Exact primitives for smallest enclosing ellipses
Bernd Gärtner, Sven Schönherr · 1997
The problem of finding the unique closed ellipsoid of smallest volume enclosing an n-point set P in d-space (known as the Loewner-John ellipsoid of P) is an instance of convex programming and can be solved by general methods in time O(n) if the dimension is fixed. The problem-specific parts of these methods are encapsulated in primitive operations that deal with subproblems of constant size. We derive explicit formulae for the primitive operations of Welzl's randomized method in dimension d=2. Compared to previous ones, these formulae are simpler and faster to evaluate, and they only contain rational expressions, allowing for an exact solution.