Why Is Bi Quadratic Interpolation for Image Resampling / Interpolation Rarely Done? Related question: What are the practically relevant differences between various image resampling methods?

Bilinear and bicubic interpolation for image resampling seem to be fairly common, but biquadratic is in my experience rarely heard of. To be sure, it's available in some programs and libraries, but generally it doesn't seem to be popular. This is further evidenced by there being Wikipedia articles for bilinear and bicubic interpolation, but none for biquadratic. Why is this the case?
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.