Subdivide Bezier Curves

Question

I want to achieve that I can subdivide a Bezier Curve with a given distance. Now it works if the Bezier is a straight line, but if I change a Controll-Point(B&C) so the Bezier get's curved, the gap between the calculated Points are no more like the given Distance!

I've look through the web, but didn't crash in to a similar Problem.

float t = Distance between subdividedParts / bezier length;

//A,B,C,D = ControllPoints of Bezier

GetPoint(A,B,C,D,t);

---

//GetPoint equation:
public static Vector3 GetPoint (Vector3 p0, Vector3 p1, Vector3 p2, Vector3 p3, float t) {
         t = Mathf.Clamp01(t);
         float OneMinusT = 1f - t;
         return
             OneMinusT * OneMinusT * OneMinusT * p0 +
             3f * OneMinusT * OneMinusT * t * p1 +
             3f * OneMinusT * t * t * p2 +
             t * t * t * p3;
     }

Answer 1

I've managed now to get quite accurate way to split the Bezier and get the Positions => but it's performance consumption is increasing as more accurate it get's. So this could be improved in this code:

 //if accuracy is 0.001 = good performance | if 0.000001 laggy performance
    public Vector3[] GetPoints (float gap,float accuracy){
     SimpsonVec sv = SV_Setup(0);
     Vector3 last_spawn = Bezier.GetPoint(sv.A,sv.B,sv.C,sv.D,0);

     List<Vector3> allPoints = new List<Vector3>();
     allPoints.Add(last_spawn);

     for(float t = accuracy;t <= 1.0f; t +=accuracy){
         Vector3 trial = Bezier.GetPoint(sv.A,sv.B,sv.C,sv.D,t);
         if(Vector3.Distance(trial,last_spawn) >= gap){
             last_spawn = trial;
             allPoints.Add(trial);
         }
     }
     return allPoints.ToArray();
 }

for more performance I did now this:

Code for Vector3[] Array Output

public Vector3[] GetAllPoints(float gap,float acc){

    SimpsonVector = SV_SETUP_ALL();
    BezierPoints bp = new BezierPoints();
    bp.bp_vector3 = new List<Vector3>();
    bp.bp_lastSpawn = new List<Vector3>();

    for(int i = 0; i<points.Length / 3;i++){

        Vector3 ls = new Vector3();
        if(i == 0){
            ls = Bezier.GetPoint(SimpsonVector[0].A,SimpsonVector[0].B,SimpsonVector[0].C,SimpsonVector[0].D,0);
        }if (i > 0){
            ls = bp.bp_lastSpawn[i-1];
        }
        BezierPoints bp_temp = GetSegmentPoints(gap,acc,i,ls);
        bp.bp_lastSpawn.Add(bp_temp.bp_lastSpawn[0]);
        bp.bp_vector3.AddRange(bp_temp.bp_vector3);
        SimpsonVector_TEMP = SimpsonVector;
    }


    return bp.bp_vector3.ToArray();
}

BezierPoints GetSegmentPoints (float gap,float acc,int index, Vector3 ls)
{
    SimpsonVec sv = SimpsonVector[index];
    Vector3 last_spawn = ls;

    BezierPoints bp = new BezierPoints();
    bp.bp_vector3 = new List<Vector3>();
    bp.bp_lastSpawn = new List<Vector3>();

    float step = 0.1f;
    float t = step;
    float lastT = new float();

    while (t >= 0 && t <= 1f)
    {
        while (t < 1f && Vector3.Distance(Bezier.GetPoint(sv.A,sv.B,sv.C,sv.D,t), last_spawn) < gap){
            t += step;}
        step /= acc;
        while (t > lastT && Vector3.Distance(Bezier.GetPoint(sv.A,sv.B,sv.C,sv.D,t), last_spawn) > gap){
            t -= step;}
        step /= acc;
        if (t > 1f || t < lastT){
            break;}
        if(step < 0.000001f){
            last_spawn = Bezier.GetPoint(sv.A,sv.B,sv.C,sv.D,t);
            bp.bp_vector3.Add(last_spawn + transform.position);
            lastT = t;
            step = 0.1f;
        }
    }
    bp.bp_lastSpawn.Add(last_spawn);
    return bp;
}

Structs:

public struct SimpsonVec{
    [SerializeField] public Vector3 A;
    [SerializeField] public Vector3 B;
    [SerializeField] public Vector3 C;
    [SerializeField] public Vector3 D;
}

public struct  BezierPoints
{
    [SerializeField] public List<Vector3> bp_vector3;
    [SerializeField] public List<Vector3> bp_lastSpawn; 
}

Helper Methods:

public SimpsonVec SV_Setup(int index){
     SimpsonVec sv;
     sv.A = points[index];
     sv.B = points[index+1];
     sv.C = points[index+2];
     sv.D = points[index+3];
     return sv;

 }
 public SimpsonVec[] SV_SETUP_ALL(){
     SimpsonVec[] sv = new SimpsonVec[points.Length / 3];
     for(int i = 0; i<points.Length / 3;i++){
         sv[i] = SV_Setup(i*3);
     }
     return sv;
 }
 public Vector3 GetPoint (Vector3 p0, Vector3 p1, Vector3 p2, Vector3 p3, float t) {
     t = Mathf.Clamp01(t);
     float OneMinusT = 1f - t;
     return
         OneMinusT * OneMinusT * OneMinusT * p0 +
         3f * OneMinusT * OneMinusT * t * p1 +
         3f * OneMinusT * t * t * p2 +
         t * t * t * p3;
 }

Answer 2

You need to draw your curve already anyway, so keep the curve's lookup table and either precalculate the distances at each entry, or binary-search your way to victory by picking a t value, computing the distance, and then moving half that t value up/down if you're off, and repeating that until you've hit the desired precision. And because binary searching is crazy efficient, you get there in a negligible number of tries.

See https://pomax.github.io/bezierinfo/#tracing for a rationale, https://pomax.github.io/bezierinfo/#arclength for computing the length of a curve (with https://pomax.github.io/bezierinfo/#splitting as obvious section on getting the value you need to determine the length of a curve at some point t), and "all of https://pomax.github.io/bezierinfo for more information", really.