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:

– 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.

Math is helpful for morphology

• Seems pretty obvious.

• This was a radical view in biology.

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.

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.

Simplest, subset of affine

Cannon-bone of ox, sheep, giraffe

Piecewise affine

Logarithmically varying: eg., tapir’s toes

Smooth: amphipods (a kind of crustacean).

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).

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

Asymmetric amounts of growth on two sides.

Related Species

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

Invention of Morphing?

• Given transformation between species, linearly interpolate intermediate transformations.

• Intermediate morphs predict intermediate species.

Pages 1070-71

Figure 537

Pages 1078-79

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?