  The exponential orthogonal polynomials encode via the theory of hyponormal
operators a shade function $g$ supported by a bounded planar shape. We prove
under natural regularity assumptions that these complex polynomials satisfy a
three term relation if and only if the underlying shape is an ellipse carrying
uniform black on white. More generally, we show that a finite term relation
among these orthogonal polynomials holds if and only if the first row in the
associated Hessenberg matrix has finite support. This rigidity phenomenon is in
sharp contrast with the theory of classical complex orthogonal polynomials. On
function theory side, we offer an effective way based on the Cauchy transforms
of $g, \overline{z}g, \ldots, \overline{z}^dg$, to decide whether a
$(d+2)$-term relation among the exponential orthogonal polynomials exists; in
that case we indicate how the shade function $g$ can be reconstructed from a
resulting polynomial of degree $d$ and the Cauchy transform of $g$. A
discussion of the relevance of the main concepts in Hele-Shaw dynamics
completes the article.
