Finding the equation of a parabola in 3d space

[Infil]

Hi everyone,

I have three points in 3D space, and I would like to find the equation of a parabola that goes between them. My final goal is to sample about 20-25 points that lie on the parabola between these three points (ie, the user of my program will provide 3 points, then I will draw a "dotted line" version of the parabola between them through more discretized sampling).

I know how to find the parabola that goes through 3 points in 2D space, and I know how to find the equation of the unique plane that runs through these 3 points.

I just don't know how to connect the two pieces (or if there is an easier way to accomplish my above goal).

Any help would be greatly appreciated!

 

[EnumaElish]

The only 3D "parabola" in the 3D space that goes through 3 points is a plane.

 

[Infil]

Right, the parabola must lie on the plane defined by those 3 points.

I guess what I'm really asking for is a way to transform between the 3D space and the local coordinate system of a plane Ax + By + Cz + D = 0. This way, I can transform the three 3D points to a local 2D coordinate system, solve my problem there, and then transform any point on the plane back to 3D.

This sounds like it should be easy, but I'm drawing a blank. :)

 

[EnumaElish]

You need to:
1. project the 3 points onto the "xy-plane"
2. fit the polynomial on the xy-plane
3. project the polynomial from the xy-plane to the (Ax + By + Cz + D = 0)-plane.