# How to find a Bezier curve without control points?

Let's say someone created a cubic Bezier curve using software and rasterised it. However, the original equation of the Bezier curve was not noted.

Since we have the image of the Bezier curve, we can know the location of every point on the curve. What can I do if I want to know the coordinates of its control points and hence its equation as accurately as possible?

The StackOverflow page below suggests that we plug in random (and appropriate) t-values for some points, but I feel that this would not be accurate and depending on what t-value we place it in (and which point we sample), we will get a different equation which might not be nearly the same as the original equation.

Or is it better that we use any other method which is not too complex instead of Bezier curves to reconstruct the parametric curves?

Note: I have read these pages but I feel that the question I ask here is slightly different.

Reconstruct Control points in a Bézier Curve?

and

from StackOverflow

There are two problems with the data:

• You only have pixels and not points.

• You don't have $t$ values for the points.

Here are natural but somewhat arbitrary solutions for these problems:

• Choose the center of the pixel as the point.

• Order the points and choose $t$ proportional to the position of the point in this order.

Now use least squares to find the Bézier curve by solving $$\min \sum_{i=0}^n \left|\gamma\left(\frac i n\right)-p_i\right|^2$$