Differential Geometry PFUG

VIGRE at Rice: http://math.rice.edu/VIGRE

Our current research project.


Curve Flattening

Begin with a simple, closed, planar curve, C.

Define

  • X: the vector which parameterizes the curve
  • T: the unit tangent to the curve, T = X' / |X'|
  • N: the inward pointing unit normal
  • k: the signed curvature of the curve (positive where C is convex)
  • Tí = k N |X'|

Let J(C) be the flatness energy of the curve,

            J(C) = ∫ k2 ds.

Allow C to evolve by making local changes in order to minimize J.

We conjecture that C will evolve into a circle of infinite radius. We recall that k is the inverse of the radius, R, of the osculating circle, so k = 0 requires R = 8.

 

What We Do

Currently, we are studying the behavior of curves under a "curve flattening" evolution.

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What We Have Done

Overviews of our past research projects.

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News and Events

Presentations and major findings.

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Many thanks to the NSF and the VIGRE program at Rice University.