Steve Sigur's Webpages

Building complicated graphs from simple objects such as lines and circles.

The method of oriented lines gives a wonderful way of building more complicated graphs with special properties. It is particularly useful for cubic curves, which have an intimate connection with straight lines anyway. The exposition here begins by first speaking of the power of thinking algebraically about graphs. Second conics are analyzed. And third some cubics are constructed. Perhaphs (not so) finally, circular cubics, and quartics.

Here is a cubic where the relevance of the signs can be easily seen. The regions occupied by the graph are those whose signs multiply to a negative.

This cubic is the product of the three orinted lines shown above. Their product in this case is -1. Several features of this cubic can be easily seen. First is that the cubic only occupies certain signed regions, those whose product is negative. Second is that the lines form the three asymptotes of this cubic. The reason for this is the same reason that for the hyperbola xy = 1, the lines x =0 and y = 0 are the two asymptotes. Each of the line functions in L1 · L1 · L3 can get very close to zero but can never equal zero. Hence the cubic can get close to the lines L1, L2, L3 but never be on them.