Linear movement over a Bézier curve

Question

Is it possible to get a constant speed movement over a Bézier curve when its points are not kept symmetrical ?

Here white square (almost) moves at constant speed. (red and green squares are control points)

Here travel time is much longer in the green part than in the red part. (control points moved)

Curve generation and white square movement is defined as follows :

    public void Update(...)
    {
        // White square movement        
        _timeline += 0.01f;
        if (_timeline >= 1.0f)
        {
            _timeline = 0.01f;
        }
        _timelineVector2 = GetBezierPoint(_timeline, _p0, _p1, _p2, _p3);
    }

    // Also used for generating curve
    public static Vector2 GetBezierPoint(float t, Vector2 p0, Vector2 p1, Vector2 p2, Vector2 p3)
    {
        float cx = 3 * (p1.X - p0.X);
        float cy = 3 * (p1.Y - p0.Y);

        float bx = 3 * (p2.X - p1.X) - cx;
        float by = 3 * (p2.Y - p1.Y) - cy;

        float ax = p3.X - p0.X - cx - bx;
        float ay = p3.Y - p0.Y - cy - @by;

        float Cube = t * t * t;
        float Square = t * t;

        float resX = (ax * Cube) + (bx * Square) + (cx * t) + p0.X;
        float resY = (ay * Cube) + (@by * Square) + (cy * t) + p0.Y;

        return new Vector2(resX, resY);
    }

Answer 1

There is a simple approximation you could apply here. Lets say the movement pace you want is 1.0 units per frame. You start with t = 0 and a stepSize = 1.0

Lets say the function getBezierPoint(t) returns a position on the bezier, now you do a binary search like approximation:

desiredStepSize = 1.0; // How much we move each frame / second ... some other time unit
acceptableError = desiredStepSize / 20.0; // How much can we deviate from the desired speed

currentT = 0.0; // Where do we start on the bezier (t = 0)
tStepSize = 0.01; // How much we guess the change in t needs to be

actualStepSize = 0; // The distance between the current position and the next one

  function compute_t_step_size_from_guess()
  // Make tStepSize big enough if needed

  actualStepSize = distance(getBezierPoint(currentT),
                              getBezierPoint(currentT + tStepSize));
  while (actualStepSize < desiredStepSize);
  {
    tStepSize *= 2.0;
    actualStepSize = distance(getBezierPoint(currentT),
                              getBezierPoint(currentT + tStepSize));
  } 


  deltaT = 0.5 * tStepSize;
  // Fine tune with binary search
  while ((actualStepSize < desiredStepSize - acceptableError 
          ||
          actualStepSize > desiredStepSize + acceptableError )
  {
    if (actualStepSize < desiredStepSize) tStepSize += deltaT;
    else tStepSize -= deltaT;

    deltaT *= 0.5;  
  }
}

This runs very fast and gets really accurate results. You can use it to compute the next position for nearly perfect linear velocity (it doesn't take the arc into account). You can vastly improve the speed by fine tuning some things, for instance, increasing the size during the growth phase by * (1 + acceptableError / desiredStepSize ), then binary searching between the the current approximation and 1 / (1 + acceptableError / desiredStepSize ).