Constructive Synthetic Geometry

In the current problem set we are investigating two problems in computational geometry using concepts from our study of the computable real numbers and calculus. The main problem we are examining is the notion of a set of points in the plane and how to compute their convex hull. In the last lecture we briefly discussed how it is possible to investigate computational geometry in a more abstract manner, in the spirit of Euclidean geometry. In such an approach we start with an abstract type of Points. We do not need to think that these will be represented as real numbers. We can imagine them as “points in the plane” and state the properties of line segments, angles, triangles, polygons, and so forth using abstract axiomatic mathematics in the spirit of Euclid, but with more precision. This is the way algebra is usually presented as we have seen in looking about the algebraic properties of the rational and reals. Both of these types of numbers satisfy the algebraic properties of a field. We can define this concept without reference to any specific instances. We are interested in the general properties of any field.