Cartoon Deformations using the Medial Axis

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Cartoon Deformations using the Medial Axis

• Eli Broadhurst
• COMP 259
• May 07, 2003

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Medial Axis Deformations

• Uses?
• Key frame animation
• Elastic deformation of a 1D line
• Dynamic deformation determined by velocity or
  collisions

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Medial Axis Deformations

• Uses?
• Key frame animation
• Elastic deformation of a 1D line
• Dynamic deformation determined by velocity or
  collisions

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Medial Axis Deformations

• Input
• Polygonal object with a medial axis from z 0 to
  z 1.
• Not a true Blum Medial Axis

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Medial Axis Deformations

• Deformation Parameters
• Curve
• Cubic Hermite, Bezier, B-Spline
• Coordinate Frame
• Frenet Frame BAD BAD BAD
• Easier Solution Gram-Schmidt orthonormalization
  of a reference vector.
• Twisting

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Medial Axis Deformations

• Sounds easy, right?

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Medial Axis Deformations

• Sounds easy, right?
• Volume preservation
• Arc length parameterization
• What if radius of curvature at a point is less
  than the radius of the object?
• Normals?
• UI? Need to edit/view curves.

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Medial Axis Deformations

• Volume preservation
• Cylinder v L ? R2
• In General v Integral from 0 to L of ? R(s)2
  ds
• If arc length changes L -gt L
• R(s) (L / L)0.5 R(s)
• Assuming an arc length parameterization

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Medial Axis Deformations

• Arc Length Parameterization
• Analytic solution does not exist for cubic
  curves.
• Approximate by doing a table lookup from sampled
  curve followed by bisection method.

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Medial Axis Deformations

• Radius of Curvature
• dC/ds, dC2/ds2, dC3/ds3, and the curvature at a
  point on the curve must be computed.
• The radius of curvature 1 / curvature.
• Curvature x x / x3
• A scale Radius of Curvature
• B 2 R2/(A R)

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Medial Axis Deformations

• Defining the Warp
• From z value compute corresponding point on
  curve.
• Rotate the x and y values of the point then
  convert to the local coordinate frame.
• Scale the point based off of its position in
  local coordinate frame.
• Normals similar except divide by scales instead
  of multiply.

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Medial Axis Deformations

• Where is the animation?
• Easier to demonstrate.

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References

• Principles of Traditional Animation Applied to 3D
  Computer Graphics. John Lasseter. SIGGRAPH 87
• Global and Local Deformations of Solid
  Primitives. Alan Barr. SIGGRAPH 84
• Simulating Cartoon Style Animation. Stephen
  Chenney, Mark Pingel, Rob Iverson, Marcin
  Szymanski. ACM Non-photorealistic animation and
  rendering 02.
• Continuous medial representations for geometric
  object modeling in 2D and 3D. Paul Yushkevich, P.
  Thomas Fletcher, Sarang Joshi, Andrew Thall,
  Shephen M. Pizer. Image and Vision Computing 21
  (2003) 17-27.