public static List <InterpolationTransform> GetTransforms_UpRef(_Curve curve, List <float> tValues, Vector3 upRef)
{
List <InterpolationTransform> orientations = new List <InterpolationTransform>();
foreach (float t in tValues)
{
InterpolationTransform transform = InterpolationTransform.GetTransform_UpRef(curve, t, upRef);
orientations.Add(transform);
}
return(orientations);
}
public static List <InterpolationTransform> GetTransforms_FrenetNormal(_Curve curve, List <float> tValues)
{
List <InterpolationTransform> transforms = new List <InterpolationTransform>();
foreach (float t in tValues)
{
InterpolationTransform transform = InterpolationTransform.GetTransform_FrenetNormal(curve, t);
transforms.Add(transform);
}
return(transforms);
}
public static List <InterpolationTransform> GetTransforms_InterpolateBetweenUpVectors(
_Curve curve, List <float> tValues, Vector3 upRefStart, Vector3 upRefEnd)
{
List <InterpolationTransform> transforms = new List <InterpolationTransform>();
foreach (float t in tValues)
{
InterpolationTransform transform = InterpolationTransform.GetTransform_InterpolateBetweenUpVectors(curve, t, upRefStart, upRefEnd);
transforms.Add(transform);
}
return(transforms);
}
public static InterpolationTransform GetTransform_FrenetNormal(_Curve curve, float t)
{
//Position on the curve at point t
Vector3 pos = curve.GetPosition(t);
//Forward direction (tangent) on the curve at point t
Vector3 forwardDir = curve.GetTangent(t);
Vector3 secondDerivativeVec = curve.GetSecondDerivativeVec(t);
MyQuaternion orientation = InterpolationTransform.GetOrientation_FrenetNormal(forwardDir, secondDerivativeVec);
InterpolationTransform trans = new InterpolationTransform(pos, orientation);
return(trans);
}
//
// 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, Vector3 upRefStart, Vector3 upRefEnd)
{
//Position on the curve at point t
Vector3 pos = curve.GetPosition(t);
//Forward direction (tangent) on the curve at point t
Vector3 forwardDir = curve.GetTangent(t);
//Interpolate between the start and end up vector to get an up vector at a t position
Vector3 interpolatedUpDir = Vector3.Normalize(BezierLinear.GetPosition(upRefStart, upRefEnd, t));
MyQuaternion orientation = InterpolationTransform.GetOrientation_UpRef(forwardDir, interpolatedUpDir);
InterpolationTransform trans = new InterpolationTransform(pos, orientation);
return(trans);
}
public static InterpolationTransform GetTransform_UpRef(_Curve curve, float t, Vector3 upRef)
{
//Position on the curve at point t
Vector3 pos = curve.GetPosition(t);
//Forward direction (tangent) on the curve at point t
Vector3 forwardDir = curve.GetTangent(t);
//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
//This idea is not working for all possible curve orientations
//MyQuaternion orientation = new MyQuaternion(forwardDir);
//Your own reference up vector
MyQuaternion orientation = InterpolationTransform.GetOrientation_UpRef(forwardDir, upRef);
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);
}
//Not defined for a single point, you always need a previous transform
//public static InterpolationTransform InterpolationTransform GetTransform_RotationMinimisingFrame()
//{
//}
public static List <InterpolationTransform> GetTransforms_RotationMinimisingFrame(_Curve curve, List <float> tValues, Vector3 upRef)
{
List <InterpolationTransform> transforms = new List <InterpolationTransform>();
for (int i = 0; i < tValues.Count; i++)
{
float t = tValues[i];
//Position on the curve at point t
Vector3 position = curve.GetPosition(t);
//Forward direction (tangent) on the curve at point t
Vector3 tangent = curve.GetTangent(t);
//At first pos we dont have a previous transform
if (i == 0)
{
//Just use one of the other algorithms available to generate a transform at a single position
MyQuaternion orientation = InterpolationTransform.GetOrientation_UpRef(tangent, upRef);
InterpolationTransform transform = new InterpolationTransform(position, orientation);
transforms.Add(transform);
}
else
{
//To calculate the orientation for this point, we need data from the previous point on the curve
InterpolationTransform previousTransform = transforms[i - 1];
MyQuaternion orientation = InterpolationTransform.GetOrientation_RotationFrame(position, tangent, previousTransform);
InterpolationTransform transform = new InterpolationTransform(position, orientation);
transforms.Add(transform);
}
}
return(transforms);
}
//Generate a mesh
public static Mesh GenerateMesh(List <InterpolationTransform> transforms, MeshProfile profile, float profileScale)
{
if (profile == null)
{
Debug.Log("You need to assign a mesh profile");
return(null);
}
if (transforms == null || transforms.Count <= 1)
{
Debug.Log("You need more transforms");
return(null);
}
//Test that the profile is correct
//InterpolationTransform testTrans = transforms[1];
//DisplayMeshProfile(profile, testTrans, profileScale);
//Vertices
List <UnityEngine.Vector3> vertices = new List <UnityEngine.Vector3>();
//Normals
List <UnityEngine.Vector3> normals = new List <UnityEngine.Vector3>();
for (int step = 0; step < transforms.Count; step++)
{
InterpolationTransform thisTransform = transforms[step];
for (int i = 0; i < profile.vertices.Length; i++)
{
MyVector2 localPos2d = profile.vertices[i].point;
Vector3 localPos = new Vector3(localPos2d.x, localPos2d.y, 0f);
Vector3 pos = thisTransform.LocalToWorld_Pos(localPos * profileScale);
vertices.Add(pos.ToVector3());
//Normals
MyVector2 localNormal2d = profile.vertices[i].normal;
Vector3 localNormal = new Vector3(localNormal2d.x, localNormal2d.y, 0f);
Vector3 normal = thisTransform.LocalToWorld_Dir(localNormal);
normals.Add(normal.ToVector3());
}
}
//Triangles
List <int> triangles = new List <int>();
//We connect the first transform with the next transform, ignoring the last transform because it doesnt have a next
for (int step = 0; step < transforms.Count - 1; step++)
{
//The index where this profile starts in the list of all vertices in the entire mesh
int profileIndexThis = step * profile.vertices.Length;
//The index where the next profile starts
int profileIndexNext = (step + 1) * profile.vertices.Length;
//Each line has 2 points
for (int line = 0; line < profile.lineIndices.Length; line++)
{
int lineIndexA = profile.lineIndices[line].x;
int lineIndexB = profile.lineIndices[line].y;
//Now we can identify the vertex we need in the list of all vertices in the entire mesh
//The current profile
int thisA = profileIndexThis + lineIndexA;
int thisB = profileIndexThis + lineIndexB;
//The next profile
int nextA = profileIndexNext + lineIndexA;
int nextB = profileIndexNext + lineIndexB;
//Build two triangles
triangles.Add(thisA);
triangles.Add(nextA);
triangles.Add(nextB);
triangles.Add(thisB);
triangles.Add(thisA);
triangles.Add(nextB);
}
}
Mesh mesh = new Mesh();
mesh.vertices = vertices.ToArray();
mesh.triangles = triangles.ToArray();
mesh.normals = normals.ToArray();
//mesh.RecalculateNormals();
return(mesh);
}
//
// 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(Vector3 position, Vector3 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
Vector3 T1 = previousTransform.Forward;
Vector3 T2 = tangent;
//You move T1 to the new position, so A is a vector going from the new position
Vector3 A = Vector3.Cross(T1, T2);
//This is the angle between T1 and T2
float alpha = Mathf.Acos(Vector3.Dot(T1, T2) / (Vector3.Magnitude(T1) * Vector3.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);
}
