Optimizing Curves and Surfaces

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Optimizing Curves and Surfaces

• Advisor Prof. Carlo H. Séquin
• Ryo Takahashi

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Minimum Variation Surfaces

• Aesthetically pleasing surfaces tend to have
  relatively little or no change in curvature, e.g.
• Spheres
• Tori
• Cones
• Minimize the variation in curvature, specifically
  integral of square of magnitude of the derivative
  of curvature
• Goal to calculate the minimum variation surface
  given a set of constraints for a surface

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Colorization

• Useful tool for visualization of surface
  properties
• Gaussian curvature product of maximum and
  minimum curvatures
• Mean curvature average of maximum and minimum
  curvatures
• Color code
• blue positive, green zero, red negative

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original mesh
Gaussian curvature
mean curvature
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Surface OptimizationSurface Area

• Idea Soap bubbles in nature are in
  configurations that minimize surface area
• Calculate gradient of surface area av at each
  vertex
• Set each component of gradient equal to 0 and
  solve to find point (x, y, z) where surface area
  is minimal

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Example Catenoid

• Starting shape cylinder, fixed end rings
• If the rings are too far apart
• catenoid is unstable - pinches off in middle

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Example Catenoid

• If the rings are close enough
• catenoid stabilizes - finds balance points,
  where mean curvature is 0

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Curve OptimizationTurning Angles

• Bending energy function
• Problem difficult to calculate integral exactly
• Solution approximate integral by using turning
  angle

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Curve OptimizationTurning Angles

• Bending energy approximation

• Algorithm
• Use finite differences to calculate direction to
  move each vertex
• Move vertex in calculated direction
• Move adjacent vertices in opposite direction

• Advantages easily to calculate, easy to use with
  finite difference methods
• Disadvantage leads to instability, circles
  collapse inward

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Example Circle

• Circles decrease in bending energy as radius
  increases, but with this algorithm
• circles radius decreases gt energy increases!

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Curve Optimization Edge Forces

• Problem Finite differences method unstable
• Solution Calculate movement on edges, not
  vertices
• Approximate MVC (minimum variation curve)

• Algorithm
• Calculate movement of each vertex so that turning
  angle equals average of adjacent vertices
• Calculate forces on each edge from each adjacent
  vertex
• Move endpoints of each edge according to force on
  edge

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Example Circle

• Calculating forces on edges
• forces cancel out gt circle remains stationary!

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Example Rectangle

• Starting shape Rectangle with fixed corners
• Result Circle through fixed points

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Example Irregular Polygon

• Starting shape Irregular 12-gon with 2 fixed
  vertices
• Result Circle through fixed points preserves
  ratio of segments on each side