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We study algebraic curves that are envelopes of families of polygonssupported on the unit circle T. We address, in particular, a characterizationof such curves of minimal class and show that all realizations of these curvesare essentially equivalent and can be described in terms of orthogonalpolynomials on the unit circle (OPUC), also known as Szeg\H{o} polynomials. Ourresults have connections to classical results from algebraic and projectivegeometry, such as theorems of Poncelet, Darboux, and Kippenhahn; numericalranges of a class of matrices; and Blaschke products and disk functions.  This paper contains new results, some old results presented from a differentperspective or with a different proof, and a formal foundation for ouranalysis. We give a rigorous definition of the Poncelet property, of curvestangent to a family of polygons, and of polygons associated with Ponceletcurves. As a result, we are able to clarify some misconceptions that appear inthe literature and present counterexamples to some existing assertions alongwith necessary modifications to their hypotheses to validate them. Forinstance, we show that curves inscribed in some families of polygons supportedon T are not necessarily convex, can have cusps, and can even intersect theunit circle.  Two ideas play a unifying role in this work. The first is the utility of OPUCand the second is the advantage of working with tangent coordinates. Thislatter idea has been previously exploited in the works of B. Mirman, whosecontribution we have tried to put in perspective.

Poncelet-Darboux, Kippenhahn, and Szeg\H{o}: interactions between projective geometry, matrices and orthogonal polynomials