  Orthogonal polynomials on quadratic curves in the plane are studied. These
include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two
lines. For an integral with respect to an appropriate weight function defined
on any quadratic curve, an explicit basis of orthogonal polynomials is
constructed in terms of two families of orthogonal polynomials in one variable.
Convergence of the Fourier orthogonal expansions is also studied in each case.
As an application, we see that the resulting bases can be used to interpolate
functions on the real line with singularities of the form $|x|$, $\sqrt{x^2+
\epsilon^2}$, or $1/x$, with exponential convergence.
