Notes On Plane Curves and Proofs

By Xah Lee. Date:

[some random, personal notes on doing plane curves]

I worked on my plane curves site a lot in the past week, mostly creating new GeoGebra files that construct curves, and in general adding links to Wikipedia and filling out missing proofs. But one spectacular construction i made is the deltoid: Deltoid Caustics. This one involves using the technique of using the “Sequence[]” in GeoGebra, and the whole construction for that file is somewhat complex in part due to constructing parallel ray caustics. For example, i had to use this input for constructing a family of refracted rays: “refractionRays=Sequence[Ray[((2 cos(t)) + cos(2 t), (2 sin(t)) - sin(2 t)), Mirror[Mirror[Intersect[bb,Perpendicular[((2 cos(t)) + cos(2 t), (2 sin(t)) - sin(2 t)),bb]],Line[(2 cos(t)*3/2, 2 sin(t)*3/2), ((2 cos(t)) + cos(2 t), (2 sin(t)) - sin(2 t))]],((2 cos(t)) + cos(2 t), (2 sin(t)) - sin(2 t))] ], t,.01,2 π+.01,2π/(3*14)]”.

The difficulty in constructing the parallel ray caustics of deltoid in GeoGebra is partly because GeoGebra does not support constructing family of lines based on a moving object. Geometer's Sketchpad and Cabri Geometry both do.

When i created my visual dict of special plane curves (most work done in 1993 to 1997), I actually have not seen the proofs of many of the curve's facts. In the mid 1990s, i mostly just read the properties and wrote programs to visualize them. As long as the graphics match the theorem, i'm satisfied at the time, both for my program's correctness and the theorem's validity. I do love reading and doing proofs, but writing programs to visualize them and visually verifying them gives me more pleasure, and in many cases is easier to do than proofs for a lot of the properties. Also, there are a lot programs to be written, learning the programing language, and a lot properties to learn and understand. All these took time. So, for a lot properties or theorem of curves, i actually have never seen their proofs, but merely explained the facts with visual demonstration. I'm doing them gradually now.

Part of the difficulty in doing proofs is that, many of curve's properties, involves elementary differential geometry and algebraic geometry. At the time i have not studied them. For example, the caustic of deltoid, by parallel rays in any direction, is a astroid. To prove this theorem needs differential geometry. In particular, the theories of envelope of curves. (not that it necessarily require the knowledge diff geo for the proof of deltoid's caustics. However, it is the standard requirement to systematically tackle such problems.) There are quite a lot curves with properties involving envelope of lines and circles.

Conics sections are algebraic curves of degree 2. Five points determines a conics. Pascal's theorem on conics. Conics represented by matrix. To do proofs of these involves basic, but solid understanding of algebraic geometry, projective geometry, linear algebra. Many of these properties belongs to studies that are rather esoteric in today's math curriculum. For example, envelope of curves, projective geometry, are not taught, or just barely touched on as part of other courses. Also, the systematic study of the conversion of polynomial equations and parametric formulas, falls under Theory Of Equations, a subject that is supplanted by abstract algebra today. Many of the curve's properties involve mechanical linkages. Theory of linkages are esoteric and not taught today as such. Many curve's properties also involve Greek constructions. As such, it is not a subject normally taught as is. It involves good understanding and application of theory of fields.