Exact primitives for smallest enclosing ellipses

Abstract

The problem of finding the unique closed ellipsoid having the smallest volume enclosing an n-point set P in d-space is an instance of convex programming and can be solved by general methods in time O(n) if the dimension is fixed. Primitive operations that deal with subproblems of constant size encapsulate the problem-specific parts of these methods. The explicit formula for the primitive of operations of Welzl's randomized method in dimension d = 2 is derived. These formulas are simpler and faster to evaluate and only contain rational expressions that allow exact solutions.