Ray Ellipsoid intersections?

In summary, the conversation discusses the vector form equation for a sphere with center c and radius R, and how it can be manipulated to find the parameter t. For ellipsoids, the coordinates can be transformed to center the ellipsoid at the origin and find the equation for t using the components of p(t). This could also be a potential band name, "Ray Ellipsoid and the intersections."
  • #1
bobthebanana
23
0
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
 
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  • #2
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.
 
  • #3
Wouldn't that make a great name for a band "Ray Ellipsoid and the intersections"?
 

1. What is a Ray Ellipsoid intersection?

A Ray Ellipsoid intersection is a mathematical concept used in physics and computer graphics to determine the point where a ray (a straight line with a specific direction) intersects with an ellipsoid (a three-dimensional shape similar to a sphere but with different radii in each direction).

2. How is a Ray Ellipsoid intersection calculated?

The calculation for a Ray Ellipsoid intersection involves finding the point where the ray equation (which describes the path of the ray) and the ellipsoid equation (which describes the shape of the ellipsoid) intersect. This is typically done using algebraic methods or numerical approximation techniques.

3. What is the significance of Ray Ellipsoid intersections?

Ray Ellipsoid intersections are important in fields such as optics, computer graphics, and geodesy. They allow us to determine the path of light rays through lenses and other optical devices, create realistic 3D graphics, and accurately model the shape of the Earth.

4. Can a ray intersect with an ellipsoid more than once?

Yes, depending on the direction and position of the ray and the size and shape of the ellipsoid, it is possible for a ray to intersect with an ellipsoid multiple times. In some cases, the ray may even intersect with the ellipsoid infinitely.

5. Are there any real-world applications of Ray Ellipsoid intersections?

Yes, Ray Ellipsoid intersections have numerous real-world applications, such as satellite imaging, navigation systems, and terrain mapping. They are also used in scientific research to study the behavior of light and to model the shape of celestial bodies.

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