# Piecewise circular curves and positivity

@inproceedings{Burelle2021PiecewiseCC, title={Piecewise circular curves and positivity}, author={Jean-Philippe Burelle and Ryan Kirk}, year={2021} }

We introduce the moduli space of generic piecewise circular n-gons in the Riemann sphere and relate it to a moduli space of Legendrian polygons. We prove that when n = 2k, this moduli space contains a connected component homeomorphic to the Fock-Goncharov space of k-tuples of positive flags for PSp(4,R) and hence is a topological ball. We characterize this component geometrically as the space of simple piecewise circular curves with decreasing curvature.

## References

SHOWING 1-10 OF 18 REFERENCES

### Moduli spaces of local systems and higher Teichmüller theory

- Mathematics
- 2003

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S…

### Symmetry sets of piecewise-circular curves

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 1993

Synopsis Piecewise-circular (PC) curves are made up of circular arcs and segments of straight lines, joined so that the (undirected) tangent line turns continuously. PC curves have arisen in various…

### On the Geometry of Piecewise Circular Curves

- Mathematics
- 1994

In this article we would like to promote a class of plane curves that have a number of special and attractive properties, the piecewise circular curves, or PC curves. (We feel constrained to point…

### Anosov flows, surface groups and curves in projective space

- Mathematics
- 2004

Note that in [10], W. Goldman gives a complete description of these connected components in the case of finite covers of PSL(2,R). In the case of PSL(2,R), two homeomorphic components, called…

### Schottky presentations of positive representations

- MathematicsMathematische Annalen
- 2022

We show that total positivity gives rise to a partial cyclic order on the set of oriented flags in $${\mathbb {R}}^n$$ R n . Using the notion of interval given by this partial cyclic order, we…

### Symplectic Frieze Patterns

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2019

We introduce a new class of frieze patterns which is related to symplectic geometry. On the algebraic side, this variant of friezes is related to the cluster algebras involving the Dynkin diagrams of…

### Lie Sphere Geometry

- Mathematics
- 2016

This chapter presents the method of moving frames in Lie sphere geometry. This involves a number of new ideas, beginning with the fact that some Lie sphere transformations are not diffeomorphisms of…

### Total Positivity in Reductive Groups

- Mathematics
- 2019

An invertible n×n matrix with real entries is said to be totally ≥0 (resp. totally >0) if all its minors are ≥0 (resp. >0). This definition appears in Schoenberg’s 1930 paper [S] and in the 1935 note…

### A survey and comparison of traditional piecewise circular approximations to the ellipse

- MathematicsComput. Aided Geom. Des.
- 1999