Why Is Bi Quadratic Interpolation for Image Resampling / Interpolation Rarely Done?
You want to have an interpolation with a equal number of weights on both sides of the points you want to interpolate in between. So you choose either one or two weights on each side, resulting in an interpolation of two (linear) or four (cubic) points.

An quadratic interpolation would need three points, which would only make sense at the border of a grid, where you can choose only one point on one side and two on the other.