public static void ComputeJacobians(ref Vector3Wide tangentX, ref Vector3Wide tangentY, ref Vector3Wide offsetA, ref Vector3Wide offsetB,
out Jacobians jacobians)
{
//Two velocity constraints:
//dot(velocity(p, A), tangentX) = dot(velocity(p, B), tangentX)
//dot(velocity(p, A), tangentY) = dot(velocity(p, B), tangentY)
//where velocity(p, A) is the velocity of a point p attached to object A.
//velocity(p, A) = linearVelocityA + angularVelocityA x (p - positionA) = linearVelocityA + angularVelocityA x offsetA
//so:
//dot(velocity(p, A), tangentX) = dot(linearVelocityA, tangentX) + dot(angularVelocityA x offsetA, tangentX)
//dot(velocity(p, A), tangentX) = dot(linearVelocityA, tangentX) + dot(offsetA x tangentX, angularVelocityA)
//Restating the two constraints:
//dot(linearVelocityA, tangentX) + dot(offsetA x tangentX, angularVelocityA) = dot(linearVelocityB, tangentX) + dot(offsetB x tangentX, angularVelocityB)
//dot(linearVelocityA, tangentY) + dot(offsetA x tangentY, angularVelocityA) = dot(linearVelocityB, tangentY) + dot(offsetB x tangentY, angularVelocityB)
//dot(linearVelocityA, tangentX) + dot(offsetA x tangentX, angularVelocityA) - dot(linearVelocityB, tangentX) - dot(offsetB x tangentX, angularVelocityB) = 0
//dot(linearVelocityA, tangentY) + dot(offsetA x tangentY, angularVelocityA) - dot(linearVelocityB, tangentY) - dot(offsetB x tangentY, angularVelocityB) = 0
//Since there are two constraints (2DOFs), there are two rows in the jacobian, which based on the above is:
//jLinearA = [ tangentX ]
// [ tangentY ]
//jAngularA = [ offsetA x tangentX ]
// [ offsetA x tangentY ]
//jLinearB = [ -tangentX ]
// [ -tangentY ]
//jAngularB = [ -offsetB x tangentX ] = [ tangentX x offsetB ]
// [ -offsetB x tangentY ] [ tangentY x offsetB ]
//TODO: there would be a minor benefit in eliminating this copy manually, since it's very likely that the compiler won't. And it's probably also introducing more locals init.
jacobians.LinearA.X = tangentX;
jacobians.LinearA.Y = tangentY;
Vector3Wide.CrossWithoutOverlap(offsetA, tangentX, out jacobians.AngularA.X);
Vector3Wide.CrossWithoutOverlap(offsetA, tangentY, out jacobians.AngularA.Y);
Vector3Wide.CrossWithoutOverlap(tangentX, offsetB, out jacobians.AngularB.X);
Vector3Wide.CrossWithoutOverlap(tangentY, offsetB, out jacobians.AngularB.Y);
}
public static void Prestep(ref BodyInertias inertiaA, ref Vector3Wide normal, ref Contact3OneBodyPrestepData prestep, float dt, float inverseDt,
out Projection projection)
{
Vector3Wide.CrossWithoutOverlap(prestep.OffsetA0, normal, out projection.Penetration0.AngularA);
Vector3Wide.CrossWithoutOverlap(prestep.OffsetA1, normal, out projection.Penetration1.AngularA);
Vector3Wide.CrossWithoutOverlap(prestep.OffsetA2, normal, out projection.Penetration2.AngularA);
//effective mass
Symmetric3x3Wide.VectorSandwich(projection.Penetration0.AngularA, inertiaA.InverseInertiaTensor, out var angularA0);
Symmetric3x3Wide.VectorSandwich(projection.Penetration1.AngularA, inertiaA.InverseInertiaTensor, out var angularA1);
Symmetric3x3Wide.VectorSandwich(projection.Penetration2.AngularA, inertiaA.InverseInertiaTensor, out var angularA2);
//Linear effective mass contribution notes:
//1) The J * M^-1 * JT can be reordered to J * JT * M^-1 for the linear components, since M^-1 is a scalar and dot(n * scalar, n) = dot(n, n) * scalar.
//2) dot(normal, normal) == 1, so the contribution from each body is just its inverse mass.
SpringSettingsWide.ComputeSpringiness(ref prestep.SpringSettings, dt, out var positionErrorToVelocity, out var effectiveMassCFMScale, out projection.SoftnessImpulseScale);
//Note that we don't precompute the JT * effectiveMass term. Since the jacobians are shared, we have to do that multiply anyway.
projection.Penetration0.EffectiveMass = effectiveMassCFMScale / (inertiaA.InverseMass + angularA0);
projection.Penetration1.EffectiveMass = effectiveMassCFMScale / (inertiaA.InverseMass + angularA1);
projection.Penetration2.EffectiveMass = effectiveMassCFMScale / (inertiaA.InverseMass + angularA2);
//If depth is negative, the bias velocity will permit motion up until the depth hits zero. This works because positionErrorToVelocity * dt will always be <=1.
var inverseDtVector = new Vector <float>(inverseDt);
projection.Penetration0.BiasVelocity = Vector.Min(prestep.PenetrationDepth0 * inverseDtVector, Vector.Min(prestep.PenetrationDepth0 * positionErrorToVelocity, prestep.MaximumRecoveryVelocity));
projection.Penetration1.BiasVelocity = Vector.Min(prestep.PenetrationDepth1 * inverseDtVector, Vector.Min(prestep.PenetrationDepth1 * positionErrorToVelocity, prestep.MaximumRecoveryVelocity));
projection.Penetration2.BiasVelocity = Vector.Min(prestep.PenetrationDepth2 * inverseDtVector, Vector.Min(prestep.PenetrationDepth2 * positionErrorToVelocity, prestep.MaximumRecoveryVelocity));
}
public static void ComputeJacobians(ref Vector3Wide tangentX, ref Vector3Wide tangentY, ref Vector3Wide offsetA, out Jacobians jacobians)
{
//TODO: there would be a minor benefit in eliminating this copy manually, since it's very likely that the compiler won't. And it's probably also introducing more locals init.
jacobians.LinearA.X = tangentX;
jacobians.LinearA.Y = tangentY;
Vector3Wide.CrossWithoutOverlap(offsetA, tangentX, out jacobians.AngularA.X);
Vector3Wide.CrossWithoutOverlap(offsetA, tangentY, out jacobians.AngularA.Y);
}
public static void Prestep(ref BodyInertias inertiaA, ref BodyInertias inertiaB,
ref Vector3Wide contactOffsetA, ref Vector3Wide contactOffsetB, ref Vector3Wide normal, ref Vector <float> depth, ref SpringSettingsWide springSettings, ref Vector <float> maximumRecoveryVelocity,
float dt, float inverseDt,
out Projection projection)
{
//We directly take the prestep data here since the jacobians and error don't undergo any processing.
//The contact penetration constraint takes the form:
//dot(positionA + offsetA, N) >= dot(positionB + offsetB, N)
//Or:
//dot(positionA + offsetA, N) - dot(positionB + offsetB, N) >= 0
//dot(positionA + offsetA - positionB - offsetB, N) >= 0
//where positionA and positionB are the center of mass positions of the bodies offsetA and offsetB are world space offsets from the center of mass to the contact,
//and N is a unit length vector calibrated to point from B to A. (The normal pointing direction is important; it changes the sign.)
//In practice, we'll use the collision detection system's penetration depth instead of trying to recompute the error here.
//So, treating the normal as constant, the velocity constraint is:
//dot(d/dt(positionA + offsetA - positionB - offsetB), N) >= 0
//dot(linearVelocityA + d/dt(offsetA) - linearVelocityB - d/dt(offsetB)), N) >= 0
//The velocity of the offsets are defined by the angular velocity.
//dot(linearVelocityA + angularVelocityA x offsetA - linearVelocityB - angularVelocityB x offsetB), N) >= 0
//dot(linearVelocityA, N) + dot(angularVelocityA x offsetA, N) - dot(linearVelocityB, N) - dot(angularVelocityB x offsetB), N) >= 0
//Use the properties of the scalar triple product:
//dot(linearVelocityA, N) + dot(offsetA x N, angularVelocityA) - dot(linearVelocityB, N) - dot(offsetB x N, angularVelocityB) >= 0
//Bake in the negations:
//dot(linearVelocityA, N) + dot(offsetA x N, angularVelocityA) + dot(linearVelocityB, -N) + dot(-offsetB x N, angularVelocityB) >= 0
//A x B = -B x A:
//dot(linearVelocityA, N) + dot(offsetA x N, angularVelocityA) + dot(linearVelocityB, -N) + dot(N x offsetB, angularVelocityB) >= 0
//And there you go, the jacobians!
//linearA: N
//angularA: offsetA x N
//linearB: -N
//angularB: N x offsetB
//Note that we leave the penetration depth as is, even when it's negative. Speculative contacts!
Vector3Wide.CrossWithoutOverlap(contactOffsetA, normal, out projection.Penetration0.AngularA);
Vector3Wide.CrossWithoutOverlap(normal, contactOffsetB, out projection.Penetration0.AngularB);
//effective mass
Symmetric3x3Wide.VectorSandwich(projection.Penetration0.AngularA, inertiaA.InverseInertiaTensor, out var angularA0);
Symmetric3x3Wide.VectorSandwich(projection.Penetration0.AngularB, inertiaB.InverseInertiaTensor, out var angularB0);
//Linear effective mass contribution notes:
//1) The J * M^-1 * JT can be reordered to J * JT * M^-1 for the linear components, since M^-1 is a scalar and dot(n * scalar, n) = dot(n, n) * scalar.
//2) dot(normal, normal) == 1, so the contribution from each body is just its inverse mass.
SpringSettingsWide.ComputeSpringiness(ref springSettings, dt, out var positionErrorToVelocity, out var effectiveMassCFMScale, out projection.SoftnessImpulseScale);
var linear = inertiaA.InverseMass + inertiaB.InverseMass;
//Note that we don't precompute the JT * effectiveMass term. Since the jacobians are shared, we have to do that multiply anyway.
projection.Penetration0.EffectiveMass = effectiveMassCFMScale / (linear + angularA0 + angularB0);
//If depth is negative, the bias velocity will permit motion up until the depth hits zero. This works because positionErrorToVelocity * dt will always be <=1.
projection.Penetration0.BiasVelocity = Vector.Min(
depth * new Vector <float>(inverseDt),
Vector.Min(depth * positionErrorToVelocity, maximumRecoveryVelocity));
}
public static void ComputeCorrectiveImpulse(ref BodyVelocities velocityA, ref BodyVelocities velocityB, ref Vector3Wide offsetA, ref Vector3Wide offsetB,
ref Vector3Wide biasVelocity, ref Symmetric3x3Wide effectiveMass, ref Vector <float> softnessImpulseScale, ref Vector3Wide accumulatedImpulse, out Vector3Wide correctiveImpulse)
{
//csi = projection.BiasImpulse - accumulatedImpulse * projection.SoftnessImpulseScale - (csiaLinear + csiaAngular + csibLinear + csibAngular);
//Note subtraction; jLinearB = -I.
Vector3Wide.Subtract(velocityA.Linear, velocityB.Linear, out var csv);
Vector3Wide.CrossWithoutOverlap(velocityA.Angular, offsetA, out var angularCSV);
Vector3Wide.Add(csv, angularCSV, out csv);
//Note reversed cross order; matches the jacobian -CrossMatrix(offsetB).
Vector3Wide.CrossWithoutOverlap(offsetB, velocityB.Angular, out angularCSV);
Vector3Wide.Add(csv, angularCSV, out csv);
Vector3Wide.Subtract(biasVelocity, csv, out csv);
Symmetric3x3Wide.TransformWithoutOverlap(csv, effectiveMass, out correctiveImpulse);
Vector3Wide.Scale(accumulatedImpulse, softnessImpulseScale, out var softness);
Vector3Wide.Subtract(correctiveImpulse, softness, out correctiveImpulse);
}
public static void ApplyImpulse(ref BodyVelocities velocityA, ref BodyVelocities velocityB,
ref Vector3Wide offsetA, ref Vector3Wide offsetB, ref BodyInertias inertiaA, ref BodyInertias inertiaB, ref Vector3Wide constraintSpaceImpulse)
{
Vector3Wide.CrossWithoutOverlap(offsetA, constraintSpaceImpulse, out var wsi);
Symmetric3x3Wide.TransformWithoutOverlap(wsi, inertiaA.InverseInertiaTensor, out var change);
Vector3Wide.Add(velocityA.Angular, change, out velocityA.Angular);
Vector3Wide.Scale(constraintSpaceImpulse, inertiaA.InverseMass, out change);
Vector3Wide.Add(velocityA.Linear, change, out velocityA.Linear);
Vector3Wide.CrossWithoutOverlap(constraintSpaceImpulse, offsetB, out wsi); //note flip-negation
Symmetric3x3Wide.TransformWithoutOverlap(wsi, inertiaB.InverseInertiaTensor, out change);
Vector3Wide.Add(velocityB.Angular, change, out velocityB.Angular);
Vector3Wide.Scale(constraintSpaceImpulse, inertiaB.InverseMass, out change);
Vector3Wide.Subtract(velocityB.Linear, change, out velocityB.Linear); //note subtraction; the jacobian is -I
}
public void Test(ref SphereWide a, ref TriangleWide b, ref Vector <float> speculativeMargin, ref Vector3Wide offsetB, ref QuaternionWide orientationB, out Convex1ContactManifoldWide manifold)
{
//Work in the local space of the triangle, since it's quicker to transform the sphere position than the vertices of the triangle.
Matrix3x3Wide.CreateFromQuaternion(orientationB, out var rB);
Matrix3x3Wide.TransformByTransposedWithoutOverlap(offsetB, rB, out var localOffsetB);
Vector3Wide.Subtract(b.B, b.A, out var ab);
Vector3Wide.Subtract(b.C, b.A, out var ac);
//localOffsetA = -localOffsetB, so pa = triangle.A + localOffsetB.
Vector3Wide.Add(b.A, localOffsetB, out var pa);
Vector3Wide.CrossWithoutOverlap(ab, ac, out var localTriangleNormal);
Vector3Wide.Dot(localTriangleNormal, pa, out var paN);
var collidingWithSolidSide = Vector.GreaterThan(paN, Vector <float> .Zero);
if (Vector.EqualsAll(collidingWithSolidSide, Vector <int> .Zero))
{
//No lanes can generate contacts due to the triangle's one sidedness.
manifold.ContactExists = Vector <int> .Zero;
return;
}
//EdgeAB plane test: (pa x ab) * (ab x ac) >= 0
//EdgeAC plane test: (ac x pa) * (ab x ac) >= 0
//Note that these are scaled versions of the barycentric coordinates.
//To normalize them such that the weights of a point within the triangle equal 1, we just need to divide by dot(ab x ac, ab x ac).
//In other words, to test the third edge plane, we can ensure that the unnormalized weights are both positive and sum to a value less than dot(ab x ac, ab x ac).
//If a point is outside of an edge plane, we know that it's not in the face region or any other edge region. It could, however, be in an adjacent vertex region.
//Vertex cases can be handled by clamping an edge case.
//Further, note that for any query location, it is sufficient to only test one edge even if the point is outside two edge planes. If it's outside two edge planes,
//that just means it's going to be on the shared vertex, so a clamped edge test captures the correct closest point.
//So, at each edge, if the point is outside the plane, cache the edge. The last edge registering an outside result will be tested.
//(pa x ab) * (ab x ac) = (pa * ab) * (ab * ac) - (pa * ac) * (ab * ab)
//(ac x pa) * (ab x ac) = (ac * ab) * (pa * ac) - (ac * ac) * (pa * ab)
//(ab x ac) * (ab x ac) = (ab * ab) * (ac * ac) - (ab * ac) * (ab * ac)
Vector3Wide.Dot(pa, ab, out var abpa);
Vector3Wide.Dot(ab, ac, out var abac);
Vector3Wide.Dot(ac, pa, out var acpa);
Vector3Wide.Dot(ac, ac, out var acac);
Vector3Wide.Dot(ab, ab, out var abab);
var edgePlaneTestAB = abpa * abac - acpa * abab;
var edgePlaneTestAC = abac * acpa - acac * abpa;
var triangleNormalLengthSquared = abab * acac - abac * abac;
var edgePlaneTestBC = triangleNormalLengthSquared - edgePlaneTestAB - edgePlaneTestAC;
var outsideAB = Vector.LessThan(edgePlaneTestAB, Vector <float> .Zero);
var outsideAC = Vector.LessThan(edgePlaneTestAC, Vector <float> .Zero);
var outsideBC = Vector.LessThan(edgePlaneTestBC, Vector <float> .Zero);
var outsideAnyEdge = Vector.BitwiseOr(outsideAB, Vector.BitwiseOr(outsideAC, outsideBC));
Vector3Wide localClosestOnTriangle;
var negativeOne = new Vector <int>(-1);
if (Vector.EqualsAny(Vector.BitwiseAnd(collidingWithSolidSide, outsideAnyEdge), negativeOne))
{
//At least one lane detected a point outside of the triangle. Choose one edge which is outside as the representative.
Vector3Wide.ConditionalSelect(outsideAC, ac, ab, out var edgeDirection);
Vector3Wide.Subtract(b.C, b.B, out var bc);
Vector3Wide.ConditionalSelect(outsideBC, bc, edgeDirection, out edgeDirection);
Vector3Wide.ConditionalSelect(outsideBC, b.B, b.A, out var edgeStart);
Vector3Wide.Add(localOffsetB, edgeStart, out var negativeEdgeStartToP);
//This does some partially redundant work if the edge is AB or AC, but given that we didn't have bcbc or bcpb, it's fine.
Vector3Wide.Dot(negativeEdgeStartToP, edgeDirection, out var negativeOffsetDotEdge);
Vector3Wide.Dot(edgeDirection, edgeDirection, out var edgeDotEdge);
var edgeScale = Vector.Max(Vector <float> .Zero, Vector.Min(Vector <float> .One, -negativeOffsetDotEdge / edgeDotEdge));
Vector3Wide.Scale(edgeDirection, edgeScale, out var pointOnEdge);
Vector3Wide.Add(edgeStart, pointOnEdge, out pointOnEdge);
Vector3Wide.ConditionalSelect(outsideAnyEdge, pointOnEdge, localClosestOnTriangle, out localClosestOnTriangle);
}
if (Vector.EqualsAny(Vector.AndNot(collidingWithSolidSide, outsideAnyEdge), negativeOne))
{
//p + N * (pa * N) / ||N||^2 = N * (pa * N) / ||N||^2 - (-p)
var nScale = paN / triangleNormalLengthSquared;
Vector3Wide.Scale(localTriangleNormal, nScale, out var offsetToPlane);
Vector3Wide.Subtract(offsetToPlane, localOffsetB, out var pointOnFace);
Vector3Wide.ConditionalSelect(outsideAnyEdge, localClosestOnTriangle, pointOnFace, out localClosestOnTriangle);
}
manifold.FeatureId = Vector.ConditionalSelect(outsideAnyEdge, Vector <int> .Zero, new Vector <int>(MeshReduction.FaceCollisionFlag));
//We'll be using the contact position to perform boundary smoothing; in order to find other triangles, the contact position has to be on the mesh surface.
Matrix3x3Wide.TransformWithoutOverlap(localClosestOnTriangle, rB, out manifold.OffsetA);
Vector3Wide.Add(manifold.OffsetA, offsetB, out manifold.OffsetA);
Vector3Wide.Length(manifold.OffsetA, out var distance);
//Note the normal is calibrated to point from B to A.
var normalScale = new Vector <float>(-1) / distance;
Vector3Wide.Scale(manifold.OffsetA, normalScale, out manifold.Normal);
manifold.Depth = a.Radius - distance;
//In the event that the sphere's center point is touching the triangle, the normal is undefined. In that case, the 'correct' normal would be the triangle's normal.
//However, given that this is a pretty rare degenerate case and that we already treat triangle backfaces as noncolliding, we'll treat zero distance as a backface non-collision.
manifold.ContactExists = Vector.BitwiseAnd(
Vector.GreaterThan(distance, Vector <float> .Zero),
Vector.BitwiseAnd(
Vector.GreaterThanOrEqual(paN, Vector <float> .Zero),
Vector.GreaterThanOrEqual(manifold.Depth, -speculativeMargin)));
}
public unsafe void Test(ref TriangleWide a, ref ConvexHullWide b, ref Vector <float> speculativeMargin, ref Vector3Wide offsetB, ref QuaternionWide orientationA, ref QuaternionWide orientationB, int pairCount, out Convex4ContactManifoldWide manifold)
{
Matrix3x3Wide.CreateFromQuaternion(orientationA, out var triangleOrientation);
Matrix3x3Wide.CreateFromQuaternion(orientationB, out var hullOrientation);
Matrix3x3Wide.MultiplyByTransposeWithoutOverlap(triangleOrientation, hullOrientation, out var hullLocalTriangleOrientation);
Matrix3x3Wide.TransformByTransposedWithoutOverlap(offsetB, hullOrientation, out var localOffsetB);
Vector3Wide.Negate(localOffsetB, out var localOffsetA);
TriangleWide triangle;
Matrix3x3Wide.TransformWithoutOverlap(a.A, hullLocalTriangleOrientation, out triangle.A);
Matrix3x3Wide.TransformWithoutOverlap(a.B, hullLocalTriangleOrientation, out triangle.B);
Matrix3x3Wide.TransformWithoutOverlap(a.C, hullLocalTriangleOrientation, out triangle.C);
Vector3Wide.Add(triangle.A, triangle.B, out var centroid);
Vector3Wide.Add(triangle.C, centroid, out centroid);
Vector3Wide.Scale(centroid, new Vector <float>(1f / 3f), out centroid);
Vector3Wide.Subtract(triangle.A, centroid, out triangle.A);
Vector3Wide.Subtract(triangle.B, centroid, out triangle.B);
Vector3Wide.Subtract(triangle.C, centroid, out triangle.C);
Vector3Wide.Subtract(centroid, localOffsetB, out var localTriangleCenter);
Vector3Wide.Subtract(triangle.B, triangle.A, out var triangleAB);
Vector3Wide.Subtract(triangle.C, triangle.B, out var triangleBC);
Vector3Wide.Subtract(triangle.A, triangle.C, out var triangleCA);
//We'll be using B-local triangle vertices quite a bit, so cache them.
Vector3Wide.Add(triangle.A, localTriangleCenter, out var triangleA);
Vector3Wide.Add(triangle.B, localTriangleCenter, out var triangleB);
Vector3Wide.Add(triangle.C, localTriangleCenter, out var triangleC);
Vector3Wide.CrossWithoutOverlap(triangleAB, triangleCA, out var triangleNormal);
Vector3Wide.Length(triangleNormal, out var triangleNormalLength);
Vector3Wide.Scale(triangleNormal, Vector <float> .One / triangleNormalLength, out triangleNormal);
//Check if the hull's position is within the triangle and below the triangle plane. If so, we can ignore it.
Vector3Wide.Dot(triangleNormal, localTriangleCenter, out var hullToTriangleCenterDot);
var hullBelowPlane = Vector.GreaterThanOrEqual(hullToTriangleCenterDot, Vector <float> .Zero);
Vector <int> hullInsideAndBelowTriangle;
Vector3Wide.CrossWithoutOverlap(triangleAB, triangleNormal, out var edgePlaneAB);
Vector3Wide.CrossWithoutOverlap(triangleBC, triangleNormal, out var edgePlaneBC);
Vector3Wide.CrossWithoutOverlap(triangleCA, triangleNormal, out var edgePlaneCA);
if (Vector.LessThanAny(hullBelowPlane, Vector <int> .Zero))
{
//Is the hull position within the triangle bounds?
Vector3Wide.Dot(edgePlaneAB, triangleA, out var abPlaneTest);
Vector3Wide.Dot(edgePlaneBC, triangleB, out var bcPlaneTest);
Vector3Wide.Dot(edgePlaneCA, triangleC, out var caPlaneTest);
hullInsideAndBelowTriangle = Vector.BitwiseAnd(
Vector.BitwiseAnd(hullBelowPlane, Vector.LessThanOrEqual(abPlaneTest, Vector <float> .Zero)),
Vector.BitwiseAnd(Vector.LessThanOrEqual(bcPlaneTest, Vector <float> .Zero), Vector.LessThanOrEqual(caPlaneTest, Vector <float> .Zero)));
}
else
{
hullInsideAndBelowTriangle = Vector <int> .Zero;
}
ManifoldCandidateHelper.CreateInactiveMask(pairCount, out var inactiveLanes);
a.EstimateEpsilonScale(out var triangleEpsilonScale);
b.EstimateEpsilonScale(inactiveLanes, out var hullEpsilonScale);
var epsilonScale = Vector.Min(triangleEpsilonScale, hullEpsilonScale);
inactiveLanes = Vector.BitwiseOr(inactiveLanes, Vector.LessThan(triangleNormalLength, epsilonScale * 1e-6f));
inactiveLanes = Vector.BitwiseOr(inactiveLanes, hullInsideAndBelowTriangle);
//Not every lane will generate contacts. Rather than requiring every lane to carefully clear all contactExists states, just clear them up front.
manifold.Contact0Exists = default;
manifold.Contact1Exists = default;
manifold.Contact2Exists = default;
manifold.Contact3Exists = default;
if (Vector.LessThanAll(inactiveLanes, Vector <int> .Zero))
{
//No contacts generated.
return;
}
//Note the use of the triangle center as the initial normal rather than the localOffsetA.
//Triangles are not guaranteed to be centered on their center of mass, and the DepthRefiner
//will converge to a depth which does not oppose the so-far best normal- which, on the early iterations,
//could be the initial normal.
Vector3Wide.Length(localTriangleCenter, out var centerDistance);
Vector3Wide.Scale(localTriangleCenter, Vector <float> .One / centerDistance, out var initialNormal);
var useInitialFallback = Vector.LessThan(centerDistance, new Vector <float>(1e-10f));
initialNormal.X = Vector.ConditionalSelect(useInitialFallback, Vector <float> .Zero, initialNormal.X);
initialNormal.Y = Vector.ConditionalSelect(useInitialFallback, Vector <float> .One, initialNormal.Y);
initialNormal.Z = Vector.ConditionalSelect(useInitialFallback, Vector <float> .Zero, initialNormal.Z);
//Check if the extreme point of the hull toward the triangle along its face normal lies inside the triangle.
//If it is, then there's no need for depth refinement.
Vector <int> triangleNormalIsMinimal;
var hullSupportFinder = default(ConvexHullSupportFinder);
DepthRefiner.SimplexWithWitness simplex;
var triangleSupportFinder = default(PretransformedTriangleSupportFinder);
//Create a simplex entry for the direction from the hull center to triangle center.
DepthRefiner.FindSupport(b, triangle, localTriangleCenter, hullLocalTriangleOrientation, ref hullSupportFinder, ref triangleSupportFinder, initialNormal, inactiveLanes, out simplex.A.Support, out simplex.A.SupportOnA);
Vector3Wide.Dot(simplex.A.Support, initialNormal, out var depth);
simplex.A.Exists = Vector.OnesComplement(inactiveLanes);
//Create a simplex entry for the triangle face normal.
Vector3Wide.Negate(triangleNormal, out var negatedTriangleNormal);
hullSupportFinder.ComputeLocalSupport(b, negatedTriangleNormal, inactiveLanes, out var hullSupportAlongTriangleNormal);
simplex.B.SupportOnA = hullSupportAlongTriangleNormal;
Vector3Wide.Subtract(simplex.B.SupportOnA, localTriangleCenter, out simplex.B.Support);
Vector3Wide.Dot(simplex.B.Support, negatedTriangleNormal, out var triangleFaceDepth);
var useTriangleFace = Vector.LessThan(triangleFaceDepth, depth);
Vector3Wide.ConditionalSelect(useTriangleFace, negatedTriangleNormal, initialNormal, out initialNormal);
depth = Vector.ConditionalSelect(useTriangleFace, triangleFaceDepth, depth);
simplex.B.Exists = simplex.A.Exists;
simplex.C.Exists = default;
//Check if the extreme point on the hull is contained within the bounds of the triangle face. If it is, there is no need for a full depth refinement.
Vector3Wide.Subtract(triangleA, hullSupportAlongTriangleNormal, out var closestToA);
Vector3Wide.Subtract(triangleB, hullSupportAlongTriangleNormal, out var closestToB);
Vector3Wide.Subtract(triangleC, hullSupportAlongTriangleNormal, out var closestToC);
Vector3Wide.Dot(edgePlaneAB, closestToA, out var extremeABPlaneTest);
Vector3Wide.Dot(edgePlaneBC, closestToB, out var extremeBCPlaneTest);
Vector3Wide.Dot(edgePlaneCA, closestToC, out var extremeCAPlaneTest);
triangleNormalIsMinimal = Vector.BitwiseAnd(Vector.LessThanOrEqual(extremeABPlaneTest, Vector <float> .Zero), Vector.BitwiseAnd(Vector.LessThanOrEqual(extremeBCPlaneTest, Vector <float> .Zero), Vector.LessThanOrEqual(extremeCAPlaneTest, Vector <float> .Zero)));
var depthThreshold = -speculativeMargin;
var skipDepthRefine = Vector.BitwiseOr(triangleNormalIsMinimal, inactiveLanes);
Vector3Wide localNormal, closestOnHull;
if (Vector.EqualsAny(skipDepthRefine, Vector <int> .Zero))
{
DepthRefiner.FindMinimumDepth(
b, triangle, localTriangleCenter, hullLocalTriangleOrientation, ref hullSupportFinder, ref triangleSupportFinder, ref simplex, initialNormal, depth, skipDepthRefine, 1e-5f * epsilonScale, depthThreshold,
out var refinedDepth, out var refinedNormal, out var refinedClosestOnHull);
Vector3Wide.ConditionalSelect(skipDepthRefine, hullSupportAlongTriangleNormal, refinedClosestOnHull, out closestOnHull);
Vector3Wide.ConditionalSelect(skipDepthRefine, initialNormal, refinedNormal, out localNormal);
depth = Vector.ConditionalSelect(skipDepthRefine, depth, refinedDepth);
}
else
{
//No depth refine ran; the extreme point prepass did everything we needed. Just use the initial normal.
localNormal = initialNormal;
closestOnHull = hullSupportAlongTriangleNormal;
}
Vector3Wide.Dot(triangleNormal, localNormal, out var triangleNormalDotLocalNormal);
inactiveLanes = Vector.BitwiseOr(inactiveLanes, Vector.BitwiseOr(Vector.GreaterThanOrEqual(triangleNormalDotLocalNormal, Vector <float> .Zero), Vector.LessThan(depth, depthThreshold)));
if (Vector.LessThanAll(inactiveLanes, Vector <int> .Zero))
{
//No contacts generated.
return;
}
//To find the contact manifold, we'll clip the triangle edges against the hull face as usual, but we're dealing with potentially
//distinct convex hulls. Rather than vectorizing over the different hulls, we vectorize within each hull.
Helpers.FillVectorWithLaneIndices(out var slotOffsetIndices);
var boundingPlaneEpsilon = 1e-3f * epsilonScale;
//There can be no more than 6 contacts (provided there are no numerical errors); 2 per triangle edge.
var candidates = stackalloc ManifoldCandidateScalar[6];
for (int slotIndex = 0; slotIndex < pairCount; ++slotIndex)
{
if (inactiveLanes[slotIndex] < 0)
{
continue;
}
ref var hull = ref b.Hulls[slotIndex];
ConvexHullTestHelper.PickRepresentativeFace(ref hull, slotIndex, ref localNormal, closestOnHull, slotOffsetIndices, ref boundingPlaneEpsilon, out var slotFaceNormal, out var slotLocalNormal, out var bestFaceIndex);
//Test each triangle edge against the hull face.
//Note that we do not use the faceNormal x edgeOffset edge plane, but rather edgeOffset x localNormal.
//The faces are wound counterclockwise.
//Note that the triangle edges are packed into a Vector4. Historically, there were some minor codegen issues with Vector3.
//May not matter anymore, but it costs ~nothing to use a dead slot.
ref var aSlot = ref GatherScatter.GetOffsetInstance(ref triangleA, slotIndex);
public void Test(
ref CapsuleWide a, ref CapsuleWide b, ref Vector <float> speculativeMargin,
ref Vector3Wide offsetB, ref QuaternionWide orientationA, ref QuaternionWide orientationB, int pairCount,
out Convex2ContactManifoldWide manifold)
{
//Compute the closest points between the two line segments. No clamping to begin with.
//We want to minimize distance = ||(a + da * ta) - (b + db * tb)||.
//Taking the derivative with respect to ta and doing some algebra (taking into account ||da|| == ||db|| == 1) to solve for ta yields:
//ta = (da * (b - a) + (db * (a - b)) * (da * db)) / (1 - ((da * db) * (da * db))
QuaternionWide.TransformUnitXY(orientationA, out var xa, out var da);
QuaternionWide.TransformUnitY(orientationB, out var db);
Vector3Wide.Dot(da, offsetB, out var daOffsetB);
Vector3Wide.Dot(db, offsetB, out var dbOffsetB);
Vector3Wide.Dot(da, db, out var dadb);
//Note potential division by zero when the axes are parallel. Arbitrarily clamp; near zero values will instead produce extreme values which get clamped to reasonable results.
var ta = (daOffsetB - dbOffsetB * dadb) / Vector.Max(new Vector <float>(1e-15f), Vector <float> .One - dadb * dadb);
//tb = ta * (da * db) - db * (b - a)
var tb = ta * dadb - dbOffsetB;
//We cannot simply clamp the ta and tb values to the capsule line segments. Instead, project each line segment onto the other line segment, clamping against the target's interval.
//That new clamped projected interval is the valid solution space on that line segment. We can clamp the t value by that interval to get the correctly bounded solution.
//The projected intervals are:
//B onto A: +-BHalfLength * (da * db) + da * offsetB
//A onto B: +-AHalfLength * (da * db) - db * offsetB
var absdadb = Vector.Abs(dadb);
var bOntoAOffset = b.HalfLength * absdadb;
var aOntoBOffset = a.HalfLength * absdadb;
var aMin = Vector.Max(-a.HalfLength, Vector.Min(a.HalfLength, daOffsetB - bOntoAOffset));
var aMax = Vector.Min(a.HalfLength, Vector.Max(-a.HalfLength, daOffsetB + bOntoAOffset));
var bMin = Vector.Max(-b.HalfLength, Vector.Min(b.HalfLength, -aOntoBOffset - dbOffsetB));
var bMax = Vector.Min(b.HalfLength, Vector.Max(-b.HalfLength, aOntoBOffset - dbOffsetB));
ta = Vector.Min(Vector.Max(ta, aMin), aMax);
tb = Vector.Min(Vector.Max(tb, bMin), bMax);
Vector3Wide.Scale(da, ta, out var closestPointOnA);
Vector3Wide.Scale(db, tb, out var closestPointOnB);
Vector3Wide.Add(closestPointOnB, offsetB, out closestPointOnB);
//Note that normals are calibrated to point from B to A by convention.
Vector3Wide.Subtract(closestPointOnA, closestPointOnB, out manifold.Normal);
Vector3Wide.Length(manifold.Normal, out var distance);
var inverseDistance = Vector <float> .One / distance;
Vector3Wide.Scale(manifold.Normal, inverseDistance, out manifold.Normal);
//In the event that the line segments are touching, the normal doesn't exist and we need an alternative. Any direction along the local horizontal (XZ) plane of either capsule
//is valid. (Normals along the local Y axes are not guaranteed to be as quick of a path to separation due to nonzero line length.)
var normalIsValid = Vector.GreaterThan(distance, new Vector <float>(1e-7f));
Vector3Wide.ConditionalSelect(normalIsValid, manifold.Normal, xa, out manifold.Normal);
//In the event that the two capsule axes are coplanar, we accept the whole interval as a source of contact.
//As the axes drift away from coplanarity, the accepted interval rapidly narrows to zero length, centered on ta and tb.
//We rate the degree of coplanarity based on the angle between the capsule axis and the plane defined by the opposing segment and contact normal:
//sin(angle) = dot(da, (db x normal)/||db x normal||)
//Finally, note that we are dealing with extremely small angles, and for small angles sin(angle) ~= angle,
//and also that fade behavior is completely arbitrary, so we can directly use squared angle without any concern.
//angle^2 ~= dot(da, (db x normal))^2 / ||db x normal||^2
//Note that if ||db x normal|| is zero, then any da should be accepted as being coplanar because there is no restriction. ConditionalSelect away the discontinuity.
Vector3Wide.CrossWithoutOverlap(db, manifold.Normal, out var planeNormal);
Vector3Wide.LengthSquared(planeNormal, out var planeNormalLengthSquared);
Vector3Wide.Dot(da, planeNormal, out var numeratorUnsquared);
var squaredAngle = Vector.ConditionalSelect(Vector.LessThan(planeNormalLengthSquared, new Vector <float>(1e-10f)), Vector <float> .Zero, numeratorUnsquared * numeratorUnsquared / planeNormalLengthSquared);
//Convert the squared angle to a lerp parameter. For squared angle from 0 to lowerThreshold, we should use the full interval (1). From lowerThreshold to upperThreshold, lerp to 0.
const float lowerThresholdAngle = 0.01f;
const float upperThresholdAngle = 0.05f;
const float lowerThreshold = lowerThresholdAngle * lowerThresholdAngle;
const float upperThreshold = upperThresholdAngle * upperThresholdAngle;
var intervalWeight = Vector.Max(Vector <float> .Zero, Vector.Min(Vector <float> .One, (new Vector <float>(upperThreshold) - squaredAngle) * new Vector <float>(1f / (upperThreshold - lowerThreshold))));
//If the line segments intersect, even if they're coplanar, we would ideally stick to using a single point. Would be easy enough,
//but we don't bother because it's such a weird and extremely temporary corner case. Not really worth handling.
var weightedTa = ta - ta * intervalWeight;
aMin = intervalWeight * aMin + weightedTa;
aMax = intervalWeight * aMax + weightedTa;
Vector3Wide.Scale(da, aMin, out manifold.OffsetA0);
Vector3Wide.Scale(da, aMax, out manifold.OffsetA1);
//In the coplanar case, there are two points. We need a method of computing depth which gives a reasonable result to the second contact.
//Note that one of the two contacts should end up with a distance equal to the previously computed segment distance, so we're doing some redundant work here.
//It's just easier to do that extra work than it would be to track which endpoint contributed the lower distance.
//Unproject the final interval endpoints from a back onto b.
//dot(offsetB + db * tb0, da) = ta0
//tb0 = (ta0 - daOffsetB) / dadb
//distance0 = dot(a0 - (offsetB + tb0 * db), normal)
//distance1 = dot(a1 - (offsetB + tb1 * db), normal)
Vector3Wide.Dot(db, manifold.Normal, out var dbNormal);
Vector3Wide.Subtract(manifold.OffsetA0, offsetB, out var offsetB0);
Vector3Wide.Subtract(manifold.OffsetA1, offsetB, out var offsetB1);
//Note potential division by zero. In that case, treat both projected points as the closest point. (Handled by the conditional select that chooses the previously computed distance.)
var inverseDadb = Vector <float> .One / dadb;
var projectedTb0 = Vector.Max(bMin, Vector.Min(bMax, (aMin - daOffsetB) * inverseDadb));
var projectedTb1 = Vector.Max(bMin, Vector.Min(bMax, (aMax - daOffsetB) * inverseDadb));
Vector3Wide.Dot(offsetB0, manifold.Normal, out var b0Normal);
Vector3Wide.Dot(offsetB1, manifold.Normal, out var b1Normal);
var capsulesArePerpendicular = Vector.LessThan(Vector.Abs(dadb), new Vector <float>(1e-7f));
var distance0 = Vector.ConditionalSelect(capsulesArePerpendicular, distance, b0Normal - dbNormal * projectedTb0);
var distance1 = Vector.ConditionalSelect(capsulesArePerpendicular, distance, b1Normal - dbNormal * projectedTb1);
var combinedRadius = a.Radius + b.Radius;
manifold.Depth0 = combinedRadius - distance0;
manifold.Depth1 = combinedRadius - distance1;
//Apply the normal offset to the contact positions.
var negativeOffsetFromA0 = manifold.Depth0 * 0.5f - a.Radius;
var negativeOffsetFromA1 = manifold.Depth1 * 0.5f - a.Radius;
Vector3Wide.Scale(manifold.Normal, negativeOffsetFromA0, out var normalPush0);
Vector3Wide.Scale(manifold.Normal, negativeOffsetFromA1, out var normalPush1);
Vector3Wide.Add(manifold.OffsetA0, normalPush0, out manifold.OffsetA0);
Vector3Wide.Add(manifold.OffsetA1, normalPush1, out manifold.OffsetA1);
manifold.FeatureId0 = Vector <int> .Zero;
manifold.FeatureId1 = Vector <int> .One;
var minimumAcceptedDepth = -speculativeMargin;
manifold.Contact0Exists = Vector.GreaterThanOrEqual(manifold.Depth0, minimumAcceptedDepth);
manifold.Contact1Exists = Vector.BitwiseAnd(
Vector.GreaterThanOrEqual(manifold.Depth1, minimumAcceptedDepth),
Vector.GreaterThan(aMax - aMin, new Vector <float>(1e-7f) * a.HalfLength));
//TODO: Since we added in the complexity of 2 contact support, this is probably large enough to benefit from working in the local space of one of the capsules.
//Worth looking into later.
}
public void Test(ref SphereWide a, ref TriangleWide b, ref Vector3Wide offsetB, ref QuaternionWide orientationA, ref QuaternionWide orientationB,
out Vector <int> intersected, out Vector <float> distance, out Vector3Wide closestA, out Vector3Wide normal)
{
//Note that we're borrowing a lot here from the SphereTriangleCollisionTask. Could share more if you find yourself needing to change things dramatically.
//Main difficulty in fully sharing is that sweep tests do not honor one sidedness, so some of the conditions change.
//Work in the local space of the triangle, since it's quicker to transform the sphere position than the vertices of the triangle.
Matrix3x3Wide.CreateFromQuaternion(orientationB, out var rB);
Matrix3x3Wide.TransformByTransposedWithoutOverlap(offsetB, rB, out var localOffsetB);
Vector3Wide.Subtract(b.B, b.A, out var ab);
Vector3Wide.Subtract(b.C, b.A, out var ac);
//localOffsetA = -localOffsetB, so pa = triangle.A + localOffsetB.
Vector3Wide.Add(b.A, localOffsetB, out var pa);
Vector3Wide.CrossWithoutOverlap(ab, ac, out var localTriangleNormal);
Vector3Wide.Dot(localTriangleNormal, pa, out var paN);
//EdgeAB plane test: (pa x ab) * (ab x ac) >= 0
//EdgeAC plane test: (ac x pa) * (ab x ac) >= 0
//Note that these are scaled versions of the barycentric coordinates.
//To normalize them such that the weights of a point within the triangle equal 1, we just need to divide by dot(ab x ac, ab x ac).
//In other words, to test the third edge plane, we can ensure that the unnormalized weights are both positive and sum to a value less than dot(ab x ac, ab x ac).
//If a point is outside of an edge plane, we know that it's not in the face region or any other edge region. It could, however, be in an adjacent vertex region.
//Vertex cases can be handled by clamping an edge case.
//Further, note that for any query location, it is sufficient to only test one edge even if the point is outside two edge planes. If it's outside two edge planes,
//that just means it's going to be on the shared vertex, so a clamped edge test captures the correct closest point.
//So, at each edge, if the point is outside the plane, cache the edge. The last edge registering an outside result will be tested.
//(pa x ab) * (ab x ac) = (pa * ab) * (ab * ac) - (pa * ac) * (ab * ab)
//(ac x pa) * (ab x ac) = (ac * ab) * (pa * ac) - (ac * ac) * (pa * ab)
//(ab x ac) * (ab x ac) = (ab * ab) * (ac * ac) - (ab * ac) * (ab * ac)
Vector3Wide.Dot(pa, ab, out var abpa);
Vector3Wide.Dot(ab, ac, out var abac);
Vector3Wide.Dot(ac, pa, out var acpa);
Vector3Wide.Dot(ac, ac, out var acac);
Vector3Wide.Dot(ab, ab, out var abab);
var edgePlaneTestAB = abpa * abac - acpa * abab;
var edgePlaneTestAC = abac * acpa - acac * abpa;
var triangleNormalLengthSquared = abab * acac - abac * abac;
var edgePlaneTestBC = triangleNormalLengthSquared - edgePlaneTestAB - edgePlaneTestAC;
var outsideAB = Vector.LessThan(edgePlaneTestAB, Vector <float> .Zero);
var outsideAC = Vector.LessThan(edgePlaneTestAC, Vector <float> .Zero);
var outsideBC = Vector.LessThan(edgePlaneTestBC, Vector <float> .Zero);
var outsideAnyEdge = Vector.BitwiseOr(outsideAB, Vector.BitwiseOr(outsideAC, outsideBC));
Vector3Wide localClosestOnTriangle;
var negativeOne = new Vector <int>(-1);
if (Vector.EqualsAny(outsideAnyEdge, negativeOne))
{
//At least one lane detected a point outside of the triangle. Choose one edge which is outside as the representative.
Vector3Wide.ConditionalSelect(outsideAC, ac, ab, out var edgeDirection);
Vector3Wide.Subtract(b.C, b.B, out var bc);
Vector3Wide.ConditionalSelect(outsideBC, bc, edgeDirection, out edgeDirection);
Vector3Wide.ConditionalSelect(outsideBC, b.B, b.A, out var edgeStart);
Vector3Wide.Add(localOffsetB, edgeStart, out var negativeEdgeStartToP);
//This does some partially redundant work if the edge is AB or AC, but given that we didn't have bcbc or bcpb, it's fine.
Vector3Wide.Dot(negativeEdgeStartToP, edgeDirection, out var negativeOffsetDotEdge);
Vector3Wide.Dot(edgeDirection, edgeDirection, out var edgeDotEdge);
var edgeScale = Vector.Max(Vector <float> .Zero, Vector.Min(Vector <float> .One, -negativeOffsetDotEdge / edgeDotEdge));
Vector3Wide.Scale(edgeDirection, edgeScale, out var pointOnEdge);
Vector3Wide.Add(edgeStart, pointOnEdge, out pointOnEdge);
Vector3Wide.ConditionalSelect(outsideAnyEdge, pointOnEdge, localClosestOnTriangle, out localClosestOnTriangle);
}
if (Vector.EqualsAny(outsideAnyEdge, Vector <int> .Zero))
{
//p + N * (pa * N) / ||N||^2 = N * (pa * N) / ||N||^2 - (-p)
var nScale = paN / triangleNormalLengthSquared;
Vector3Wide.Scale(localTriangleNormal, nScale, out var offsetToPlane);
Vector3Wide.Subtract(offsetToPlane, localOffsetB, out var pointOnFace);
Vector3Wide.ConditionalSelect(outsideAnyEdge, localClosestOnTriangle, pointOnFace, out localClosestOnTriangle);
}
//normal = normalize(localOffsetA - localClosestOnTriangle) = (localOffsetB + localClosestOnTriangle) / (-||localOffsetB + localClosestOnTriangle||)
Vector3Wide.Add(localOffsetB, localClosestOnTriangle, out var localNormal);
Vector3Wide.Length(localNormal, out var localNormalLength);
Vector3Wide.Scale(localNormal, new Vector <float>(-1f) / localNormalLength, out localNormal);
Matrix3x3Wide.TransformWithoutOverlap(localNormal, rB, out normal);
Vector3Wide.Scale(normal, -a.Radius, out closestA);
distance = localNormalLength - a.Radius;
intersected = Vector.LessThanOrEqual(distance, Vector <float> .Zero);
}