/// <summary>
/// Creates a quadratic bezier path connecting two points.
/// </summary>
/// <param name="startPoint">The start of the path.</param>
/// <param name="controlPoint">The control point.</param>
/// <param name="endPoint">The end of the path.</param>
public NormalizedBezierPath(Vector3 startPoint, Vector3 controlPoint, Vector3 endPoint)
{
_segment = new CubicBezierSegment(startPoint, controlPoint, endPoint);
_coefficients = _segment.CalculateTangentCoefficients();
CalculateSamples();
}
private static float CalculateApproximateLength(this CubicBezierSegment segment)
{
// Approximate the curve using a quadratic bezier.
Vector3 controlPoint = segment.GetQuadraticApproximationControlPoint();
Vector3 v1 = controlPoint - segment.StartPoint;
Vector3 v2 = segment.StartPoint - 2 * controlPoint + segment.EndPoint;
// Calculate the exact length of the quadratic bezier by evaluating it's integral.
// http://stackoverflow.com/a/11857788/814164
if (v2 != Vector3.zero)
{
double c = 4 * Vector3.Dot(v2, v2);
double b = 8 * Vector3.Dot(v1, v2);
double a = 4 * Vector3.Dot(v1, v1);
double q = 4 * a * c - b * b;
double twoCpB = 2 * c + b;
double sumCBA = c + b + a;
double mult0 = 0.25 / c;
double mult1 = q / (8 * Math.Pow(c, 1.5));
double length =
mult0 * (twoCpB * Math.Sqrt(sumCBA) - b * Math.Sqrt(a)) +
mult1 * (Math.Log(2 * Math.Sqrt(c * sumCBA) + twoCpB) - Math.Log(2 * Math.Sqrt(c * a) + b));
return(double.IsNaN(length) ? v1.magnitude + (segment.EndPoint - controlPoint).magnitude : (float)length);
}
else
{
return(2 * v1.magnitude);
};
}
/// <summary>
/// Subdivides a bezier curve segment at a specific point.
/// </summary>
/// <param name="segment">The segment to subdivide.</param>
/// <param name="leftSegment">The segment to write the left segment data into. </param>
/// <param name="rightSegment">The segment to write the right segment data into.</param>
/// <param name="t">The time along the curve to subdivide at, with range [0, 1]</param>
public static void Subdivide(this CubicBezierSegment segment, CubicBezierSegment startSegment,
CubicBezierSegment leftSegment, double t)
{
if (t < 0 || t > 1)
{
throw new ArgumentOutOfRangeException("t");
}
float posT = (float)t;
Vector3 e = Vector3.Lerp(segment.StartPoint, segment.StartControlPoint, posT);
Vector3 f = Vector3.Lerp(segment.StartControlPoint, segment.EndControlPoint, posT);
Vector3 g = Vector3.Lerp(segment.EndControlPoint, segment.EndPoint, posT);
Vector3 h = Vector3.Lerp(e, f, posT);
Vector3 j = Vector3.Lerp(f, g, posT);
Vector3 k = Vector3.Lerp(h, j, posT);
startSegment.StartPoint = segment.StartPoint;
startSegment.StartControlPoint = e;
startSegment.EndControlPoint = h;
startSegment.EndPoint = k;
leftSegment.StartPoint = k;
leftSegment.StartControlPoint = j;
leftSegment.EndControlPoint = g;
leftSegment.EndPoint = segment.EndPoint;
}
private void CalculateSamples()
{
var workingSegment = new CubicBezierSegment();
workingSegment.StartPoint = _segment.StartPoint;
workingSegment.StartControlPoint = _segment.StartControlPoint;
workingSegment.EndControlPoint = _segment.EndControlPoint;
workingSegment.EndPoint = _segment.EndPoint;
_length = 0.0f;
// First, break the working segment down into a discrete number of segments. We only care about the approximate
// length of these segments, not the curve segments themselves.
CubicBezierSegment leftSegment = new CubicBezierSegment(), rightSegment = new CubicBezierSegment();
for (int i = 0; i < NumberOfSamples; ++i)
{
// Subdivide another part off the curve. We take 1/64 of the curve on the first loop, and keep the
// remaining 63/64 of the curve. On the next loop, we want 1/63 of the curve and so on, to keep the
// segment size consistent.
double splitT = 1 / (double)(NumberOfSamples - i);
workingSegment.Subdivide(leftSegment, rightSegment, splitT);
float nodeLength = leftSegment.CalculateLength();
_length += nodeLength;
SegmentSample sample = new SegmentSample();
sample.Length = nodeLength;
sample.StartTime = 0.0f;
sample.EndTime = 1.0f;
_samples.Add(sample);
// Copy the right curve segment into the working curve segment.
workingSegment.StartPoint = rightSegment.StartPoint;
workingSegment.StartControlPoint = rightSegment.StartControlPoint;
workingSegment.EndControlPoint = rightSegment.EndControlPoint;
workingSegment.EndPoint = rightSegment.EndPoint;
}
// Knowing what the length of the curve segments are, and a total approximate length, we can calculate the
// the start/end times of each segment. These will be the values we use for our runtime lookup.
float runningLength = _samples[0].Length;
for (int i = 1; i < _samples.Count; ++i)
{
float t = (float)runningLength / _length;
SegmentSample sample = _samples[i];
sample.StartTime = t;
_samples[i] = sample;
SegmentSample oldSample = _samples[i - 1];
oldSample.EndTime = t;
_samples[i - 1] = oldSample;
runningLength += sample.Length;
}
}
/// <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 point along the curve.
/// </summary>
/// <returns>The point.</returns>
/// <param name="segment">The segment to calculate the point of.</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 GetPoint(this CubicBezierSegment segment, double t)
{
double t2 = t * t;
double t3 = t2 * t;
double mt = 1.0 - t;
double mt2 = mt * mt;
double mt3 = mt2 * mt;
return
((float)mt3 * segment.StartPoint +
(float)(3 * mt2 * t) * segment.StartControlPoint +
(float)(3 * mt * t2) * segment.EndControlPoint +
(float)t3 * segment.EndPoint);
}
/// <summary>
/// Calculates the length of a CubicBezierSegment. Note that because this uses a numerical method to get the length,
/// it won't be completely accurate.
/// </summary>
/// <returns>The length of the bezier segment.</returns>
/// <param name="segment">The segment to calculate the length of.</param>
public static float CalculateLength(this CubicBezierSegment segment)
{
// Subdividing the segment into 4 pieces, then getting an approximate length on those 4 pieces, gives
// a fairly accurate length (< 1% error), and is very quick.
CubicBezierSegment l = new CubicBezierSegment(), r = new CubicBezierSegment(),
ll = new CubicBezierSegment(), lr = new CubicBezierSegment(), rl = new CubicBezierSegment(),
rr = new CubicBezierSegment();
segment.Subdivide(l, r, 0.5);
l.Subdivide(ll, lr, 0.5);
r.Subdivide(rl, rr, 0.5);
return(ll.CalculateApproximateLength() + lr.CalculateApproximateLength() + rl.CalculateApproximateLength() +
rr.CalculateApproximateLength());
}
/// <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>
/// 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>
/// Creates a cubic bezier path connecting two points.
/// </summary>
/// <param name="startPoint">The start of the path.</param>
/// <param name="startControlPoint">The first control point.</param>
/// <param name="startControlPoint">The second control point.</param>
/// <param name="endPoint">The end of the path.</param>
public BezierPath(Vector3 startPoint, Vector3 startControlPoint, Vector3 endControlPoint, Vector3 endPoint)
{
_segment = new CubicBezierSegment(startPoint, startControlPoint, endControlPoint, endPoint);
_length = -1f;
}
private static Vector3 GetQuadraticApproximationControlPoint(this CubicBezierSegment segment)
{
return
((3 * segment.EndControlPoint - segment.EndPoint + 3 * segment.StartControlPoint - segment.StartPoint) * 0.25f);
}
