Support Mapping of an Arbitrary Ellipsoid

In summary, the support mapping of a convex geometry is any point on the geometry that results in the largest dot product to a direction vector. In order to computationally find the support mapping of an arbitrary ellipsoid, it is best to orient the ellipsoid to a "zero rotation" and center it at the origin before applying transformations and rotations. Solutions and recommendations for completing this task are welcome.
  • #1
Fruitless
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In this context, the support mapping of any convex geometry is any point on the geometry which results in the largest dot product to some direction vector.

I would appreciate some help in computationally finding the support mapping of an arbitrary ellipsoid (some arbitrary orthonormal basis and extents on each axis, centered anywhere).

So far, I have concluded that the support mapping is the maximum of the bounded curve between some direction vector and it's perpendicular hyperplane towards the "positive" side of the plane.

I am having trouble finding this point computationally. I decided that it would be best to orient the ellipsoid to a "zero rotation" and center it at the origin, and then apply the transformations and rotations to the support mapping after the calculation. Does anyone here know how to complete this problem?

Any solutions or recommendations are greatly appreciated.
 
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  • #2
Is no one able to help me with finding the solution?
 

1. What is support mapping of an arbitrary ellipsoid?

Support mapping of an arbitrary ellipsoid is a method used in computer graphics and geometry to determine the closest point on the surface of an ellipsoid to a given point in space. It is commonly used in collision detection algorithms and optimization problems.

2. How is support mapping different from other mapping techniques?

Support mapping is different from other mapping techniques because it takes into account the shape of the ellipsoid, rather than just its bounding box or sphere. This allows for more accurate calculations and can be used in a wider range of applications.

3. What information is needed for support mapping of an arbitrary ellipsoid?

To perform support mapping on an arbitrary ellipsoid, you will need the position of the ellipsoid in 3D space, its size and shape parameters, and the direction vector of the point of interest. This information can then be used to calculate the closest point on the ellipsoid's surface to the given point.

4. Can support mapping be used for any shape?

Support mapping can be used for any convex shape, including ellipsoids. However, for non-convex shapes, other techniques such as Minkowski sum or GJK algorithm may be more suitable.

5. What are the practical applications of support mapping of an arbitrary ellipsoid?

Support mapping of an arbitrary ellipsoid has many practical applications, including collision detection in video games and simulations, optimization problems in engineering and physics, and ray tracing in computer graphics. It is also used in robotics for motion planning and path optimization.

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