[Paper Review] Real projective space as a space of planar polygons

Donald M. Davis|arXiv (Cornell University)|Jan 12, 2015

Computational Geometry and Mesh Generation4 references3 citations

TL;DR

This paper establishes a homeomorphism between real projective space $\mathbb{RP}^{n-3}$ and the moduli space $\overline{M}_{n,n-2}$ of isometry classes of planar $n$-gons with one side of length $n-2$ and all others of length 1. By parameterizing polygon configurations via angular coordinates on circles centered at the far vertex, the authors construct a $\mathbb{Z}/2$-equivariant homeomorphism to the sphere $S^{n-3}$, whose quotient yields the desired identification with $\mathbb{RP}^{n-3}$, linking topological complexity to robotic motion planning.

ABSTRACT

We prove that real projective space RP^{n-3} is homeomorphic to the space of all isometry classes of n-gons in the plane with one side of length n-2 and all other sides of length 1. This makes the topological complexity of real projective space more relevant to robotics.

Motivation & Objective

• To establish a topological equivalence between real projective space $\mathbb{RP}^{n-3}$ and the moduli space of planar $n$-gons with one long side and unit-length others.
• To provide a geometric realization of $\mathbb{RP}^{n-3}$ as a space of polygon configurations, making abstract topological invariants more tangible in robotics.
• To clarify the $\mathbb{Z}/2$-equivariance of the homeomorphism between polygon spaces and spheres, resolving ambiguity in prior work.
• To connect the topological complexity of $\mathbb{RP}^{n-3}$ to the motion planning problem of polygonal robotic arms.

Proposed method

• Parameterize each polygon in $M_{n,r}$ by angular coordinates $t_i$ derived from the intersection of circles centered at previous vertices and the far vertex.
• Define $i_0$ as the smallest index where the distance from $x_i$ to $x_{n-1}$ achieves the maximum possible under the triangle inequality.
• Use the arc of intersection between $C(x_{i-1},1)$ and $C(x_{n-1},n-1-i)$ to define a parameter $t_i \in [-1,1]$ for each vertex $x_i$, with $t_i = \pm 1$ when the arc degenerates.
• Construct a $\mathbb{Z}/2$-equivariant homeomorphism $\Phi: M_{n,r} \to S^{n-3}$ by mapping the polygon configuration to a point in the iterated unreduced suspension of $S^0$, respecting reflection and antipodal symmetries.
• Take the quotient under the $\mathbb{Z}/2$-action to obtain a homeomorphism between $\overline{M}_{n,r}$ and $\mathbb{RP}^{n-3}$.
• Verify that the model for $S^{n-3}$ via $J^{n-3} \times S^0$ with the specified equivalence relation matches the standard sphere, enabling explicit coordinate transformation.

Experimental results

Research questions

• RQ1 Can real projective space $\mathbb{RP}^{n-3}$ be geometrically realized as a moduli space of planar polygons with fixed side lengths?
• RQ2 Is there a $\mathbb{Z}/2$-equivariant homeomorphism between the space of oriented $n$-gons with one long side and the sphere $S^{n-3}$?
• RQ3 Does the quotient of this polygon space under reflection yield a space homeomorphic to $\mathbb{RP}^{n-3}$?
• RQ4 How does this construction clarify the topological complexity of $\mathbb{RP}^{n-3}$ in the context of robotic motion planning?
• RQ5 Can the topological invariants of $\mathbb{RP}^{n-3}$, such as cohomology, be understood through the geometry of polygon configurations?

Key findings

• The space $\overline{M}_{n,n-2}$ of isometry classes of planar $n$-gons with one side of length $n-2$ and all others of length 1 is homeomorphic to $\mathbb{RP}^{n-3}$ for $n \geq 3$.
• A $\mathbb{Z}/2$-equivariant homeomorphism $\Phi: M_{n,r} \to S^{n-3}$ is constructed explicitly for $n-2 \leq r < n-1$, resolving ambiguity in earlier work.
• The construction uses angular parameters derived from circle intersections to parameterize polygon configurations, with $t_i = \pm 1$ corresponding to degenerate arcs.
• The identification of $\overline{M}_{n,n-2}$ with $\mathbb{RP}^{n-3}$ provides a geometric interpretation of topological complexity in robotic motion planning.
• The result confirms that the mod-2 cohomology rings of $\overline{M}_{n,n-2}$ and $\mathbb{RP}^{n-3}$ are isomorphic, as previously suggested by other work.
• The model for $S^{n-3}$ via iterated suspension of $S^0$ with a collapse relation at $\pm 1$ is shown to be homeomorphic to the standard sphere, enabling the coordinate transformation.