/// <summary>
/// Calculates the tangent coefficents of a segment. This is used in tangent and acceleration calculations.
/// </summary>
/// <returns>The tangent coefficients.</returns>
/// <param name="segment">The segment to calculate the coefficients of.</param>
public static CubicBezierTangentCoefficients CalculateTangentCoefficients(this CubicBezierSegment segment)
{
var coeffs = new CubicBezierTangentCoefficients();
coeffs.Coeff1 = -3f * segment.StartPoint + 9f * segment.StartControlPoint - 9 * segment.EndControlPoint + 3f * segment.EndPoint;
coeffs.Coeff2 = 6f * segment.StartPoint - 12f * segment.StartControlPoint + 6 * segment.EndControlPoint;
coeffs.Coeff3 = -3f * segment.StartPoint + 3f * segment.StartControlPoint;
return(coeffs);
}
/// <summary>
/// Gets a normal for a given segment. Note that there are an infinite number of possible normals, this method just
/// produces a locally consistent normal.
/// </summary>
/// <returns>A normal at a given point.</returns>
/// <param name="segment">The segment to calculate the tangent of.</param>
/// <param name="segment">The precalculated coefficients of the curve.</param>
/// <param name="t">
/// The time along the curve, where t=0 is the start point, and t=1 is the end point. Note that t can be extrapolated
/// beyond these bounds.
/// </param>
/// <param name="up">
/// The direction the normal should try to match.
/// </param>
public static Vector3 GetNormal(this CubicBezierSegment segment, CubicBezierTangentCoefficients coeffs, double t, Vector3 up)
{
Vector3 baseNormal = segment.GetNormal(coeffs, t);
Vector3 tangent = coeffs.GetTangent(t);
Vector3 upNormal = Vector3.Cross(Vector3.Cross(tangent, up), tangent).normalized;
if (Vector3.Angle(tangent, up) == 0f)
{
return(baseNormal);
}
float angle = Mathf.Min(Mathf.Abs(Vector3.Angle(tangent, up)), Mathf.Abs(Vector3.Angle(tangent, -up)));
float lerp = Mathf.Clamp01((angle + 5) / 45f);
return(AngleLerp(baseNormal, upNormal, lerp));
}
/// <summary>
/// Finds the points of inflection of the curve.
/// </summary>
/// <returns>The points of inflection times.</returns>
/// <param name="segment">Segment.</param>
public static double[] GetPointsOfInflectionTimes(this CubicBezierTangentCoefficients coeffs)
{
// Solve the x,y and z axis separately.
IEnumerable <double> xRoots = GetCubicExtremities(
coeffs.Coeff1.x, coeffs.Coeff2.x, coeffs.Coeff3.x);
IEnumerable <double> yRoots = GetCubicExtremities(
coeffs.Coeff1.y, coeffs.Coeff2.y, coeffs.Coeff3.y);
IEnumerable <double> zRoots = GetCubicExtremities(
coeffs.Coeff1.z, coeffs.Coeff2.z, coeffs.Coeff3.z);
// Join them together, and resort.
return(xRoots
.Union(yRoots)
.Union(zRoots)
.Where(t => 0.0 <= t && t <= 1.0) // In Range of bezier.
.OrderBy(root => root)
.ToArray());
}
/// <summary>
/// Gets the tangent at a point along the curve.
/// </summary>
/// <returns>The tangent at the given time.</returns>
/// <param name="coeffs">The precalculated coefficients of the curve.</param>
/// <param name="t">
/// The time along the curve, where t=0 is the start point, and t=1 is the end point. Note that t can be extrapolated
/// beyond these bounds.
/// </param>
public static Vector3 GetTangent(this CubicBezierTangentCoefficients coeffs, double t)
{
Vector3 velocity = (float)(t * t) * coeffs.Coeff1 + ((float)t) * coeffs.Coeff2 + coeffs.Coeff3;
// If the velocity is 0, we look at the acceleration of the path to determine which direction the path is about
// to travel to.
if (velocity.magnitude < float.Epsilon)
{
Vector3 acceleration = coeffs.GetAcceleration(t);
// If acceleration is 0, the entire path is probably a single point.
if (acceleration.magnitude < float.Epsilon)
{
if (coeffs.Coeff1.magnitude < float.Epsilon)
{
return(Vector3.forward);
}
return(coeffs.Coeff1.normalized);
}
return(acceleration.normalized);
}
return(velocity.normalized);
}
/// <summary>
/// Gets a normal for a given segment. Note that there are an infinite number of possible normals, this method just
/// produces a locally consistent normal.
/// </summary>
/// <returns>A normal at a given point.</returns>
/// <param name="segment">The segment to calculate the tangent of.</param>
/// <param name="segment">The precalculated coefficients of the curve.</param>
/// <param name="t">
/// The time along the curve, where t=0 is the start point, and t=1 is the end point. Note that t can be extrapolated
/// beyond these bounds.
/// </param>
public static Vector3 GetNormal(this CubicBezierSegment segment, CubicBezierTangentCoefficients coeffs, double t)
{
Vector3 currentTangent = coeffs.GetTangent(t);
Vector3 normal;
if (currentTangent == Vector3.zero)
{
Vector3 acceleration = coeffs.GetAcceleration(t);
currentTangent = acceleration;
}
// The tangent doesn't change, (such in the case of a straight line).
// We pick a vector which shouldn't be parallel to the tangent, and
// use the cross product with the tangent to find a normal.
normal = currentTangent;
normal.x = -currentTangent.y;
normal.y = -currentTangent.z;
normal.z = currentTangent.x;
normal = Vector3.Cross(currentTangent, normal).normalized;
return(normal);
}
/// <summary>
/// Get the acceleration at a point along the curve.
/// </summary>
/// <returns>The acceleration at a given time.</returns>
/// <param name="coeffs">The precalculated coefficients of the curve.</param>
/// <param name="t">
/// The time along the curve, where t=0 is the start point, and t=1 is the end point. Note that t can be extrapolated
/// beyond these bounds.
/// </param>
public static Vector3 GetAcceleration(this CubicBezierTangentCoefficients coeffs, double t)
{
return(((float)t) * coeffs.Coeff1 * 2 + coeffs.Coeff2);
}
/// <summary>
/// Gets the velocity at a point of the curve.
/// </summary>
/// <returns>The tangent at the given time.</returns>
/// <param name="coeffs">The precalculated coefficients of the curve.</param>
/// <param name="t">
/// The time along the curve, where t=0 is the start point, and t=1 is the end point. Note that t can be extrapolated
/// beyond these bounds.
/// </param>
public static Vector3 GetVelocity(this CubicBezierTangentCoefficients coeffs, double t)
{
return((float)(t * t) * coeffs.Coeff1 + ((float)t) * coeffs.Coeff2 + coeffs.Coeff3);
}
