I am trying to create a set of parabolas going through some points. My need is to know if in a plane (2D) three points determine uniquely an analytical curve (parabola, circle, hyperbola etc.). My Analytical Geometry knowledge is kind of rusty.
You need 3 points and two angles to define the parabola as well as RHO. So, 3 points is correct, if you are allowed to specify the angles and RHO seperately. Two points are required to define a circle and 3 points are required to define an arc.