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Kempe's Universality Theorem: An Example

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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
.
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