Projective Geometry and Collisions

A few years ago a series of papers by Frank Coffman Bell was published in this Journal suggesting how collision avoidance manoeuvres could be analysed using projective geometry. However it is probably true that most non-mathematicians have little experience of any type of geometry other than the Euclidean metric geometry which is taught in schools. Euclid's approach was to elaborate a science of the measurement of physical space and for this purpose he deduced a number of geometrical theorems starting from the intuitional ideas of point, line, distance and length. To Euclid and his followers the notion of distance was completely obvious and basic and underlay everything in geometrical science, hence the terminology that is given to this work ‘metric geometry’. Another concept of geometry was however put forward by a group of geometers of whom Pappus is one of the best known. His work shows that he was interested in theorems concerned not with distance but with such things as concurrence of lines and collinearity of points. This type of theorem may be thought of as the projective type. If for example it is assumed that any two lines in a plane meet in a point, then two parallel lines must also meet in a point which may be termed a point at infinity. In projective geometry no distinction is then made between ordinary points and points at infinity.