Kempe's Universality Theorem: An Example

x coordinate of d

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Roughly, Kempe's universality theorem states that any finite part of a plane algebraic curve can be traced out by a vertex of some linkage. This Demonstration illustrates a particular example: there exists a linkage such that, if the point

d

is forced to move on the straight line

y=c

, the point

k

moves on the hyperbola

xy=c

.

On the curve,

x=cosα+cosβ,

y=sinα+sinβ

, so that

xy=sin(α+β)+1/2sin(2α)+1/2sin(2β)=

2

.