Dupin's Indicatrix

A pair of conics obtained by expanding an equation in Monge's Form $z=F(x,y)$ in a Maclaurin Series

$$\begin{eqnarray*} z&=&z(0,0)+z_1 x+z_2 y+{\textstyle{1\over 2}}(z_{11}x^2+2z_{1... ...ts\\ &=&{\textstyle{1\over 2}}(b_{11}x^2+2b_{12}xy+b_{22}y^2). \end{eqnarray*}$$

This gives the equation

$$b_{11}x^2+2b_{12}xy+b_{22}y^2=\pm 1.$$

Amazingly, the radius of the indicatrix in any direction is equal to the Square Root of the Radius of Curvature in that direction (Coxeter 1969).

References

Coxeter, H. S. M. ``Dupin's Indicatrix'' §19.8 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 363-365, 1969.