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Catastrophe Set of the Plane Projection of a Movable Surface

This Demonstration shows the catastrophe set of the plane projection from the surface. You can rotate the surface by dragging the locator. Observe that the cusp type is the only shape that appears for the catastrophe sets.
Initially, the surface is given by the image of the map . The projection is given by ). The catastrophe set (red curve) is defined by the image of the projection of the discriminant set (blue curve) on the surface, where the surface has vertical tangent lines, or, to be more colloquial, the places on the surface where a walker would fall to a fold directly below or could jump up to a fold directly above.

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DETAILS

The theory of Catastrophe Set of the Plane Projection of a Movable Surface in [1] and [2].
References
[1] V. I. Arnold, S. M. Gusein–Zade, and A. N. Varchenko, Singularities of Differentiable Maps, Vol. 1, Boston: Birkhäuser, 1985.
[2] T. Bröcker, Differentiable Germs and Catastrophes, London: Cambridge University Press, 1975.
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