Type of Document Dissertation Author StClair, Jessica Lindsey Author's Email Address firstname.lastname@example.org URN etd-04152011-110946 Title Geometry of Spaces of Planar Quadrilaterals Degree PhD Department Mathematics Advisory Committee
Advisor Name Title Haskell, Peter E. Committee Chair Day, Martin V. Committee Member Floyd, William J. Committee Member Thomson, James E. Committee Member Keywords
- Riemannian Metric
- Moduli Space
- Pre-Moduli Space
- Differential Geometry
Date of Defense 2011-04-14 Availability unrestricted AbstractThe purpose of this dissertation is to investigate the geometry of spaces of planar quadrilaterals.
The topology of moduli spaces of planar quadrilaterals (the set of all distinct planar
quadrilaterals with fixed side lengths) has been well-studied , , . The symplectic
geometry of these spaces has been studied by Kapovich and Millson , but the Riemannian
geometry of these spaces has not been thoroughly examined. We study paths in the moduli
space and the pre-moduli space. We compare intraplanar paths between points in the moduli
space to extraplanar paths between those same points. We give conditions on side lengths
to guarantee that intraplanar motion is shorter between some points. Direct applications of
this result could be applied to motion-planning of a robot arm. We show that horizontal lifts
to the pre-moduli space of paths in the moduli space can exhibit holonomy. We determine
exactly which collections of side lengths allow holonomy.
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