# Kempe's Universality Theorem: An Example

Kempe's Universality Theorem: An Example

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 is forced to move on the straight line , the point moves on the hyperbola .

d

y=c

k

xy=c

On the curve, , so that .

x=cosα+cosβ,

y=sinα+sinβ

xy=sin(α+β)+1/2sin(2α)+1/2sin(2β)=

2