The nephroid curve (see inside the coffee cup on [url=http://www.geom.uiuc.edu/~fjw/calc-init/nephroid/]Stuart Levy's photo[/url] above) is a sixth order algebraic curve defined by the envelope of a set of mirrored light rays as a family of curves. Unfortunately, computationally it is rather complex to solve the corresponding equation system, thus it is inconvenient to use the Envelope command with the recent version of GeoGebra.[br][br]From the optical point of view, there are two approaches. One possibility is to assume that the source of the light is a point. In this case the rays will be concurrent. The other possibility is to assume that the source is infinitely distant, in this case the rays will be parallel. Clearly, the second case is the mathematical "limit" of the first one since if the point converges to infinity, the models will be closer and closer to each other.[br][br]The first approach is computationally easier. In the following figure we can investigate the model of the concurrent rays by using the Java desktop version of GeoGebra.

GeoGebra applets in GeoGebraBooks use Giac, but newest versions of GeoGebra can be configured to use faster methods than Giac has. In the following video we can learn how the Java desktop version can be started to use or not use the external computation machine SingularWS with the embedded Gröbner cover algorithm.

Finally, the approach of the parallel rays is computationally the most difficult one. It is impossible for Giac to compute the envelope equation in a reasonable time, thus we have to force using SingularWS and the Gröbner cover method by using the command line:

As we can see in the above videos, in the parallel case there are two extra components which can be separated by factorization. But also in the concurrent case when the source of the light is a perimeter point of the circle there is an extra component.

As [url=http://www.phikwadraat.nl/huygens_cusp_of_tea/]Sander Wildeman[/url] remarks, [i]once you have written an article about caustics you start to see them everywhere[/i]. His illustration is mathematics on the bottom of an empty water bucket caused by a light bulb on the ceiling. We can only agree: mathematics is everywhere, not only in geometric forms of basic objects but various loci and envelopes. Mathematics is indeed everywhere -- so maths teachers can build motivation on emphasizing these not well known facts in the modern era of education.