// // Tangent // public static MyVector3 GetTangent(MyVector3 posA, MyVector3 posB, MyVector3 handleA, MyVector3 handleB, float t) { //The tangent is also the derivative vector MyVector3 tangent = MyVector3.Normalize(GetDerivativeVec(posA, posB, handleA, handleB, t)); return(tangent); }
//Calculate the angle between two vectors //This angle should be measured in 360 degrees (Vector3.Angle is measured in 180 degrees) //Should maybe be moved to _Geometry?? //In 3d space [radians] //https://stackoverflow.com/questions/5188561/signed-angle-between-two-3d-vectors-with-same-origin-within-the-same-plane //https://math.stackexchange.com/questions/2906314/how-to-calculate-angle-between-two-vectors-in-3d-with-clockwise-or-counter-clock public static float AngleFromToCCW(MyVector3 from, MyVector3 to, MyVector3 upRef) { //This is only working in 2d space //float angleDegrees = Quaternion.FromToRotation(to.ToVector3(), from.ToVector3()).eulerAngles.y; from = MyVector3.Normalize(from); to = MyVector3.Normalize(to); upRef = MyVector3.Normalize(upRef); float angleRad = AngleBetween(from, to, shouldNormalize: false); //To get 0-2pi (360 degrees) we can use the determinant [a, b, u] = (a x b) dot u //Where u is a reference up vector //Remember that the cross product is not alwayspointing up - it can change to down depending on how the vectors are aligned //Which is why we need a fixed reference up MyVector3 cross = MyVector3.Cross(from, to); float determinant = MyVector3.Dot(MyVector3.Cross(from, to), upRef); //Debug.Log(determinant); if (determinant >= 0f) { return(angleRad); } else { return((Mathf.PI * 2f) - angleRad); } }
//Get the forward direction do the Bezier Quadratic //This direction is always tangent to the curve public static MyVector3 BezierQuadraticForwardDir(MyVector3 posA, MyVector3 posB, MyVector3 handlePos, float t) { //Same as when we calculate t MyVector3 interpolation_posA_handlePos = BezierLinear(posA, handlePos, t); MyVector3 interpolation_handlePos_posB = BezierLinear(handlePos, posB, t); MyVector3 forwardDir = MyVector3.Normalize(interpolation_handlePos_posB - interpolation_posA_handlePos); return(forwardDir); }
//3d public static MyVector3 GetLinePlaneIntersectionPoint(Plane3 plane, Edge3 line) { MyVector3 lineDir = MyVector3.Normalize(line.p1 - line.p2); Ray3 ray = new Ray3(line.p1, lineDir); MyVector3 intersectionPoint = GetIntersectionCoordinate(plane, ray); return(intersectionPoint); }
//If we have a forward and an up reference vector //So this is not going to work if we have loops //tangent is same as forward public InterpolationTransform(MyVector3 position, MyVector3 tangent, MyVector3 up) { this.position = position; MyVector3 biNormal = MyVector3.Normalize(MyVector3.Cross(up, tangent)); MyVector3 normal = MyVector3.Normalize(MyVector3.Cross(tangent, biNormal)); this.orientation = Quaternion.LookRotation(tangent.ToVector3(), normal.ToVector3()); }
// // Calculate the normal of a clock-wise oriented triangle in 3d space // public static MyVector3 CalculateTriangleNormal(MyVector3 p1, MyVector3 p2, MyVector3 p3, bool shouldNormalize = true) { MyVector3 normal = MyVector3.Cross(p3 - p2, p1 - p2); if (shouldNormalize) { normal = MyVector3.Normalize(normal); } return(normal); }
// // Get transforms (position and orientation) at point t // //The position and the tangent are easy to find //what's difficult to find is the normal because a line doesn't have a single normal //To get the normal in 2d, we can just flip two coordinates in the forward vector and set one to negative //MyVector3 normal = new MyVector3(-forwardDir.z, 0f, forwardDir.x); //In 3d there are multiple alternatives: //You can read about these methods here: //https://pomax.github.io/bezierinfo/#pointvectors3d //Game Programming Gems 2: The Parallel Transport Frame (p. 215) //Unite 2015 - A coder's guide to spline-based procedural geometry https://www.youtube.com/watch?v=o9RK6O2kOKo // // Alternative 1. Fixed up // //Use ref vector to know which direction is up //Is not going to work if we have loops, but should work if you make "2d" roads like in cities skylines so no roller coasters public static MyQuaternion GetOrientation_UpRef(MyVector3 tangent, MyVector3 upRef) { tangent = MyVector3.Normalize(tangent); MyVector3 biNormal = MyVector3.Normalize(MyVector3.Cross(upRef, tangent)); MyVector3 normal = MyVector3.Normalize(MyVector3.Cross(tangent, biNormal)); MyQuaternion orientation = new MyQuaternion(tangent, normal); return(orientation); }
// // Add a triangle to this mesh // //We dont have a normal so we have to calculate it, so make sure v1-v2-v3 is clock-wise public HalfEdgeFace3 AddTriangle(MyVector3 p1, MyVector3 p2, MyVector3 p3, bool findOppositeEdge = false) { MyVector3 normal = MyVector3.Normalize(MyVector3.Cross(p3 - p2, p1 - p2)); MyMeshVertex v1 = new MyMeshVertex(p1, normal); MyMeshVertex v2 = new MyMeshVertex(p2, normal); MyMeshVertex v3 = new MyMeshVertex(p3, normal); HalfEdgeFace3 f = AddTriangle(v1, v2, v3); return(f); }
// // Tangent at point t (Forward direction if we travel along the curve) // public static MyVector3 GetTangent(MyVector3 posA, MyVector3 posB, MyVector3 handlePos, float t) { t = Mathf.Clamp01(t); //Alternative 1 //Same as when we calculate position from t //MyVector3 interpolation_posA_handlePos = BezierLinear.GetPosition(posA, handlePos, t); //MyVector3 interpolation_handlePos_posB = BezierLinear.GetPosition(handlePos, posB, t); //MyVector3 tangent = MyVector3.Normalize(interpolation_handlePos_posB - interpolation_posA_handlePos); //Alternative 2 //The tangent is also the derivative vector MyVector3 tangent = MyVector3.Normalize(GetDerivativeVec(posA, posB, handlePos, t)); return(tangent); }
// // Alternative 2. Frenet normal (also known as Frenet Frame) // //Use the tagent we have and a tangent next to it //Works in many cases (but sometimes the frame may flip because of changes in the second derivative) public static MyQuaternion GetOrientation_FrenetNormal(MyVector3 tangent, MyVector3 secondDerivativeVec) { MyVector3 a = MyVector3.Normalize(tangent); //What a next point's tangent would be if the curve stopped changing at our point and just had the same derivative and second derivative from that point on MyVector3 b = MyVector3.Normalize(a + secondDerivativeVec); //A vector that we use as the "axis of rotation" for turning the tangent a quarter circle to get the normal MyVector3 r = MyVector3.Normalize(MyVector3.Cross(a, b)); //The normal vector should be perpendicular to the plane that the tangent and the axis of rotation lie in MyVector3 normal = MyVector3.Normalize(MyVector3.Cross(r, a)); MyQuaternion orientation = new MyQuaternion(tangent, normal); return(orientation); }
// // Alternative 1.5. Similar to Alternative 1, but we know the up vector at both the start and end position // public static InterpolationTransform GetTransform_InterpolateBetweenUpVectors( _Curve curve, float t, MyVector3 upRefStart, MyVector3 upRefEnd) { //Position on the curve at point t MyVector3 pos = curve.GetPosition(t); //Forward direction (tangent) on the curve at point t MyVector3 forwardDir = curve.GetTangent(t); //Interpolate between the start and end up vector to get an up vector at a t position MyVector3 interpolatedUpDir = MyVector3.Normalize(BezierLinear.GetPosition(upRefStart, upRefEnd, t)); MyQuaternion orientation = InterpolationTransform.GetOrientation_UpRef(forwardDir, interpolatedUpDir); InterpolationTransform trans = new InterpolationTransform(pos, orientation); return(trans); }
// // Get a Transform (includes position and orientation) at point t // public InterpolationTransform GetTransform(float t) { //Same as when we calculate t MyVector3 interpolation_1_2 = _Interpolation.BezierQuadratic(posA, handleB, handleA, t); MyVector3 interpolation_2_3 = _Interpolation.BezierQuadratic(posA, posB, handleB, t); MyVector3 finalInterpolation = _Interpolation.BezierLinear(interpolation_1_2, interpolation_2_3, t); //This direction is always tangent to the curve MyVector3 forwardDir = MyVector3.Normalize(interpolation_2_3 - interpolation_1_2); //A simple way to get the other directions is to use LookRotation with just forward dir as parameter //Then the up direction will always be the world up direction, and it calculates the right direction Quaternion orientation = Quaternion.LookRotation(forwardDir.ToVector3()); InterpolationTransform trans = new InterpolationTransform(finalInterpolation, orientation); return(trans); }
//The angle between two vectors 0 <= angle <= 180 //Same as Vector3.Angle() but we are using MyVector3 public static float AngleBetween(MyVector3 from, MyVector3 to, bool shouldNormalize = true) { //from and to should be normalized //But sometimes they are already normalized and then we dont need to do it again if (shouldNormalize) { from = MyVector3.Normalize(from); to = MyVector3.Normalize(to); } //dot(a_normalized, b_normalized) = cos(alpha) -> acos(dot(a_normalized, b_normalized)) = alpha float dot = MyVector3.Dot(from, to); //This shouldn't happen but may happen because of floating point precision issues dot = Mathf.Clamp(dot, -1f, 1f); float angleRad = Mathf.Acos(dot); return(angleRad); }
//Cut a triangle where two vertices are inside and the other vertex is outside //Make sure they are sorted clockwise: O1-O2-I1 //F means that this vertex is outside the plane private static void CutTriangleTwoOutside(MyMeshVertex O1, MyMeshVertex O2, MyMeshVertex I1, HalfEdgeData3 newMeshO, HalfEdgeData3 newMeshI, HashSet <HalfEdge3> newEdgesI, HashSet <HalfEdge3> newEdgesO, Plane3 cutPlane) { //Cut the triangle by using edge-plane intersection //Triangles in Unity are ordered clockwise, so form edges that intersects with the plane: Edge3 e_O2I1 = new Edge3(O2.position, I1.position); //Edge3 e_F1F2 = new Edge3(F1, F2); //Not needed because never intersects with the plane Edge3 e_I1O1 = new Edge3(I1.position, O1.position); //The positions of the intersection vertices MyVector3 pos_O2I1 = _Intersections.GetLinePlaneIntersectionPoint(cutPlane, e_O2I1); MyVector3 pos_I1O1 = _Intersections.GetLinePlaneIntersectionPoint(cutPlane, e_I1O1); //The normals of the intersection vertices float percentageBetween_O2I1 = MyVector3.Distance(O2.position, pos_O2I1) / MyVector3.Distance(O2.position, I1.position); float percentageBetween_I1O1 = MyVector3.Distance(I1.position, pos_I1O1) / MyVector3.Distance(I1.position, O1.position); MyVector3 normal_O2I1 = _Interpolation.Lerp(O2.normal, I1.normal, percentageBetween_O2I1); MyVector3 normal_I1O1 = _Interpolation.Lerp(I1.normal, O1.normal, percentageBetween_I1O1); //MyVector3 normal_F2B1 = Vector3.Slerp(F2.normal.ToVector3(), B1.normal.ToVector3(), percentageBetween_F2B1).ToMyVector3(); //MyVector3 normal_B1F1 = Vector3.Slerp(B1.normal.ToVector3(), F1.normal.ToVector3(), percentageBetween_B1F1).ToMyVector3(); normal_O2I1 = MyVector3.Normalize(normal_O2I1); normal_I1O1 = MyVector3.Normalize(normal_I1O1); //The intersection vertices MyMeshVertex v_O2I1 = new MyMeshVertex(pos_O2I1, normal_O2I1); MyMeshVertex v_I1O1 = new MyMeshVertex(pos_I1O1, normal_I1O1); //Form 3 new triangles //Outside AddTriangleToMesh(v_O2I1, v_I1O1, O2, newMeshO, newEdgesO); AddTriangleToMesh(O2, v_I1O1, O1, newMeshO, null); //Inside AddTriangleToMesh(v_I1O1, v_O2I1, I1, newMeshI, newEdgesI); }
//Remove flat tetrahedrons (a vertex in a triangle) private static bool RemoveFlatTetrahedrons(HalfEdgeData3 meshData, Normalizer3 normalizer = null) { HashSet <HalfEdgeVertex3> vertices = meshData.verts; bool foundFlatTetrahedron = false; foreach (HalfEdgeVertex3 vertex in vertices) { HashSet <HalfEdge3> edgesGoingToVertex = vertex.GetEdgesPointingToVertex(meshData); if (edgesGoingToVertex.Count == 3) { //Find the vertices of the triangle covering this vertex clock-wise HalfEdgeVertex3 v1 = vertex.edge.v; HalfEdgeVertex3 v2 = vertex.edge.prevEdge.oppositeEdge.v; HalfEdgeVertex3 v3 = vertex.edge.oppositeEdge.nextEdge.v; //Build a plane MyVector3 normal = MyVector3.Normalize(MyVector3.Cross(v3.position - v2.position, v1.position - v2.position)); Plane3 plane = new Plane3(v1.position, normal); //Find the distance from the vertex to the plane float distance = _Geometry.GetSignedDistanceFromPointToPlane(vertex.position, plane); distance = Mathf.Abs(distance); if (distance < FLAT_TETRAHEDRON_DISTANCE) { //Debug.Log("Found flat tetrahedron"); Vector3 p1 = normalizer.UnNormalize(v1.position).ToVector3(); Vector3 p2 = normalizer.UnNormalize(v2.position).ToVector3(); Vector3 p3 = normalizer.UnNormalize(v3.position).ToVector3(); TestAlgorithmsHelpMethods.DebugDrawTriangle(p1, p2, p3, normal.ToVector3(), Color.blue, Color.red); foundFlatTetrahedron = true; //Save the opposite edges HashSet <HalfEdge3> oppositeEdges = new HashSet <HalfEdge3>(); oppositeEdges.Add(v1.edge.oppositeEdge); oppositeEdges.Add(v2.edge.oppositeEdge); oppositeEdges.Add(v3.edge.oppositeEdge); //Remove the three triangles foreach (HalfEdge3 e in edgesGoingToVertex) { meshData.DeleteFace(e.face); } //Add the new triangle (could maybe connect it ourselves) HalfEdgeFace3 newTriangle = meshData.AddTriangle(v1.position, v2.position, v3.position, findOppositeEdge: false); meshData.TryFindOppositeEdge(newTriangle.edge, oppositeEdges); meshData.TryFindOppositeEdge(newTriangle.edge.nextEdge, oppositeEdges); meshData.TryFindOppositeEdge(newTriangle.edge.nextEdge.nextEdge, oppositeEdges); break; } } } return(foundFlatTetrahedron); }