Bezier Length Estimation

Bezier.cs

using UnityEngine;
using Unity.Mathematics;

using static Unity.Mathematics.math;
[ExecuteInEditMode]
public class Bezier : MonoBehaviour {

    public Vector3 a;
    public Vector3 b;
    public Vector3 c;

    
    public Vector3 Evaluate(float t) {
        Vector3 b = this.b - a;
        Vector3 c = this.c - a;
        return Vector3.Lerp(b*t,
                            Vector3.Lerp(b, c, t),
                            t) + a;
    }

    public void OnDrawGizmos() {
        const float DENSITY = 100f;
        Vector3 p0, p1;
        for (int i = 1; i < DENSITY; i++) {
            p0 = Evaluate(i / DENSITY);
            p1 = Evaluate((i - 1f) / DENSITY);
            Gizmos.DrawLine(p0, p1);
        }
    }
    
    public float ComputeLengthAnalytic(float3 p0, float3 p1, float3 p2) {
        float3 v0 = p0 - 2f*p1 + p2;
        float3 v1 = 2f*p1 - 2f*p0;
        
        float A = 4f*lengthsq(v0);
        float B = 4f*dot(v0, v1);
        float C = lengthsq(v1);
        
        float sqrtABC = 2f*sqrt(A + B + C);
        float A2 = sqrt(A);
        float A32 = 2f*A*A2;
        float C2 = 2f*sqrt(C);
        float BA = B / A2;

        const float EPSILON = 1e-4f;
        bool isLinear = A2 <= EPSILON || 
                        C2 <= EPSILON || 
                        BA + C2 <= EPSILON;
        if (isLinear) {
            return length(p0 - p1) + length(p2 - p1);
        }
        else {
            return (A32 * sqrtABC +
                    A2 * B * (sqrtABC - C2) +
                    (4f * C * A - B * B) * log((2f*A2 + BA + sqrtABC) / (BA + C2))) / 
                   (4f*A32);    
        }
    }

    public float ComputeLengthDiscrete() {
        float discreteLength = 0f;
        const float DENSITY = 200f;
        for (int i = 1; i <= DENSITY; i++) {
            discreteLength += (Evaluate(i / DENSITY) - Evaluate((i - 1f) / DENSITY)).magnitude;
        }
        return discreteLength;
    }

    public void Update() {
        Debug.Log($"analytic length: {ComputeLengthAnalytic(a, b, c)} discreteLength: {ComputeLengthDiscrete()}");
    }
}

BezierEditor.cs

using UnityEngine;
using UnityEditor;

[CustomEditor(typeof(Bezier)), CanEditMultipleObjects]
public class BezierEditor : Editor  {
    protected virtual void OnSceneGUI() {
        Bezier bezier = (Bezier)target;
        
        EditorGUI.BeginChangeCheck();
        bezier.a = Handles.PositionHandle(bezier.a, Quaternion.identity);
        bezier.b = Handles.PositionHandle(bezier.b, Quaternion.identity);
        bezier.c = Handles.PositionHandle(bezier.c, Quaternion.identity);
        if (EditorGUI.EndChangeCheck()) {
            Undo.RecordObject(bezier, "Moved Bezier Handle");
            bezier.Update();
        }
    }
}