//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);
}
}
//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);
}
//
// Alternative 3. Rotation Minimising Frame (also known as "Parallel Transport Frame" or "Bishop Frame")
//
//Gets its stability by incrementally rotating a coordinate system (= frame) as it is translate along the curve
//Has to be computed for the entire curve because we need the previous frame (previousTransform) belonging to a point before this point
//Is initalized by using "Fixed Up" or "Frenet Normal"
public static MyQuaternion GetOrientation_RotationFrame(MyVector3 position, MyVector3 tangent, InterpolationTransform previousTransform)
{
/*
* //This version is from https://pomax.github.io/bezierinfo/#pointvectors3d
* //Reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous points as a "mirror"
* MyVector3 v1 = position - previousTransform.position;
*
* float c1 = MyVector3.Dot(v1, v1);
*
* MyVector3 riL = previousTransform.Right - v1 * (2f / c1) * MyVector3.Dot(v1, previousTransform.Right);
*
* MyVector3 tiL = previousTransform.Forward - v1 * (2f / c1) * MyVector3.Dot(v1, previousTransform.Forward);
*
* //This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal that's slightly off-kilter
* //reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror".
* MyVector3 v2 = tangent - tiL;
*
* float c2 = MyVector3.Dot(v2, v2);
*
* //Now we can calculate the normal and right vector belonging to this orientation
* MyVector3 right = riL - v2 * (2f / c2) * MyVector3.Dot(v2, riL);
*
* //The source has right x tangent, but then every second normal is flipped
* MyVector3 normal = MyVector3.Cross(tangent, right);
*
* MyQuaternion orientation = new MyQuaternion(tangent, normal);
*/
//This version is from Game Programming Gems 2: The Parallel Transport Frame
//They generate the same result and this one is easier to understand
//The two tangents
MyVector3 T1 = previousTransform.Forward;
MyVector3 T2 = tangent;
//You move T1 to the new position, so A is a vector going from the new position
MyVector3 A = MyVector3.Cross(T1, T2);
//This is the angle between T1 and T2
float alpha = Mathf.Acos(MyVector3.Dot(T1, T2) / (MyVector3.Magnitude(T1) * MyVector3.Magnitude(T2)));
//Now rotate the previous frame around axis A with angle alpha
MyQuaternion F1 = previousTransform.orientation;
MyQuaternion F2 = MyQuaternion.RotateQuaternion(F1, alpha * Mathf.Rad2Deg, A);
MyQuaternion orientation = F2;
return(orientation);
}
//
// 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);
}
//
// Calculate the center of circle in 3d space given three coordinates
//
//From https://gamedev.stackexchange.com/questions/60630/how-do-i-find-the-circumcenter-of-a-triangle-in-3d
public static MyVector3 CalculateCircleCenter(MyVector3 a, MyVector3 b, MyVector3 c)
{
MyVector3 ac = c - a;
MyVector3 ab = b - a;
MyVector3 abXac = MyVector3.Cross(ab, ac);
//This is the vector from a to the circumsphere center
MyVector3 toCircumsphereCenter = MyVector3.Cross(abXac, ab) * Mathf.Pow(MyVector3.Magnitude(ac), 2f);
toCircumsphereCenter += MyVector3.Cross(ac, abXac) * Mathf.Pow(MyVector3.Magnitude(ab), 2f);
toCircumsphereCenter *= (1f / (2f * Mathf.Pow(MyVector3.Magnitude(abXac), 2f)));
float circumsphereRadius = MyVector3.Magnitude(toCircumsphereCenter);
//The circumsphere center becomes
MyVector3 ccs = a + toCircumsphereCenter;
return(ccs);
}
//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);
}
