Generalizations of Kempe’s Universality Theorem

Abstract 

In 1876, A. B. Kempe presented a flawed proof of what is now called Kempe’s Universality Theorem: that the intersection of a closed disk with any curve in R 2 defined by a polynomial equation can be drawn by a linkage. 

Kapovich and Millson published the first correct proof of this claim in 2002, but their argument relied on different, more complex constructions. We provide a corrected version of Kempe’s proof, using a novel contraparallelogram bracing. The resulting historical proof of Kempe’s Universality Theorem uses simpler gadgets than those of Kapovich and Millson.

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