Ellipsoid Equation for Object Motion?

[jclar1701d]

Homework Statement

Hey, I'm doing some physics programming for a game, and could use some general help getting a formula. I'm not great with mathematics/physics, but I know enough to comprehend any feedback. Any help is greatly appreciated!

So I have an object free-floating in 3D space at velocity 'v' (an (x,y,z) vector). Then, a nearby point is selected, 'P' ((x,y,z) point).
I need to calculate an ellipsoid using 'v' as a tangent vector, and 'P' as the center point. The object will need to move along the surface of the ellipsoid. Initially, it will move in an elliptical motion around 'P', but input will allow the object to change direction along the full ellipsoid's surface.

I guess my first question is: do I need any other information to calculate the ellipsoid?

Homework Equations

To give an idea, at the end of the day, I need to have a single (x,y,z) vector to act as my object's new velocity at every new moment. My new equation needs to provide this.

The Attempt at a Solution

I'm not really even sure where to start. It's been a while since I've dealt with math/physics of this level, so any help is appreciated.

 

[haruspex]

An ellipsoid has 8 degrees of freedom: 3 for the coordinates of the centre, two for the orientation and 3 more for the shape. P nails the first 3, but the tangent requirement only fixes one more (I think). So yes, you need a fair bit more information.