How Do You Calculate Ray and Ellipsoid Intersections?

[bobthebanana]

So I have an array p(t) = e + td, where e is the start position, t is some parameter, and d is the direction of the ray

For a sphere with center c and radius R, the vector form equation is (p-c).(p-c)-R^2=0

This can be algebraically manipulated into:

t = (-d.(e-c) +- sqrt((d.(e-c))^2 - (d.d)((e-c).(e-c)-R^2))) / (d.d)



How can I express t for ellipsoids? I know there's an xRadius, yRadius, and a zRadius instead of radius R

 

[mathman]

The most direct way is to transform the coordinates so that the ellipsoid is centered at the origin and its three axes coincide with the coordinate axes. The equation for the ellipsoid is then (x/a)²+(y/b)²+(z/c)²=1. Then substitute the components of p(t) for x,y,and z to get the equation for t.

 

[mgb_phys]

Wouldn't that make a great name for a band "Ray Ellipsoid and the intersections"?