WOLFRAM|DEMONSTRATIONS PROJECT

Kempe's Universality Theorem: An Example

​
x coordinate of d
show axes
show angles
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
.