public void Test2(
ref CapsuleWide a, ref BoxWide b,
ref Vector3Wide offsetB, ref QuaternionWide orientationA, ref QuaternionWide orientationB,
out Convex2ContactManifoldWide manifold)
{
QuaternionWide.Conjugate(ref orientationB, out var toLocalB);
QuaternionWide.TransformWithoutOverlap(ref offsetB, ref toLocalB, out var localOffsetA);
Vector3Wide.Negate(ref localOffsetA);
QuaternionWide.ConcatenateWithoutOverlap(ref orientationA, ref toLocalB, out var boxLocalOrientationA);
QuaternionWide.TransformUnitY(ref boxLocalOrientationA, out var capsuleAxis);
//For each face of the box, clip the capsule axis against its bounding planes and then clamp the result to the box's extent along that axis.
//The face which has the largest clipped interval of the capsule line segment will be picked as the representative.
var positive = Vector <float> .One;
var negative = -positive;
var divisionGuard = new Vector <float>(1e-15f);
//Division by zero is hacked away by clamping the magnitude to some nonzero value and recovering the sign in the numerator.
//The resulting interval will be properly signed and enormous, so it serves the necessary purpose.
//Note the negation: t = dot(N, +-halfExtent - offsetA) / dot(N, capsuleAxis) = dot(N, offsetA +- halfExtent) / -dot(N, capsuleAxis)
var scaleX = Vector.ConditionalSelect(Vector.LessThan(capsuleAxis.X, Vector <float> .Zero), positive, negative) / Vector.Max(Vector.Abs(capsuleAxis.X), divisionGuard);
var scaleY = Vector.ConditionalSelect(Vector.LessThan(capsuleAxis.Y, Vector <float> .Zero), positive, negative) / Vector.Max(Vector.Abs(capsuleAxis.Y), divisionGuard);
var scaleZ = Vector.ConditionalSelect(Vector.LessThan(capsuleAxis.Z, Vector <float> .Zero), positive, negative) / Vector.Max(Vector.Abs(capsuleAxis.Z), divisionGuard);
Vector3Wide scaledExtents, scaledOffset;
scaledExtents.X = b.HalfWidth * Vector.Abs(scaleX);
scaledExtents.Y = b.HalfHeight * Vector.Abs(scaleY);
scaledExtents.Z = b.HalfLength * Vector.Abs(scaleZ);
scaledOffset.X = localOffsetA.X * scaleX;
scaledOffset.Y = localOffsetA.Y * scaleY;
scaledOffset.Z = localOffsetA.Z * scaleZ;
Vector3Wide.Subtract(ref scaledOffset, ref scaledExtents, out var tCandidateMin);
var negativeHalfLength = -a.HalfLength;
Vector3Wide.Max(ref negativeHalfLength, ref tCandidateMin, out tCandidateMin);
Vector3Wide.Add(ref scaledOffset, ref scaledExtents, out var tCandidateMax);
Vector3Wide.Min(ref a.HalfLength, ref tCandidateMax, out tCandidateMax);
//We now have clipped intervals for all faces. The faces along axis X, for example, use interval max(tMin.Y, tMin.Z) to min(tMax.Y, tMax.Z).
//Find the longest interval. Note that it's possible for the interval length to be negative when a line segment does not enter the face's voronoi region.
//It may be negative for all axes, even; that can happen when the line segment is outside all face voronoi regions. That's fine, though.
//It's simply a signal that the line segment is in one of the face-adjacent voronoi regions, and clamping the endpoints will provide the closest point on the box to the segment.
var intervalMaxX = Vector.Min(tCandidateMax.Y, tCandidateMax.Z);
var intervalMaxY = Vector.Min(tCandidateMax.X, tCandidateMax.Z);
var intervalMaxZ = Vector.Min(tCandidateMax.X, tCandidateMax.Y);
var intervalMinX = Vector.Max(tCandidateMin.Y, tCandidateMin.Z);
var intervalMinY = Vector.Max(tCandidateMin.X, tCandidateMin.Z);
var intervalMinZ = Vector.Max(tCandidateMin.X, tCandidateMin.Y);
var intervalLengthX = Vector.Max(Vector <float> .Zero, intervalMaxX - intervalMinX);
var intervalLengthY = Vector.Max(Vector <float> .Zero, intervalMaxY - intervalMinY);
var intervalLengthZ = Vector.Max(Vector <float> .Zero, intervalMaxZ - intervalMinZ);
var x2 = capsuleAxis.X * capsuleAxis.X;
var y2 = capsuleAxis.Y * capsuleAxis.Y;
var z2 = capsuleAxis.Z * capsuleAxis.Z;
var projectedLengthSquaredX = (y2 + z2) * intervalLengthX * intervalLengthX;
var projectedLengthSquaredY = (x2 + z2) * intervalLengthY * intervalLengthY;
var projectedLengthSquaredZ = (x2 + y2) * intervalLengthZ * intervalLengthZ;
var maxLengthSquared = Vector.Max(projectedLengthSquaredX, Vector.Max(projectedLengthSquaredY, projectedLengthSquaredZ));
var useX = Vector.Equals(projectedLengthSquaredX, maxLengthSquared);
var useY = Vector.Equals(projectedLengthSquaredY, maxLengthSquared);
var tMin = Vector.ConditionalSelect(useX, intervalMinX, Vector.ConditionalSelect(useY, intervalMinY, intervalMinZ));
var tMax = Vector.ConditionalSelect(useX, intervalMaxX, Vector.ConditionalSelect(useY, intervalMaxY, intervalMaxZ));
Vector3Wide.Scale(ref capsuleAxis, ref tMin, out var capsuleStart);
Vector3Wide.Scale(ref capsuleAxis, ref tMax, out var capsuleEnd);
Vector3Wide.Add(ref localOffsetA, ref capsuleStart, out capsuleStart);
Vector3Wide.Add(ref localOffsetA, ref capsuleEnd, out capsuleEnd);
//Clamp the start and end to the box. Note that we haven't kept track of which axis we're working on, so we just do clamping on all three axes. Two axes will be wasted,
//but it's easier to do this than to try to reconstruct which axis is the one we're supposed to work on.
Vector3Wide boxStart, boxEnd;
boxStart.X = Vector.Max(Vector.Min(capsuleStart.X, b.HalfWidth), -b.HalfWidth);
boxStart.Y = Vector.Max(Vector.Min(capsuleStart.Y, b.HalfHeight), -b.HalfHeight);
boxStart.Z = Vector.Max(Vector.Min(capsuleStart.Z, b.HalfLength), -b.HalfLength);
boxEnd.X = Vector.Max(Vector.Min(capsuleEnd.X, b.HalfWidth), -b.HalfWidth);
boxEnd.Y = Vector.Max(Vector.Min(capsuleEnd.Y, b.HalfHeight), -b.HalfHeight);
boxEnd.Z = Vector.Max(Vector.Min(capsuleEnd.Z, b.HalfLength), -b.HalfLength);
//Compute the closest point on the line segment to the clamped start and end.
Vector3Wide.Subtract(ref boxStart, ref localOffsetA, out var aToBoxStart);
Vector3Wide.Subtract(ref boxEnd, ref localOffsetA, out var aToBoxEnd);
Vector3Wide.Dot(ref capsuleAxis, ref aToBoxStart, out var dotStart);
Vector3Wide.Dot(ref capsuleAxis, ref aToBoxEnd, out var dotEnd);
dotStart = Vector.Max(Vector.Min(dotStart, a.HalfLength), negativeHalfLength);
dotEnd = Vector.Max(Vector.Min(dotEnd, a.HalfLength), negativeHalfLength);
//Now compute the distances. The normal will be defined by the shorter of the two offsets.
Vector3Wide.Scale(ref capsuleAxis, ref dotStart, out var closestFromBoxStart);
Vector3Wide.Scale(ref capsuleAxis, ref dotEnd, out var closestFromBoxEnd);
Vector3Wide.Add(ref closestFromBoxStart, ref localOffsetA, out closestFromBoxStart);
Vector3Wide.Add(ref closestFromBoxEnd, ref localOffsetA, out closestFromBoxEnd);
//Note that these offsets are pointing from the box to the capsule (B to A), matching contact normal convention.
Vector3Wide.Subtract(ref closestFromBoxStart, ref boxStart, out var startOffset);
Vector3Wide.Subtract(ref closestFromBoxEnd, ref boxEnd, out var endOffset);
Vector3Wide.LengthSquared(ref startOffset, out var startLengthSquared);
Vector3Wide.LengthSquared(ref endOffset, out var endLengthSquared);
var minLengthSquared = Vector.Min(startLengthSquared, endLengthSquared);
var useStart = Vector.Equals(minLengthSquared, startLengthSquared);
Vector3Wide.ConditionalSelect(ref useStart, ref startOffset, ref endOffset, out var localNormal);
var inverseLength = Vector <float> .One / Vector.SquareRoot(minLengthSquared);
Vector3Wide.Scale(ref localNormal, ref inverseLength, out localNormal);
//Note that we defer contact position offseting by the normal * depth until later when we know the normal isn't going to change.
Vector3Wide.Subtract(ref closestFromBoxStart, ref localOffsetA, out var localA0);
Vector3Wide.Subtract(ref closestFromBoxEnd, ref localOffsetA, out var localA1);
Vector3Wide.Dot(ref localNormal, ref startOffset, out var dot0);
Vector3Wide.Dot(ref localNormal, ref endOffset, out var dot1);
manifold.Depth0 = a.Radius - dot0;
manifold.Depth1 = a.Radius - dot1;
manifold.FeatureId0 = Vector <int> .Zero;
manifold.FeatureId1 = Vector <int> .One;
//Capsule-box has an extremely important property:
//Capsule internal line segments almost never intersect the box due to the capsule's radius.
//We can effectively split this implementation into two tests- exterior and interior.
//The interior test only needs to be run if one of the subpairs happens to be deeply intersecting, which should be ~never in practice.
//That's fortunate, because computing an accurate interior normal is painful!
var useInterior = Vector.LessThan(minLengthSquared, new Vector <float>(1e-10f));
if (Vector.EqualsAny(useInterior, new Vector <int>(-1)))
{
//There are six possible separating axes when the capsule's internal line segment intersects the box:
//box.X
//box.Y
//box.Z
//box.X x capsule.Y
//box.Y x capsule.Y
//box.Z x capsule.Y
//Choose the axis with the minimum penetration depth.
//By working in the box's local space, box.X/Y/Z become simply (1,0,0), (0,1,0), and (0,0,1). This cancels out a whole bunch of arithmetic.
//Note that these depths will not take into account the radius until the very end. We pretend that the capsule is simply a line until then, since the radius changes nothing.
var faceDepthX = b.HalfWidth - localOffsetA.X + Vector.Abs(capsuleAxis.X * a.HalfLength);
var faceDepthY = b.HalfHeight - localOffsetA.Y + Vector.Abs(capsuleAxis.Y * a.HalfLength);
var faceDepthZ = b.HalfLength - localOffsetA.Z + Vector.Abs(capsuleAxis.Z * a.HalfLength);
var faceDepth = Vector.Min(Vector.Min(faceDepthX, faceDepthY), faceDepthZ);
var useFaceX = Vector.Equals(faceDepth, faceDepthX);
var useFaceY = Vector.AndNot(Vector.Equals(faceDepth, faceDepthY), useFaceX);
var useFaceZ = Vector.OnesComplement(Vector.BitwiseOr(useFaceX, useFaceY));
//Note that signs are calibrated as a last step so that edge normals do not need their own calibration.
Vector3Wide faceNormal;
faceNormal.X = Vector.ConditionalSelect(useFaceX, Vector <float> .One, Vector <float> .Zero);
faceNormal.Y = Vector.ConditionalSelect(useFaceY, Vector <float> .One, Vector <float> .Zero);
faceNormal.Z = Vector.ConditionalSelect(useFaceZ, Vector <float> .One, Vector <float> .Zero);
//For each edge, given the zero components, the cross product boils down to creating a perpendicular 2d vector on the plane of the box axis from the cylinder axis.
var infiniteDepth = new Vector <float>(float.MaxValue);
var bestDepth = infiniteDepth;
var edgeXLength = Vector.SquareRoot(y2 + z2);
var edgeYLength = Vector.SquareRoot(x2 + z2);
var edgeZLength = Vector.SquareRoot(x2 + y2);
//depth = dot(N, halfExtents - offsetA), where N = (edgeAxis x capsuleAxis) / ||(edgeAxis x capsuleAxis)||, and both N and halfExtents have their signs calibrated.
//Using abs allows us to handle the sign calibration inline at the cost of splitting the dot product.
var inverseLengthX = Vector <float> .One / edgeXLength;
var inverseLengthY = Vector <float> .One / edgeYLength;
var inverseLengthZ = Vector <float> .One / edgeZLength;
var edgeXDepth = (Vector.Abs(b.HalfHeight * capsuleAxis.Z - b.HalfLength * capsuleAxis.Y) - Vector.Abs(offsetB.Y * capsuleAxis.Z - offsetB.Z * capsuleAxis.Y)) * inverseLengthX;
var edgeYDepth = (Vector.Abs(b.HalfWidth * capsuleAxis.Z - b.HalfLength * capsuleAxis.X) - Vector.Abs(offsetB.X * capsuleAxis.Z - offsetB.Z * capsuleAxis.X)) * inverseLengthY;
var edgeZDepth = (Vector.Abs(b.HalfWidth * capsuleAxis.Y - b.HalfHeight * capsuleAxis.X) - Vector.Abs(offsetB.X * capsuleAxis.Y - offsetB.Y * capsuleAxis.X)) * inverseLengthZ;
var epsilon = new Vector <float>(1e-9f);
//If the capsule internal segment and the edge are parallel, we ignore the edge by replacing its depth with ~infinity.
//Such parallel axes are guaranteed to do no better than one of the face directions since this path is only relevant during segment-box deep collisions.
edgeXDepth = Vector.ConditionalSelect(Vector.LessThan(edgeXLength, epsilon), infiniteDepth, edgeXDepth);
edgeYDepth = Vector.ConditionalSelect(Vector.LessThan(edgeYLength, epsilon), infiniteDepth, edgeYDepth);
edgeZDepth = Vector.ConditionalSelect(Vector.LessThan(edgeZLength, epsilon), infiniteDepth, edgeZDepth);
var edgeDepth = Vector.Min(Vector.Min(edgeXDepth, edgeYDepth), edgeZDepth);
var useEdgeX = Vector.Equals(edgeDepth, edgeXDepth);
var useEdgeY = Vector.Equals(edgeDepth, edgeYDepth);
Vector3Wide edgeNormal;
edgeNormal.X = Vector.ConditionalSelect(useEdgeX, Vector <float> .Zero, Vector.ConditionalSelect(useEdgeY, capsuleAxis.Z, capsuleAxis.Y));
edgeNormal.Y = Vector.ConditionalSelect(useEdgeX, capsuleAxis.Z, Vector.ConditionalSelect(useEdgeY, Vector <float> .Zero, -capsuleAxis.X));
edgeNormal.Z = Vector.ConditionalSelect(useEdgeX, -capsuleAxis.Y, Vector.ConditionalSelect(useEdgeY, -capsuleAxis.X, Vector <float> .Zero));
var useEdge = Vector.LessThan(edgeDepth, faceDepth);
Vector3Wide.ConditionalSelect(ref useEdge, ref edgeNormal, ref faceNormal, out var interiorNormal);
//Calibrate the normal such that it points from B to A.
Vector3Wide.Dot(ref interiorNormal, ref offsetB, out var dot);
var shouldNegateNormal = Vector.GreaterThan(dot, Vector <float> .Zero);
interiorNormal.X = Vector.ConditionalSelect(shouldNegateNormal, -interiorNormal.X, interiorNormal.X);
interiorNormal.Y = Vector.ConditionalSelect(shouldNegateNormal, -interiorNormal.Y, interiorNormal.Y);
interiorNormal.Z = Vector.ConditionalSelect(shouldNegateNormal, -interiorNormal.Z, interiorNormal.Z);
//Each contact may have its own depth.
//Imagine a face collision- if the capsule axis isn't fully parallel with the plane's surface, it would be strange to use the same depth for both contacts.
//Compute the interval of the box on the normal. Note that the normal is already calibrated to point from B to A (box to capsule).
var boxExtreme = Vector.Abs(localNormal.X * b.HalfWidth) + Vector.Abs(localNormal.Y * b.HalfHeight) + Vector.Abs(localNormal.Z * b.HalfLength);
Vector3Wide.Dot(ref localNormal, ref closestFromBoxStart, out dot0);
Vector3Wide.Dot(ref localNormal, ref closestFromBoxEnd, out dot1);
manifold.Depth0 = Vector.ConditionalSelect(useInterior, a.Radius + boxExtreme - dot0, manifold.Depth0);
manifold.Depth1 = Vector.ConditionalSelect(useInterior, a.Radius + boxExtreme - dot1, manifold.Depth1);
//The interior normal doesn't check all possible separating axes in the non-segment-intersecting case.
//It would be wrong to overwrite the initial test if the interior test was unnecessary.
Vector3Wide.ConditionalSelect(ref useInterior, ref interiorNormal, ref localNormal, out localNormal);
}
//Transform A0, A1, and the normal into world space.
Matrix3x3Wide.CreateFromQuaternion(ref orientationB, out var orientationMatrixB);
Matrix3x3Wide.TransformWithoutOverlap(ref localNormal, ref orientationMatrixB, out manifold.Normal);
Matrix3x3Wide.TransformWithoutOverlap(ref localA0, ref orientationMatrixB, out manifold.OffsetA0);
Matrix3x3Wide.TransformWithoutOverlap(ref localA1, ref orientationMatrixB, out manifold.OffsetA1);
//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(ref manifold.Normal, ref negativeOffsetFromA0, out var normalPush0);
Vector3Wide.Scale(ref manifold.Normal, ref negativeOffsetFromA1, out var normalPush1);
Vector3Wide.Add(ref manifold.OffsetA0, ref normalPush0, out manifold.OffsetA0);
Vector3Wide.Add(ref manifold.OffsetA1, ref normalPush1, out manifold.OffsetA1);
//Note that it is possible that the two contacts is redundant. Only keep them both if they're separated.
manifold.Count = Vector.ConditionalSelect(Vector.LessThan(Vector.Abs(dotStart - dotEnd), a.HalfLength * new Vector <float>(1e-7f)), Vector <int> .One, new Vector <int>(2));
}
public static void ExpandBoundingBoxes(ref Vector3Wide min, ref Vector3Wide max, ref BodyVelocities velocities, float dt,
ref Vector <float> maximumRadius, ref Vector <float> maximumAngularExpansion, ref Vector <float> maximumExpansion)
{
/*
* If an object sitting on a plane had a raw (unexpanded) AABB that is just barely above the plane, no contacts would be generated.
* If the velocity of the object would shove it down into the plane in the next frame, then it would generate contacts in the next frame- and,
* potentially, this cycle would repeat and cause jitter.
*
* To solve this, there are a couple of options:
* 1) Introduce an 'allowed penetration' so that objects can overlap a little bit. This tends to confuse people a little bit when they notice it,
* and in some circumstances objects can be seen settling into the allowed penetration slowly. It looks a bit odd.
* 2) Make contact constraints fight to maintain zero penetration depth, but expand the bounding box with velocity and allow contacts to be generated speculatively-
* contacts with negative penetration depth.
*
#2 is a form of continuous collision detection, but it's handy for general contact stability too.
* In this version of the engine, all objects generate speculative contacts by default, though only within a per-collidable-tuned 'speculative margin'.
* It's kind of like BEPUphysics v1's AllowedPenetration, except inverted. Speculative contacts that fall within the speculative margin-
* that is, those with negative depth of a magnitude less than the margin- are kept.
*
* So, a user could choose to have a very large speculative margin, and the speculative contact generation would provide a form of continuous collision detection.
* The main purpose, though, is just contact stability. With this in isolation, there's no strong reason to expand the bounding box more than the speculative margin.
* This is the 'discrete' mode.
*
* However, consider what would happen if an object A with high velocity and this 'discrete' mode was headed towards an object B in a 'continuous' mode.
* Object B only expands its bounding box by its own velocity, and object A doesn't expand beyond its speculative margin. The collision between A and B could easily be missed.
* To account for this, there is an intermediate mode- 'passive'- where the bounding box is allowed to expand beyond the margin,
* but no further continuous collision detection is performed.
*
* The fully continuous modes fully expand the bounding boxes. Notably, the inner sphere continuous collision detection mode could get by with less,
* but it would be pretty confusing to have the same kind of missed collision possibility if the other object in the pair was a substepping object.
* Two different inner sphere modes could be offered, but I'm unsure about the usefulness versus the complexity.
*
* (Note that there ARE situations where a bounding box which contains the full unconstrained motion will fail to capture constrained motion.
* Consider object A flying at high speed to impact the stationary object B, which sits next to another stationary object C.
* Object B's bounding box doesn't overlap with object C's bounding box- they're both stationary, so there's no velocity expansion. But during one frame,
* object A slams into B, and object B's velocity during that frame now forces it to tunnel all the way through C unimpeded, because no contacts were generated between B and C.
* There are ways to address this- all of which are a bit expensive- but CCD as implemented is not a hard guarantee.
* It's a 'best effort' that compromises with performance. Later on, if it's really necessary, we could consider harder guarantees with higher costs, but...
* given that no one seemed to have much of an issue with v1's rather limited CCD, it'll probably be fine.)
*
* So, how is the velocity expansion calculated?
* There's two parts, linear and angular.
*
* Linear is pretty simple- expand the bounding box in the direction of linear displacement (linearVelocity * dt).
*/
Vector <float> vectorDt = new Vector <float>(dt);
Vector3Wide.Scale(ref velocities.LinearVelocity, ref vectorDt, out var linearDisplacement);
var zero = Vector <float> .Zero;
Vector3Wide.Min(ref zero, ref linearDisplacement, out var minDisplacement);
Vector3Wide.Max(ref zero, ref linearDisplacement, out var maxDisplacement);
/*
* Angular requires a bit more care. Since the goal is to create a tight bound, simply using a v = w * r approximation isn't ideal. A slightly tighter can be found:
* 1) The maximum displacement along ANY axis during an intermediate time is equal to the distance from a starting position at MaximumRadius
* to the position of that point at the intermediate time.
* 2) The expansion cannot exceed the maximum radius, so angular deltas greater than pi/3 do not need to be considered.
* (An expansion equal to the maximum radius would result in an equilateral triangle, which has an angle of 60 degrees in each corner.)
* Larger values can simply be clamped.
* 3) The largest displacement along any axis, at any time, is the distance from the starting position to the position at dt. Note that this only holds because of the clamp:
* if the angle was allowed to wrap around, it the distance would start to go down again.
* 4) position(time) = {radius * sin(angular speed * time), radius * cos(angular speed * time)}
* 5) largest expansion required = ||position(dt) - position(0)|| = sqrt(2 * radius^2 * (1 - cos(dt * w)))
* 6) Don't have any true SIMD sin function, but we can approximate it using a taylor series, like: cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6!
* 7) Note that the cosine approximation should stop at a degree where it is smaller than the true value of cosine for the interval 0 to pi/3: this guarantees that the distance,
* which is larger when the cosine is smaller, is conservative and fully bounds the angular motion.
*
* Why do this extra work?
* 1) The bounding box calculation phase, as a part of the pose integration phase, tends to be severely memory bound.
* Spending a little of ALU time to get a smaller bounding box isn't a big concern, even though it includes a couple of sqrts.
* An extra few dozen ALU cycles is unlikely to meaningfully change the execution time.
* 2) Shrinking the bounding box reduces the number of collision pairs. Collision pairs are expensive- many times more expensive than the cost of shrinking the bounding box.
*/
Vector3Wide.Length(ref velocities.AngularVelocity, out var angularVelocityMagnitude);
var a = Vector.Min(angularVelocityMagnitude * vectorDt, new Vector <float>(MathHelper.Pi / 3f));
var a2 = a * a;
var a4 = a2 * a2;
var a6 = a4 * a2;
var cosAngleMinusOne = a2 * new Vector <float>(-1f / 2f) + a4 * new Vector <float>(1f / 24f) - a6 * new Vector <float>(1f / 720f);
//Note that it's impossible for angular motion to cause an increase in bounding box size beyond (maximumRadius-minimumRadius) on any given axis.
//That value, or a conservative approximation, is stored as the maximum angular expansion.
var angularExpansion = Vector.Min(maximumAngularExpansion,
Vector.SquareRoot(new Vector <float>(-2f) * maximumRadius * maximumRadius * cosAngleMinusOne));
Vector3Wide.Subtract(ref minDisplacement, ref angularExpansion, out minDisplacement);
Vector3Wide.Add(ref maxDisplacement, ref angularExpansion, out maxDisplacement);
//The maximum expansion passed into this function is the speculative margin for discrete mode collidables, and ~infinity for passive or continuous ones.
var negativeMaximum = -maximumExpansion;
Vector3Wide.Max(ref negativeMaximum, ref minDisplacement, out minDisplacement);
Vector3Wide.Min(ref maximumExpansion, ref maxDisplacement, out maxDisplacement);
Vector3Wide.Add(ref min, ref minDisplacement, out min);
Vector3Wide.Add(ref max, ref maxDisplacement, out max);
//Note that this is an area that will need to adapt to changes to the body pose representation. If you use a 32 bit fixed point position or a 64 bit double position,
//the bounding box calculation needs to be aware. (In the more extreme cases, so does the broad phase. The amount of conservative padding needed for a 32 bit bounding box
//with 64 bit positions is silly.)
}
public void Test(
ref CapsuleWide a, ref BoxWide b,
ref Vector3Wide offsetB, ref QuaternionWide orientationA, ref QuaternionWide orientationB,
out Convex2ContactManifoldWide manifold)
{
QuaternionWide.Conjugate(ref orientationB, out var toLocalB);
QuaternionWide.TransformWithoutOverlap(ref offsetB, ref toLocalB, out var localOffsetA);
Vector3Wide.Negate(ref localOffsetA);
QuaternionWide.ConcatenateWithoutOverlap(ref orientationA, ref toLocalB, out var boxLocalOrientationA);
QuaternionWide.TransformUnitY(ref boxLocalOrientationA, out var capsuleAxis);
//Get the capsule-axis-perpendicular offset from the box to the capsule and use it to choose which edges to test.
//(Pointless to test the other 9; they're guaranteed to be further away.)
Vector3Wide.Dot(ref localOffsetA, ref capsuleAxis, out var axisOffsetADot);
Vector3Wide.Scale(ref capsuleAxis, ref axisOffsetADot, out var toRemove);
Vector3Wide.Subtract(ref localOffsetA, ref toRemove, out var perpendicularOffset);
Vector3Wide edgeCenters;
edgeCenters.X = Vector.ConditionalSelect(Vector.LessThan(perpendicularOffset.X, Vector <float> .Zero), -b.HalfWidth, b.HalfWidth);
edgeCenters.Y = Vector.ConditionalSelect(Vector.LessThan(perpendicularOffset.Y, Vector <float> .Zero), -b.HalfHeight, b.HalfHeight);
edgeCenters.Z = Vector.ConditionalSelect(Vector.LessThan(perpendicularOffset.Z, Vector <float> .Zero), -b.HalfLength, b.HalfLength);
//Swizzle XYZ -> YZX
Vector3Wide localNormal;
TestBoxEdge(ref localOffsetA.Y, ref localOffsetA.Z, ref localOffsetA.X,
ref capsuleAxis.Y, ref capsuleAxis.Z, ref capsuleAxis.X,
ref a.HalfLength,
ref edgeCenters.Y, ref edgeCenters.Z,
ref b.HalfHeight, ref b.HalfLength, ref b.HalfWidth,
out var taMin, out var taMax, out var depth, out localNormal.Y, out localNormal.Z, out localNormal.X);
//Swizzle XYZ -> ZXY
TestBoxEdge(ref localOffsetA.Z, ref localOffsetA.X, ref localOffsetA.Y,
ref capsuleAxis.Z, ref capsuleAxis.X, ref capsuleAxis.Y,
ref a.HalfLength,
ref edgeCenters.Z, ref edgeCenters.X,
ref b.HalfLength, ref b.HalfWidth, ref b.HalfHeight,
out var eytaMin, out var eytaMax, out var eyDepth, out var eynZ, out var eynX, out var eynY);
Select(ref depth, ref taMin, ref taMax, ref localNormal.X, ref localNormal.Y, ref localNormal.Z,
ref eyDepth, ref eytaMin, ref eytaMax, ref eynX, ref eynY, ref eynZ);
//Swizzle XYZ -> XYZ
TestBoxEdge(ref localOffsetA.X, ref localOffsetA.Y, ref localOffsetA.Z,
ref capsuleAxis.X, ref capsuleAxis.Y, ref capsuleAxis.Z,
ref a.HalfLength,
ref edgeCenters.X, ref edgeCenters.Y,
ref b.HalfWidth, ref b.HalfHeight, ref b.HalfLength,
out var eztaMin, out var eztaMax, out var ezDepth, out var eznX, out var eznY, out var eznZ);
Select(ref depth, ref taMin, ref taMax, ref localNormal.X, ref localNormal.Y, ref localNormal.Z,
ref ezDepth, ref eztaMin, ref eztaMax, ref eznX, ref eznY, ref eznZ);
//For each face, clip the capsule axis against the face bounds. The closest offset between the clipped capsule axis endpoints and the box is the normal candidate.
var positive = Vector <float> .One;
var negative = -positive;
var divisionGuard = new Vector <float>(1e-15f);
//Division by zero is hacked away by clamping the magnitude to some nonzero value and recovering the sign in the numerator.
//The resulting interval will be properly signed and enormous, so it serves the necessary purpose.
//Note the negation: t = dot(N, +-halfExtent - offsetA) / dot(N, capsuleAxis) = dot(N, offsetA +- halfExtent) / -dot(N, capsuleAxis)
var scaleX = Vector.ConditionalSelect(Vector.LessThan(capsuleAxis.X, Vector <float> .Zero), positive, negative) / Vector.Max(Vector.Abs(capsuleAxis.X), divisionGuard);
var scaleY = Vector.ConditionalSelect(Vector.LessThan(capsuleAxis.Y, Vector <float> .Zero), positive, negative) / Vector.Max(Vector.Abs(capsuleAxis.Y), divisionGuard);
var scaleZ = Vector.ConditionalSelect(Vector.LessThan(capsuleAxis.Z, Vector <float> .Zero), positive, negative) / Vector.Max(Vector.Abs(capsuleAxis.Z), divisionGuard);
Vector3Wide scaledExtents, scaledOffset;
//Clip slightly beyond the actual face limit to avoid pointlessly cutting the interval for near-edge-parallel capsules.
var epsilonScale = new Vector <float>(1f + 1e-6f);
scaledExtents.X = epsilonScale * b.HalfWidth * Vector.Abs(scaleX);
scaledExtents.Y = epsilonScale * b.HalfHeight * Vector.Abs(scaleY);
scaledExtents.Z = epsilonScale * b.HalfLength * Vector.Abs(scaleZ);
scaledOffset.X = localOffsetA.X * scaleX;
scaledOffset.Y = localOffsetA.Y * scaleY;
scaledOffset.Z = localOffsetA.Z * scaleZ;
Vector3Wide.Subtract(ref scaledOffset, ref scaledExtents, out var tCandidateMin);
var negativeHalfLength = -a.HalfLength;
Vector3Wide.Min(ref a.HalfLength, ref tCandidateMin, out tCandidateMin);
Vector3Wide.Max(ref negativeHalfLength, ref tCandidateMin, out tCandidateMin);
Vector3Wide.Add(ref scaledOffset, ref scaledExtents, out var tCandidateMax);
Vector3Wide.Min(ref a.HalfLength, ref tCandidateMax, out tCandidateMax);
Vector3Wide.Max(ref negativeHalfLength, ref tCandidateMax, out tCandidateMax);
var zero = Vector <float> .Zero;
//Face X
TestBoxFace(ref localOffsetA.X,
ref capsuleAxis.X, ref a.HalfLength,
ref tCandidateMin.Y, ref tCandidateMax.Y, ref tCandidateMin.Z, ref tCandidateMax.Z,
ref b.HalfWidth,
out var fxDepth, out var fxtaMin, out var fxtaMax, out var fxn);
Select(ref depth, ref taMin, ref taMax, ref localNormal.X, ref localNormal.Y, ref localNormal.Z,
ref fxDepth, ref fxtaMin, ref fxtaMax, ref fxn, ref zero, ref zero);
//Face Y
TestBoxFace(ref localOffsetA.Y,
ref capsuleAxis.Y, ref a.HalfLength,
ref tCandidateMin.X, ref tCandidateMax.X, ref tCandidateMin.Z, ref tCandidateMax.Z,
ref b.HalfHeight,
out var fyDepth, out var fytaMin, out var fytaMax, out var fyn);
Select(ref depth, ref taMin, ref taMax, ref localNormal.X, ref localNormal.Y, ref localNormal.Z,
ref fyDepth, ref fytaMin, ref fytaMax, ref zero, ref fyn, ref zero);
//Face Z
TestBoxFace(ref localOffsetA.Z,
ref capsuleAxis.Z, ref a.HalfLength,
ref tCandidateMin.X, ref tCandidateMax.X, ref tCandidateMin.Y, ref tCandidateMax.Y,
ref b.HalfLength,
out var fzDepth, out var fztaMin, out var fztaMax, out var fzn);
Select(ref depth, ref taMin, ref taMax, ref localNormal.X, ref localNormal.Y, ref localNormal.Z,
ref fzDepth, ref fztaMin, ref fztaMax, ref zero, ref zero, ref fzn);
//Each contact may have its own depth.
//Imagine a face collision- if the capsule axis isn't fully parallel with the plane's surface, it would be strange to use the same depth for both contacts.
//Compute the interval of the box on the normal. Note that the normal is already calibrated to point from B to A (box to capsule).
//(This is partially redundant with the per-case calculations, but simply redoing some cheap ALU work is easier than trying to keep track of per-contact depths across all cases.)
Vector3Wide.Scale(ref capsuleAxis, ref taMin, out var localA0);
Vector3Wide.Scale(ref capsuleAxis, ref taMax, out var localA1);
Vector3Wide.Add(ref localOffsetA, ref localA0, out var bToA0);
Vector3Wide.Add(ref localOffsetA, ref localA1, out var bToA1);
var boxExtreme = Vector.Abs(localNormal.X * b.HalfWidth) + Vector.Abs(localNormal.Y * b.HalfHeight) + Vector.Abs(localNormal.Z * b.HalfLength);
Vector3Wide.Dot(ref localNormal, ref bToA0, out var dot0);
Vector3Wide.Dot(ref localNormal, ref bToA1, out var dot1);
manifold.Depth0 = a.Radius + boxExtreme - dot0;
manifold.Depth1 = a.Radius + boxExtreme - dot1;
manifold.FeatureId0 = Vector <int> .Zero;
manifold.FeatureId1 = Vector <int> .One;
//Transform A0, A1, and the normal into world space.
Matrix3x3Wide.CreateFromQuaternion(ref orientationB, out var orientationMatrixB);
Matrix3x3Wide.TransformWithoutOverlap(ref localNormal, ref orientationMatrixB, out manifold.Normal);
Matrix3x3Wide.TransformWithoutOverlap(ref localA0, ref orientationMatrixB, out manifold.OffsetA0);
Matrix3x3Wide.TransformWithoutOverlap(ref localA1, ref orientationMatrixB, out manifold.OffsetA1);
//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(ref manifold.Normal, ref negativeOffsetFromA0, out var normalPush0);
Vector3Wide.Scale(ref manifold.Normal, ref negativeOffsetFromA1, out var normalPush1);
Vector3Wide.Add(ref manifold.OffsetA0, ref normalPush0, out manifold.OffsetA0);
Vector3Wide.Add(ref manifold.OffsetA1, ref normalPush1, out manifold.OffsetA1);
manifold.Count = Vector.ConditionalSelect(Vector.LessThan(taMax - taMin, new Vector <float>(1e-7f) * a.HalfLength), Vector <int> .One, new Vector <int>(2));
}
public static void Merge(ref BoundingBoxWide a, ref BoundingBoxWide b, out BoundingBoxWide merged)
{
Vector3Wide.Min(ref a.Min, ref b.Min, out merged.Min);
Vector3Wide.Max(ref a.Max, ref b.Max, out merged.Max);
}
