# How to define a parabola in 3d coordinate system.

1. Jul 18, 2010

### hangainlover

Currently im using a graphying application called "Autograph" and modeling a building with a dome shaped roof on top. I need to define parabolic shapes in 3d system.
But i cant do it ( my math knowledge is pretty elementary)
What would be the basic parabolic function in 3d that i can base my model on?
I have three sets of coordinates that have to be on the parabola.

2. Jul 18, 2010

### hangainlover

in the x y z system, i want this parabolic functionto go through (-75, 36, 0) (0,36, 36) (75,-36,0)

3. Jul 18, 2010

### disregardthat

If we start out with $$z=ax^2$$ in the xz-plane, we can translate and rotate it to the parabola in 3d. We can obviously get all 3-dimensional parabolas by varying a and doing rotations and translations.

The parameterization is the following: $$P(t) = [t,0,at^2]$$.

http://en.wikipedia.org/wiki/Rotation_matrix#Three_dimensions

Now rotate it around the x,y and z axis by angles v,u and w respectively. We do this by matrix multiplication, so we must multiply $$P(t)$$ by $$R_z(w)R_y(u)R_x(v)$$. After that we perform a translation by the arbitrary vector r.

So the general parabola is $$G(t) = R_z(w)R_y(u)R_x(v)P(t)+r$$. You can find this on vector-form by multiplying out the matrices.