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Announcements. Mid-term given out the week after next. Send powerpoint to me after presentation. Thompson: Comparison of Related Forms. Key Points. Math is helpful for morphology. Homologous structures necessary: correspondence. Given these, compute transformations of plane. Uses:

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  1. Announcements • Mid-term given out the week after next. • Send powerpoint to me after presentation.

  2. Thompson: Comparison of Related Forms

  3. Key Points • Math is helpful for morphology. • Homologous structures necessary: correspondence. • Given these, compute transformations of plane. • Uses: • Nature of transformation gives clues to forces of growth. • Shapes related by simple transformation -> species are related. Many compelling examples. • Morph between species, predict intermediate species. • Can predict missing parts of skeleton.

  4. Math is helpful for morphology • Seems pretty obvious. • This was a radical view in biology.

  5. Homologies • Had a long tradition • Aristotle: Save only for a difference in the way of excess or defect, the parts are identical in the case of such animals as are of one and the same genus. • In biology, study of homologous structures in species preceded and provided background for Darwin. • Homologous structures explained by God creating different species according to a common plan. • Ontogeny provided clues to homology.

  6. Transformations • Given matching points in two images, we find a transformation of plane. • Homeomorphism (continuous, one-to-one) • This is underconstrained problem • Implicitly, seeks simple transformation. • Not well defined here, will be subject of much future research. • Intuitively pretty clear in examples considered.

  7. Cannon-bone of ox, sheep, giraffe Simplest, subset of affine

  8. Piecewise affine

  9. Logarithmically varying: eg., tapir’s toes

  10. Smooth: amphipods (a kind of crustacean).

  11. Descriptions of shape: Clues to Growth • Somewhat different topic, shape descriptions relevant even without comparison. • Introduces fourier descriptors. • Equal growth in all directions leads to circle (or sphere).

  12. No growth in one direction (as in a leaf on a stem), growth increases in directions away from this so r = sin(q/2).

  13. Asymmetric amounts of growth on two sides.

  14. Related Species

  15. Lack of transformation -> no straight line of descent.

  16. Invention of Morphing? • Given transformation between species, linearly interpolate intermediate transformations. • Intermediate morphs predict intermediate species.

  17. Pages 1070-71

  18. Figure 537

  19. Pages 1078-79

  20. Conclusions • Stress on homologies. • Shape comparison through non-trivial transformations. • Simplicity of transformation -> similarity of shape. • What is the simplest transformation? How do we find it? • Transformation may leave some deviations, how are these handled?

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