/// <summary> /// Modifies a contribution using a transform, position, and weight. /// </summary> /// <param name="transform">Transform to use to modify the contribution.</param> /// <param name="center">Center to use to modify the contribution.</param> /// <param name="baseContribution">Original unmodified contribution.</param> /// <param name="weight">Weight of the contribution.</param> /// <param name="contribution">Transformed contribution.</param> public static void TransformContribution(ref RigidTransform transform, ref Vector3 center, ref Matrix3x3 baseContribution, float weight, out Matrix3x3 contribution) { Matrix3x3 rotation; Matrix3x3.CreateFromQuaternion(ref transform.Orientation, out rotation); Matrix3x3 temp; //Do angular transformed contribution first... Matrix3x3.MultiplyTransposed(ref rotation, ref baseContribution, out temp); Matrix3x3.Multiply(ref temp, ref rotation, out temp); contribution = temp; //Now add in the offset from the origin. Vector3 offset; Vector3.Subtract(ref transform.Position, ref center, out offset); Matrix3x3 innerProduct; Matrix3x3.CreateScale(offset.LengthSquared(), out innerProduct); Matrix3x3 outerProduct; Matrix3x3.CreateOuterProduct(ref offset, ref offset, out outerProduct); Matrix3x3.Subtract(ref innerProduct, ref outerProduct, out temp); Matrix3x3.Add(ref contribution, ref temp, out contribution); Matrix3x3.Multiply(ref contribution, weight, out contribution); }
void ComputeShapeInformation(TransformableMeshData data, out ShapeDistributionInformation shapeInformation) { //Compute the surface vertices of the shape. surfaceVertices.Clear(); try { ConvexHullHelper.GetConvexHull(data.vertices, surfaceVertices); for (int i = 0; i < surfaceVertices.Count; i++) { AffineTransform.Transform(ref surfaceVertices.Elements[i], ref data.worldTransform, out surfaceVertices.Elements[i]); } } catch { surfaceVertices.Clear(); //If the convex hull failed, then the point set has no volume. A mobile mesh is allowed to have zero volume, however. //In this case, compute the bounding box of all points. BoundingBox box = new BoundingBox(); for (int i = 0; i < data.vertices.Length; i++) { Vector3 v; data.GetVertexPosition(i, out v); if (v.X > box.Max.X) { box.Max.X = v.X; } if (v.X < box.Min.X) { box.Min.X = v.X; } if (v.Y > box.Max.Y) { box.Max.Y = v.Y; } if (v.Y < box.Min.Y) { box.Min.Y = v.Y; } if (v.Z > box.Max.Z) { box.Max.Z = v.Z; } if (v.Z < box.Min.Z) { box.Min.Z = v.Z; } } //Add the corners. This will overestimate the size of the surface a bit. surfaceVertices.Add(box.Min); surfaceVertices.Add(box.Max); surfaceVertices.Add(new Vector3(box.Min.X, box.Min.Y, box.Max.Z)); surfaceVertices.Add(new Vector3(box.Min.X, box.Max.Y, box.Min.Z)); surfaceVertices.Add(new Vector3(box.Max.X, box.Min.Y, box.Min.Z)); surfaceVertices.Add(new Vector3(box.Min.X, box.Max.Y, box.Max.Z)); surfaceVertices.Add(new Vector3(box.Max.X, box.Max.Y, box.Min.Z)); surfaceVertices.Add(new Vector3(box.Max.X, box.Min.Y, box.Max.Z)); } shapeInformation.Center = new Vector3(); if (solidity == MobileMeshSolidity.Solid) { //The following inertia tensor calculation assumes a closed mesh. shapeInformation.Volume = 0; for (int i = 0; i < data.indices.Length; i += 3) { Vector3 v2, v3, v4; data.GetTriangle(i, out v2, out v3, out v4); //Determinant is 6 * volume. It's signed, though; this is because the mesh isn't necessarily convex nor centered on the origin. float tetrahedronVolume = v2.X * (v3.Y * v4.Z - v3.Z * v4.Y) - v3.X * (v2.Y * v4.Z - v2.Z * v4.Y) + v4.X * (v2.Y * v3.Z - v2.Z * v3.Y); shapeInformation.Volume += tetrahedronVolume; shapeInformation.Center += tetrahedronVolume * (v2 + v3 + v4); } shapeInformation.Center /= shapeInformation.Volume * 4; shapeInformation.Volume /= 6; shapeInformation.Volume = Math.Abs(shapeInformation.Volume); data.worldTransform.Translation -= shapeInformation.Center; //Source: Explicit Exact Formulas for the 3-D Tetrahedron Inertia Tensor in Terms of its Vertex Coordinates //http://www.scipub.org/fulltext/jms2/jms2118-11.pdf //x1, x2, x3, x4 are origin, triangle1, triangle2, triangle3 //Looking to find inertia tensor matrix of the form // [ a -b' -c' ] // [ -b' b -a' ] // [ -c' -a' c ] float a = 0, b = 0, c = 0, ao = 0, bo = 0, co = 0; float totalWeight = 0; for (int i = 0; i < data.indices.Length; i += 3) { Vector3 v2, v3, v4; data.GetTriangle(i, out v2, out v3, out v4); //Determinant is 6 * volume. It's signed, though; this is because the mesh isn't necessarily convex nor centered on the origin. float tetrahedronVolume = v2.X * (v3.Y * v4.Z - v3.Z * v4.Y) - v3.X * (v2.Y * v4.Z - v2.Z * v4.Y) + v4.X * (v2.Y * v3.Z - v2.Z * v3.Y); totalWeight += tetrahedronVolume; a += tetrahedronVolume * (v2.Y * v2.Y + v2.Y * v3.Y + v3.Y * v3.Y + v2.Y * v4.Y + v3.Y * v4.Y + v4.Y * v4.Y + v2.Z * v2.Z + v2.Z * v3.Z + v3.Z * v3.Z + v2.Z * v4.Z + v3.Z * v4.Z + v4.Z * v4.Z); b += tetrahedronVolume * (v2.X * v2.X + v2.X * v3.X + v3.X * v3.X + v2.X * v4.X + v3.X * v4.X + v4.X * v4.X + v2.Z * v2.Z + v2.Z * v3.Z + v3.Z * v3.Z + v2.Z * v4.Z + v3.Z * v4.Z + v4.Z * v4.Z); c += tetrahedronVolume * (v2.X * v2.X + v2.X * v3.X + v3.X * v3.X + v2.X * v4.X + v3.X * v4.X + v4.X * v4.X + v2.Y * v2.Y + v2.Y * v3.Y + v3.Y * v3.Y + v2.Y * v4.Y + v3.Y * v4.Y + v4.Y * v4.Y); ao += tetrahedronVolume * (2 * v2.Y * v2.Z + v3.Y * v2.Z + v4.Y * v2.Z + v2.Y * v3.Z + 2 * v3.Y * v3.Z + v4.Y * v3.Z + v2.Y * v4.Z + v3.Y * v4.Z + 2 * v4.Y * v4.Z); bo += tetrahedronVolume * (2 * v2.X * v2.Z + v3.X * v2.Z + v4.X * v2.Z + v2.X * v3.Z + 2 * v3.X * v3.Z + v4.X * v3.Z + v2.X * v4.Z + v3.X * v4.Z + 2 * v4.X * v4.Z); co += tetrahedronVolume * (2 * v2.X * v2.Y + v3.X * v2.Y + v4.X * v2.Y + v2.X * v3.Y + 2 * v3.X * v3.Y + v4.X * v3.Y + v2.X * v4.Y + v3.X * v4.Y + 2 * v4.X * v4.Y); } float density = 1 / totalWeight; float diagonalFactor = density / 10; float offFactor = -density / 20; a *= diagonalFactor; b *= diagonalFactor; c *= diagonalFactor; ao *= offFactor; bo *= offFactor; co *= offFactor; shapeInformation.VolumeDistribution = new Matrix3x3(a, bo, co, bo, b, ao, co, ao, c); } else { shapeInformation.Center = new Vector3(); float totalWeight = 0; for (int i = 0; i < data.indices.Length; i += 3) { //Configure the inertia tensor to be local. Vector3 vA, vB, vC; data.GetTriangle(i, out vA, out vB, out vC); Vector3 vAvB; Vector3 vAvC; Vector3.Subtract(ref vB, ref vA, out vAvB); Vector3.Subtract(ref vC, ref vA, out vAvC); Vector3 cross; Vector3.Cross(ref vAvB, ref vAvC, out cross); float weight = cross.Length(); totalWeight += weight; shapeInformation.Center += weight * (vA + vB + vC) / 3; } shapeInformation.Center /= totalWeight; shapeInformation.Volume = 0; data.worldTransform.Translation -= shapeInformation.Center; shapeInformation.VolumeDistribution = new Matrix3x3(); for (int i = 0; i < data.indices.Length; i += 3) { //Configure the inertia tensor to be local. Vector3 vA, vB, vC; data.GetTriangle(i, out vA, out vB, out vC); Vector3 vAvB; Vector3 vAvC; Vector3.Subtract(ref vB, ref vA, out vAvB); Vector3.Subtract(ref vC, ref vA, out vAvC); Vector3 cross; Vector3.Cross(ref vAvB, ref vAvC, out cross); float weight = cross.Length(); totalWeight += weight; Matrix3x3 innerProduct; Matrix3x3.CreateScale(vA.LengthSquared(), out innerProduct); Matrix3x3 outerProduct; Matrix3x3.CreateOuterProduct(ref vA, ref vA, out outerProduct); Matrix3x3 contribution; Matrix3x3.Subtract(ref innerProduct, ref outerProduct, out contribution); Matrix3x3.Multiply(ref contribution, weight, out contribution); Matrix3x3.Add(ref shapeInformation.VolumeDistribution, ref contribution, out shapeInformation.VolumeDistribution); Matrix3x3.CreateScale(vB.LengthSquared(), out innerProduct); Matrix3x3.CreateOuterProduct(ref vB, ref vB, out outerProduct); Matrix3x3.Subtract(ref innerProduct, ref outerProduct, out outerProduct); Matrix3x3.Multiply(ref contribution, weight, out contribution); Matrix3x3.Add(ref shapeInformation.VolumeDistribution, ref contribution, out shapeInformation.VolumeDistribution); Matrix3x3.CreateScale(vC.LengthSquared(), out innerProduct); Matrix3x3.CreateOuterProduct(ref vC, ref vC, out outerProduct); Matrix3x3.Subtract(ref innerProduct, ref outerProduct, out contribution); Matrix3x3.Multiply(ref contribution, weight, out contribution); Matrix3x3.Add(ref shapeInformation.VolumeDistribution, ref contribution, out shapeInformation.VolumeDistribution); } Matrix3x3.Multiply(ref shapeInformation.VolumeDistribution, 1 / (6 * totalWeight), out shapeInformation.VolumeDistribution); } ////Configure the inertia tensor to be local. //Vector3 finalOffset = shapeInformation.Center; //Matrix3X3 finalInnerProduct; //Matrix3X3.CreateScale(finalOffset.LengthSquared(), out finalInnerProduct); //Matrix3X3 finalOuterProduct; //Matrix3X3.CreateOuterProduct(ref finalOffset, ref finalOffset, out finalOuterProduct); //Matrix3X3 finalContribution; //Matrix3X3.Subtract(ref finalInnerProduct, ref finalOuterProduct, out finalContribution); //Matrix3X3.Subtract(ref shapeInformation.VolumeDistribution, ref finalContribution, out shapeInformation.VolumeDistribution); }