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Consider the straightening phi of a Beltrami form that is constant on asquare, with the corresponding ellipses having a vertical major axis, and nulloutside. A generalized Schwarz-Christoffel formula is used to express theinverse of phi. The formula is found by introducing an affine Riemann surface.This formula is used to draw on a computer the image of the square by phi, andpractical aspects are discussed. The resulting shapes are shown for differentvalues of the constant dilatation ratio of the ellipses (=major axis/minoraxis). The limit when this ratio tends to infinity is surprising. A model ofthis limit is proposed, produced by an affine surface uniformization.

Beltrami forms, affine surfaces and the Schwarz-Christoffel formula: a worked out example of straightening