Arc Parameterization

Since Bernstein’s or DeCasteljau’s cubic bezier curve would not give us evenly distributed points along with a curve.

For example, when BezierCurvePointAt(float t) is 0.25, it does not give us the position along with the curve at 25%!

Arc-Length Parameterization comes to rescue! to get evenly distributed point.

This lead artist to use accurate object animation along with a curve using different ease e.g. Linear, EaseIn, EaseOut and etc.

C#, Unity

Left is Bernstein's cubic bezier curve distribution and Right is set of evenly distributed points calculated with Arc Parameterization.

Get position at t which is normalized dist

public void GetPositionAtDistance(float t, float[] arcLengthSet, float curveLength, ref Vector3 position)

    // assume that t is [0~1]

    float targetLength = t * curveLength;

    int index = 0;

    int low = 0;

    int high = arcLengthSet.Length;

    while (low < high)

        //binary search on our stored lengths to find the largest length that's smaller then targetLength

        index = low + ((high - low) / 2 | 0);

        if (arcLengthSet[index] < targetLength)

            low = index + 1;

        else

            high = index;

    if (arcLengthSet[index] > targetLength && index > 0 && index < arcLengthSet.Length)

        index--;

    float lengthBefore = arcLengthSet[index];

    float k = 0f; // just return or do the interpolation and return

    if (lengthBefore == targetLength || (index + 1) >= arcLengthSet.Length) // TODO: use .AboutEqual instead?

        k = index / arcLengthSet.Length;

    else

        k = (index + (targetLength - lengthBefore) / (arcLengthSet[index + 1] - lengthBefore)) / arcLengthSet.Length;

    GetPositionAt(k, ref position);

Get arc length set

public float[] GetArcLengthSet(int resolution, int segmentID = 0)

    float[] set = new float[resolution + 1];

    float lengthSq = 0f;

    set[0] = 0f;

    Vector3 prev = CurveSegments[segmentID].P0.transform.position; //     BezierCurveMath.GetPoint(0f, CurveSegments[segmentID].P0.transform.position,     CurveSegments[segmentID].P0_Tangent.transform.position, CurveSegments[segmentID].P1_Tangent.transform.position,     CurveSegments[segmentID].P1.transform.position);

    for (int i = 1; i <= resolution; i++)

        float t = (float)i / resolution;

        Vector3 curr = BezierCurveMath.GetPoint(t,     CurveSegments[segmentID].P0.transform.position, CurveSegments[segmentID].P0_Tangent.transform.position, CurveSegments[segmentID].P1_Tangent.transform.position, CurveSegments[segmentID].P1.transform.position);             // TODO: optimize with sq dist?                     lengthSq += Vector3.Distance(prev, curr);

        prev = curr;

        set[i] = lengthSq;

    return set;