Consider a real plane curve described parametrically, and for simplicity let's assume that it is convex. For any fixed point on the affine plane, it is always possible to apply a euclidean transformation so that the image of the curve will pass through the given point. Intuitively, there will be $2$ degrees of freedom in the choice of transformation (the point on the curve that is mapped on the fixed point and the angle of rotation).
Similarly if we want to transform the curve to pass through two points, this will be possible (if the points are sufficiently close) and there is in general $1$ degree of freedom in the choice (the point on the curve mapped on the first point). If instead we are given three points, there are presumably in general a finite number of possible transformations (unless the curve is a circle or a line).
Is there an efficient way to find such transformations (other than brute-force or solving a system of polynomial equations)?
What happens if the curve is (plane) projective and we are allowed to use projective transformations? I would think a transformation exists in general for up to five fixed points. Is there a good way to find such a transformation?
Thanks,