Page 8

Page 8

Finding the IHS polyhedron of N half-spaces from a convex hull using the power of computational geometry.
Some useful definitions:

Geometric inversion
in 3D Euclidean space is a point-to-point transformation of E³ which maps a point with spherical polar coordinates (R, theta , phi ) to the point with coordinates (1/R, theta , phi ).
Convex hull.
Given a finite subset L of points of E³, the convex hull of L, is the smallest convex polyhedron containing L.

Page 8, Image 1

Step 1: find distance vectors (red) to every halfspace boundary and transform them by geometric inversion (blue).

Step 2: find convex hull (blue) of transformed points. Find distance vectors (green) to the faces of convex hull.

Step 3: geometric inversion of green vectors yields vertices (black) of the IHS polyhedron (red).

Robert Fraczkiewicz
Thu Sep 26 15:52:24 CDT 1996

Finding the IHS polyhedron of N half-spaces from a convex hull using the power of computational geometry. Some useful definitions: Geometric inversion in 3D Euclidean space is a point-to-point transformation of E³ which maps a point with spherical polar coordinates (R,, ) to the point with coordinates (1/R,,). Convex hull. Given a finite subset L of points of E³, the convex hull of L, is the smallest convex polyhedron containing L.

	Step 1: find distance vectors (red) to every halfspace boundary and transform them by geometric inversion (blue).
	Step 2: find convex hull (blue) of transformed points. Find distance vectors (green) to the faces of convex hull.
	Step 3: geometric inversion of green vectors yields vertices (black) of the IHS polyhedron (red).