Mechanics of particles and systems Kinematics Theorem

Kempe's universality theorem

Builds on Algebraic curve Rigidity of frameworks

Statement

Let $f \in \R[x, y]$ be a nonzero polynomial with zero set $C = \set{(x, y) \in \R^2 : f(x, y) = 0}$ , a real plane algebraic curve, and let $D \subset \R^2$ be a closed disc. A planar linkage is a framework of rigid bars joined at revolute joints, with some joints pinned to the plane. Then there are a planar linkage and a joint $v$ of it with this property. The set of positions of $v$ over all configurations of the linkage is exactly $C \cap D$ .

Remarks

Idea of the proof: write the position of a point $v$ with two arms of fixed length from a pinned joint, through two angles. Then $f$ becomes a trigonometric polynomial in the two angles, that is, a sum of cosines of integer combinations of them. Gadget linkages copy angles, add angles, multiply angles by integers and add vectors, and together they build a linkage that forces the sum to be $0$ . Alfred Kempe published the argument in 1876; his gadgets could move into degenerate positions, and complete proofs were given later with braced gadgets.

Example: a single bar of length $1$ pinned at the origin realises the circle $x^2 + y^2 = 1$ ; the Peaucellier–Lipkin linkage realises a segment of a straight line.

The disc $D$ is needed: a linkage has bars of bounded length, so all its joints stay in a bounded region. So an unbounded curve such as a line cannot be traced entirely.

Going deeper

For a student

The theorem says that planar linkages are as expressive as real algebraic geometry in the plane. Extensions show that every compact real algebraic set arises, up to suitable equivalence, from configuration spaces of linkages. It connects mechanism design with rigidity of frameworks and with the study of configuration spaces in topology.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`YD5Y39065F`
URI	`m2://YD5Y39065F`
Kind	statement, theorem
Writing	W0 stub
MSC	70B15, 14P05, 52C25
Status	This copy of the site has no build data. Run `m2 status YD5Y39065F`.
About	Algebraic curve
Uses	Algebraic curve, Rigidity of frameworks
Same as	Wikidata Q30692901 Wikipedia Kempe's universality theorem
Changed	3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

an inventory row: the row of the inventory that the node accounts for (0 of 1)
a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.