point . Hence the reflected ray is part of the line with equation which is tangent to the nephroid of the previous section at point (see above). The evolute and involute of a nephroid Evolute The evolute of a curve is the locus of centers of curvature. In detail: For a curve with radius of curvature the evolute has the representation with the suitably oriented unit normal. For a nephroid one gets: The evolute of a nephroid is another nephroid half as large and rotated 90 degrees (see diagram). Proof The nephroid as shown in the picture has the parametric representation the unit normal vector pointing to the center of curvature (see section above) and the radius of curvature (s. section on metric properties). Hence the evolute has the representation: which is a nephroid half as large and rotated 90 degrees (see diagram and section above) Involute Because the evolute of a nephroid is another nephroid, the involute of the nephroid is also another nephroid. The original nephroid in the image is the involute of the smaller nephroid. Inversion of a nephroid The inversion across the circle with midpoint and radius maps the nephroid with equation onto the curve of degree 6 with equation (see diagram) . References Arganbright, D., Practical Handbook of Spreadsheet Curves and Geometric Constructions, CRC Press, 1939, , p. 54. Borceux, F., A Differential Approach to Geometry: Geometric Trilogy III, Springer, 2014, , p. 148. Lockwood, E. H., A Book of Curves, Cambridge University Press, 1961, , p. 7. External links Mathworld: nephroid Xahlee: nephroid Roulettes (curve)