# Finding the equation of a parabola

1. Feb 7, 2008

### 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!

2. Feb 7, 2008

### EnumaElish

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

3. Feb 7, 2008

### 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. :)

4. Feb 7, 2008

### 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.