Support Mapping of an Arbitrary Ellipsoid

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SUMMARY

The discussion focuses on the computational determination of the support mapping for an arbitrary ellipsoid, defined as the point on the geometry that yields the maximum dot product with a specified direction vector. The user has identified that the support mapping corresponds to the maximum bounded curve between the direction vector and its perpendicular hyperplane. To simplify calculations, the ellipsoid should be oriented to a "zero rotation" and centered at the origin before applying any transformations. The user seeks assistance in completing this computational task.

PREREQUISITES
  • Understanding of convex geometry principles
  • Familiarity with ellipsoid properties and transformations
  • Knowledge of vector mathematics and dot products
  • Experience with computational geometry algorithms
NEXT STEPS
  • Research algorithms for computing support functions of convex bodies
  • Explore methods for transforming and rotating ellipsoids in computational geometry
  • Learn about optimization techniques for maximizing dot products in vector spaces
  • Investigate libraries or tools for computational geometry, such as CGAL or Eigen
USEFUL FOR

This discussion is beneficial for computational geometers, mathematicians, and software developers working on 3D modeling, physics simulations, or any applications requiring the manipulation of ellipsoidal shapes.

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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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Is no one able to help me with finding the solution?
 

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