The Nephroid Lab

The immediate goal of this lab is to show that two curves defined by the pictures below are the same. Both are called the nephroid, because of their kidney shape.

Figure 1: An optical and mathematical nephroid curve.

The apparent curve in the first image is the pattern formed on the bottom of an empty flat-bottomed coffee mug when parallel light rays come in and bounce off the circular sides.

The similar looking curve in the second image is the path traced out by a point on the boundary of a small disk as the disk rolls along the boundary of a larger disk having twice the diameter.

In figuring out why the two nephroid curves are the same, we will encounter the following mathematical ideas:

• How to find a parametrization in terms of trigonometric functions for curves (like the nephroid) that may be generated by rolling one circle along the outside of another circle.
• How to convert trigonometric parametrizations into algebraic parametrizations.
• How to use elimination of variables to
  • find implicit polynomial equations for parametrized curves
  • compute profiles of projections of algebraic surfaces, profiles of surfaces, and envelopes of families of curves

Lab Outline

• Parametrizing the Nephroid
• Transforming a Parametrization into an Implicit Algebraic Equation
• The Geometry of Reflecting Light Rays
• Computing the Envelope of a Family of Curves
• Conclusions

Other Topics

• Gröbner Bases
• Projection, Profiles, and Envelopes
• Singular Sets

References and sources of inspiration

• Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms: an introduction to computational algebraic geometry and commutative algebra , Springer-Verlag 1992.
• Bruce and Giblin, Curves and Singularities, Cambridge University Press 1984.