Coordinate-Free Geometric Computing

Programmers are tempted to extract the coordinates of a geometric object in intermediate stages. They frequently do so because they need to implement some function that is not available in the geometry library they use. Coordinates are necessary for input and for output, but extracting coordinates in intermediate stages is dangerous since it easily becomes the source of many bugs that are hard to find.

This chapter discusses coordinate-free geometry, or the premise that one does not need (nor wants) to manipulate coordinates in the intermediate steps of a geometric system. An input to a geometric system must, by necessity, include coordinates since these would ordinarily be read from a file. The output also must include coordinates since they are the only way to interface with the graphical subsystem. But it is possible and desirable to ensure that all intermediate computations are performed without coordinate manipulation.

The irony of calling for coordinate freedom is that before Fermat and Descartes all geometry was coordinate-free. Here is how E.T. Bell chose to describe their impact in his inimitable style:

…The idea behind it all is childishly simple, yet the method of analytic geometry is so powerful that very ordinary boys of seventeen can use it to prove results which would have baffled the greatest of the Greek geometers …[8]

This chapter tries to convince the reader that we may wish after all to do geometry as ancient geometers would have—even in geometric computing. Taking this purist approach means ignoring the existence of coordinates and stepping on higher ground while only calling underlying “oracles.” That underlying predicates would be resolved via coordinates should be left as a detail and ignored during development of noncore code.