Sample thesis 2 - Natural Computing Group, LIACS, Leiden University

Figure 3: Without the need for manywords this image makes clear that thesmallest enclosing circle of a set ofpoints is uniquely determined [7].Uniqueness of the Smallest Enclosing CircleWe already kind of implicitly stated that the smallest enclosing circle of a set of points is uniquelydetermined, but we do so now explicitly in the context of Welzl's Algorithm. Figure 3, which wasborrowed from [7], illustrates the concept very well without the need for many words.We provide a proof by contradiction: Suppose that the smallest enclosing circle is not uniquelydetermined and that we have a collection of points which has distinct smallest enclosing circles D 1and D 2 (by definition of equal radius), then all of the points are contained within the area where D 1and D 2 overlap. However, as the green line in Figure 3 indicates, we can draw a circle of smallerradius through the points where D 1 and D 2 intersect. This contradicts the fact that D 1 and D 2 are thesmallest enclosing circles and we can therefore conclude that our assumption was wrong and thatthe smallest enclosing circle must be uniquely determined.A Useful LemmaAnother piece of information that we will rehash from the O(n 4 ) smallest enclosing circle algorithm,now comes in the form of a technical lemma that states: Given a set of points P, there is a subset Sof P with |S|