//3d //Same math as in 2d case public static MyVector3 GetClosestPointOnLine(Edge3 e, MyVector3 p, bool withinSegment) { MyVector3 a = e.p1; MyVector3 b = e.p2; //Assume the line goes from a to b MyVector3 ab = b - a; //Vector from start of the line to the point outside of line MyVector3 ap = p - a; //The normalized "distance" from a to the closest point, so [0,1] if we are within the line segment float distance = MyVector3.Dot(ap, ab) / MyVector3.SqrMagnitude(ab); ///This point may not be on the line segment, if so return one of the end points float epsilon = MathUtility.EPSILON; if (withinSegment && distance < 0f - epsilon) { return(a); } else if (withinSegment && distance > 1f + epsilon) { return(b); } else { //This works because a_b is not normalized and distance is [0,1] if distance is within ab return(a + ab * distance); } }
//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); } }
//3d private static MyVector3 GetIntersectionCoordinate(Plane3 plane, Ray3 ray) { float denominator = MyVector3.Dot(-plane.normal, ray.dir); MyVector3 vecBetween = plane.pos - ray.origin; float t = MyVector3.Dot(vecBetween, -plane.normal) / denominator; MyVector3 intersectionPoint = ray.origin + ray.dir * t; return(intersectionPoint); }
// // 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); }
//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); }
// // Point-plane relations // //https://gamedevelopment.tutsplus.com/tutorials/understanding-sutherland-hodgman-clipping-for-physics-engines--gamedev-11917 //Notice that the plane normal doesnt have to be normalized //The signed distance from a point to a plane //- Positive distance denotes that the point p is on the front side of the plane (in the direction of the plane normal) //- Negative means it's on the back side //3d public static float GetSignedDistanceFromPointToPlane(Plane3 plane, MyVector3 pointPos) { float distance = MyVector3.Dot(plane.normal, pointPos - plane.pos); return(distance); }