Revival of a classical topic in differential geometry: the exploration of envelopes in a computerized environment

Abstract

Learning mathematics in a technology-rich environment enables us to revive classical topics which have been removed from the curriculum a long time ago. Both theoretical issues and applications can be studied with an experimental process. We present how envelopes of 1-parameter families of plane curves and some of their applications can be presented early in the curriculum either for pre-service teachers or for in-service teachers. This approach may be useful for students in an engineering curriculum. Working with technology yields important effects, such as reviving classical topics, broadening perspectives on already known topics, and enhancing the learner's experimental skills, where conversion between various registers of representation is an important issue.

Keywords:

• differential geometry
• Computer Algebra System
• registers of representation
• slider bar
• experimental

Acknowledgements

The authors wish to thank the referee for his/her very important remarks and hints.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Duke Physicists See The Cosmos In A Coffee Cup, retrieved on July 16 [12] from http://today.duke.edu/2009/04/caustics.html

2. When a rational parametrization is given for the curve, it is easy to transform it into a polynomial one. The case of a non-rational parametrization may be harder, but not always. For example, when sin t and cos t may be replaced by rational expression. That is what we did here.

3. Actually the software works over the field of rational numbers. The algorithms at work here are based on computations of Gröbner bases.

4. For similar families of circles, for example with radius 1, or with a non-rational parametrization of the ellipse, the equations obtained may be more complicated and involve some special features of the software. See note 2.

5. A nice example is as follows: take a circle centred at the origin with the given radius R. Parametrize the circle according to the angle t, i.e. M(t) = (x(t), y(t)) = (Rcos t, Rsin t) M(t) = (x(t), y(t)) = (Rcos t, Rsin t). Consider the family of lines passing through the points M(t) and M(3t). This family has an envelope and this envelope is a nephroid. A GeoGebra animation is available at http://www.geogebra.org/material/show/id/90990

6. We chose length 1 for the sake of simplicity. The same computations may be performed for any length l; this introduces an extra parameter into the equations, whose role is different from the role of t. Prudence is needed when using afterwards the computations of Gröbner bases with elimination order.

7. Of course, this is not standard taxonomy. The reason is not to use the same word as for the registers of representation of a mathematical object (algebraic, numerical, etc.). Here both the parametric and the implicit presentation belong to the algebraic register of representation.