * (1 - c) + A.x * s, A.z * A.z * (1 - c) + c);
}
//-----------------------------------------------------------------------------
// Solver
//-----------------------------------------------------------------------------
// Solves the quadratic equation of the form: a*t^2 + b*t + c = 0.
// Returns 'false' if there are no real roots, 'true' otherwise.
// Ensures that roots.x <= roots.y.
bool SolveQuadraticEquation(float a, float b, float c, out float2 roots)
{
float det = Sq(b) - 4.0 * a * c;
float sqrtDet = sqrt(det);
roots.x = (-b - sign(a) * sqrtDet) / (2.0 * a);
roots.y = (-b + sign(a) * sqrtDet) / (2.0 * a);
return (det >= 0.0);
}
//-----------------------------------------------------------------------------
// Intersection functions
//-----------------------------------------------------------------------------
bool IntersectRayAABB(float3 rayOrigin, float3 rayDirection,
float3 boxMin, float3 boxMax,
float tMin, float tMax,
out float tEntr, out float tExit)
{
// Could be precomputed. Clamp to avoid INF. clamp() is a single ALU on GCN.
// rcp(FLT_EPS) = 16,777,216, which is large enough for our purposes,
// yet doesn't cause a lot of numerical issues associated with FLT_MAX.
float3 rayDirInv = clamp(rcp(rayDirection), -rcp(FLT_EPS), rcp(FLT_EPS));
// Perform ray-slab intersection (component-wise).
float3 t0 = boxMin * rayDirInv - (rayOrigin * rayDirInv);
float3 t1 = boxMax * rayDirInv - (rayOrigin * rayDirInv);
// Find the closest/farthest distance (component-wise).
float3 tSlabEntr = min(t0, t1);
float3 tSlabExit = max(t0, t1);
// Find the farthest entry and the nearest exit.
tEntr = Max3(tSlabEntr.x, tSlabEntr.y, tSlabEntr.z);
tExit = Min3(tSlabExit.x, tSlabExit.y, tSlabExit.z);
// Clamp to the range.
tEntr = max(tEntr, tMin);
tExit = min(tExit, tMax);
return tEntr < tExit;
}
// This simplified version assume that we care about the result only when we are inside the box
float IntersectRayAABBSimple(float3 start, float3 dir, float3 boxMin, float3 boxMax)
{
float3 invDir = rcp(dir);
// Find the ray intersection with box plane
float3 rbmin = (boxMin - start) * invDir;
float3 rbmax = (boxMax - start) * invDir;
float3 rbminmax = float3((dir.x > 0.0) ? rbmax.x : rbmin.x, (dir.y > 0.0) ? rbmax.y : rbmin.y, (dir.z > 0.0) ? rbmax.z : rbmin.z);
return min(min(rbminmax.x, rbminmax.y), rbminmax.z);
}
// Assume Sphere is at the origin (i.e start = position - spherePosition)
bool IntersectRaySphere(float3 start, float3 dir, float radius, out float2 intersections)
{
float a = dot(dir, dir);
float b = dot(dir, start) * 2.0;
float c = dot(start, start) - radius * radius;
return SolveQuadraticEquation(a, b, c, intersections);
}
// This simplified version assume that we care about the result only when we are inside the sphere
// Assume Sphere is at the origin (i.e start = position - spherePosition) and dir is normalized
// Ref: http://http.developer.nvidia.com/GPUGems/gpugems_ch19.html
float IntersectRaySphereSimple(float3 start, float3 dir, float radius)
{
float b = dot(dir, start) * 2.0;
float c = dot(start, start) - radius * radius;
float discriminant = b * b - 4.0 * c;
return abs(sqrt(discriminant) - b) * 0.5;
}
float3 IntersectRayPlane(float3 rayOrigin, float3 rayDirection, float3 planeOrigin, float3 planeNormal)
{
float dist = dot(planeNormal, planeOrigin - rayOrigin) / dot(planeNormal, rayDirection);
return rayOrigin + rayDirection * dist;
}
// Same as above but return intersection distance and true / false if the ray hit/miss
bool IntersectRayPlane(float3 rayOrigin, float3 rayDirection, float3 planePosition, float3 planeNormal, out float t)
{
bool res = false;
t = -1.0;
float denom = dot(planeNormal, rayDirection);
if (abs(denom) > 1e-5)
{
float3 d = planePosition - rayOrigin;
t = dot(d, planeNormal) / denom;
res = (t >= 0);
}
return res;
}
// Can support cones with an elliptic base: pre-scale 'coneAxisX' and 'coneAxisY' by (h/r_x) and (h/r_y).
// Returns parametric distances 'tEntr' and 'tExit' along the ray,
// subject to constraints 'tMin' and 'tMax'.
bool IntersectRayCone(float3 rayOrigin, float3 rayDirection,
float3 coneOrigin, float3 coneDirection,
float3 coneAxisX, float3 coneAxisY,
float tMin, float tMax,
out float tEntr, out float tExit)
{
// Inverse transform the ray into a coordinate system with the cone at the origin facing along the Z axis.
float3x3 rotMat = float3x3(coneAxisX, coneAxisY, coneDirection);
float3 o = mul(rotMat, rayOrigin - coneOrigin);
float3 d = mul(rotMat, rayDirection);
// Cone equation (facing along Z): (h/r*x)^2 + (h/r*y)^2 - z^2 = 0.
// Cone axes are premultiplied with (h/r).
// Set up the quadratic equation: a*t^2 + b*t + c = 0.
float a = d.x * d.x + d.y * d.y - d.z * d.z;
float b = o.x * d.x + o.y * d.y - o.z * d.z;
float c = o.x * o.x + o.y * o.y - o.z * o.z;
float2 roots;
// Check whether we have at least 1 root.
bool hit = SolveQuadraticEquation(a, 2 * b, c, roots);
tEntr = roots.x;
tExit = roots.y;
float3 pEntr = o + tEntr * d;
float3 pExit = o + tExit * d;
// Clip the negative cone.
bool pEntrNeg = pEntr.z < 0;
bool pExitNeg = pExit.z < 0;
if (pEntrNeg && pExitNeg) { hit = false; }
if (pEntrNeg) { tEntr = tExit; tExit = tMax; }
if (pExitNeg) { tExit = tEntr; tEntr = tMin; }
// Clamp using the values passed into the function.
tEntr = clamp(tEntr, tMin, tMax);
tExit = clamp(tExit, tMin, tMax);
// Check for grazing intersections.
if (tEntr == tExit) { hit = false; }
return hit;
}
bool IntersectSphereAABB(float3 position, float radius, float3 aabbMin, float3 aabbMax)
{
float x = max(aabbMin.x, min(position.x, aabbMax.x));
float y = max(aabbMin.y, min(position.y, aabbMax.y));
float z = max(aabbMin.z, min(position.z, aabbMax.z));
float distance2 = ((x - position.x) * (x - position.x) + (y - position.y) * (y - position.y) + (z - position.z) * (z - position.z));
return distance2 < radius * radius;
}
//-----------------------------------------------------------------------------
// Miscellaneous functions
//-----------------------------------------------------------------------------
// Box is AABB
float DistancePointBox(float3 position, float3 boxMin, float3 boxMax)
{
return length(max(max(position - boxMax, boxMin - position), float3(0.0, 0.0, 0.0)));
}
float3 ProjectPointOnPlane(float3 position, float3 planePosition, float3 planeNormal)
{
return position - (dot(position - planePosition, planeNormal) * planeNormal);
}
// Plane equation: {(a, b, c) = N, d = -dot(N, P)}.
// Returns the distance from the plane to the point 'p' along the normal.
// Positive -> in front (above), negative -> behind (below).
float DistanceFromPlane(float3 p, float4 plane)
{
return dot(float4(p, 1.0), plane);
}
// Returns 'true' if the triangle is outside of the frustum.
// 'epsilon' is the (negative) distance to (outside of) the frustum below which we cull the triangle.
bool CullTriangleFrustum(float3 p0, float3 p1, float3 p2, float epsilon, float4 frustumPlanes[6], int numPlanes)
{
bool outside = false;
for (int i = 0; i < numPlanes; i++)
{
// If all 3 points are behind any of the planes, we cull.
outside = outside || Max3(DistanceFromPlane(p0, frustumPlanes[i]),
DistanceFromPlane(p1, frustumPlanes[i]),
DistanceFromPlane(p2, frustumPlanes[i])) < epsilon;
}
return outside;
}
// Returns 'true' if the edge of the triangle is outside of the frustum.
// The edges are defined s.t. they are on the opposite side of the point with the given index.
// 'epsilon' is the (negative) distance to (outside of) the frustum below which we cull the triangle.
//output packing:
// x,y,z - one component per triangle edge, true if outside, false otherwise
// w - true if entire triangle is outside of at least 1 plane of the frustum, false otherwise
bool4 CullFullTriangleAndEdgesFrustum(float3 p0, float3 p1, float3 p2, float epsilon, float4 frustumPlanes[6], int numPlanes)
{
bool4 edgesOutsideXYZ_triangleOutsideW = false;
for (int i = 0; i < numPlanes; i++)
{
bool3 pointsOutside = bool3(DistanceFromPlane(p0, frustumPlanes[i]) < epsilon,
DistanceFromPlane(p1, frustumPlanes[i]) < epsilon,
DistanceFromPlane(p2, frustumPlanes[i]) < epsilon);
bool3 edgesOutside;
// If both points of the edge are behind any of the planes, we cull.
edgesOutside.x = pointsOutside.y && pointsOutside.z;
edgesOutside.y = pointsOutside.x && pointsOutside.z;
edgesOutside.z = pointsOutside.x && pointsOutside.y;
edgesOutsideXYZ_triangleOutsideW = bool4(edgesOutsideXYZ_triangleOutsideW.x || edgesOutside.x,
edgesOutsideXYZ_triangleOutsideW.y || edgesOutside.y,
edgesOutsideXYZ_triangleOutsideW.z || edgesOutside.z,
all(pointsOutside));
}
return edgesOutsideXYZ_triangleOutsideW;
}
// Returns 'true' if the edge of the triangle is outside of the frustum.
// The edges are defined s.t. they are on the opposite side of the point with the given index.
// 'epsilon' is the (negative) distance to (outside of) the frustum below which we cull the triangle.
//output packing:
// x,y,z - one component per triangle edge, true if outside, false otherwise
bool3 CullTriangleEdgesFrustum(float3 p0, float3 p1, float3 p2, float epsilon, float4 frustumPlanes[6], int numPlanes)
{
return CullFullTriangleAndEdgesFrustum(p0, p1, p2, epsilon, frustumPlanes, numPlanes).xyz;
}
bool CullTriangleBackFaceView(float3 p0, float3 p1, float3 p2, float epsilon, float3 V, float winding)
{
float3 edge1 = p1 - p0;
float3 edge2 = p2 - p0;
float3 N = cross(edge1, edge2);
float NdotV = dot(N, V) * winding;
// Optimize:
// NdotV / (length(N) * length(V)) < Epsilon
// NdotV < Epsilon * length(N) * length(V)
// NdotV < Epsilon * sqrt(dot(N, N)) * sqrt(dot(V, V))
// NdotV < Epsilon * sqrt(dot(N, N) * dot(V, V))
return NdotV < epsilon * sqrt(dot(N, N) * dot(V, V));
}
// Returns 'true' if a triangle defined by 3 vertices is back-facing.
// 'epsilon' is the (negative) value of dot(N, V) below which we cull the triangle.
// 'winding' can be used to change the order: pass 1 for (p0 -> p1 -> p2), or -1 for (p0 -> p2 -> p1).
bool CullTriangleBackFace(float3 p0, float3 p1, float3 p2, float epsilon, float3 viewPos, float winding)
{
float3 V = viewPos - p0;
return CullTriangleBackFaceView(p0, p1, p2, epsilon, V, winding);
}
#endif // UNITY_GEOMETRICTOOLS_INCLUDED