The Measurement of Biological Shape and Shape Change [Paperback]

One. Introduction: On the Absence of Geometry from Morphometrics.- First Part. The Measurement of Biological Shape.- Two. Shapes and Measures of Shape.- A. Properties of the Euclidean Plane and Euclidean Space.- B. Outlines and Homologous Landmarks.- C. Definitions of Shape, Shape Change, Shape Measurement.- D. Shapes as Data.- Three. Critique of an Applied Field: Conventional Cephalometrics.- A. Landmarks, Curvature, and Growth.- B. Registration.- Four. New Statistical Methods for Shape.- A. Analysis of the Tangent Angle Function.- 1. History.- 2. Sampling from the tangent angle function.- 3. Conic replacement curves and their estimation Fit of a conic to a point  Geometric interpretation  Estimator for a circle  Estimation for the general conic  Computation of the extremum -Linear constraints.- 4. Conic splining Joint conic fitting under constraint -An example  Application  Analysis of parameters.- B. Extension to Three Dimensions: A Sketch.- C. Skeletons Definition  A multivariate statistical method  Bibliographic note.- Second Part. The Measurement of Shape Change Using Biorthogonal Grids.- Five. The Study of Shape Transformation after DArcy Thompson.- A. The Original Method.- 1. Thompsons own work.- 2. Later examples.- 3. Difficulties.- B. Analysis of Growth Gradients.- C. Simulations.- D. Other Morphometric Schemes Vector displacements  Multivariate morphometrics.- Six. The Method of Biorthogonal Grids.- A. Representation of Affine Transformations.- B. General Lines of Growth and Biorthogonal Grids.- C. Summarizing the Grids.- Technical Note 1. Existence and Form of Biorthogonal Grids.- Technical Note 2. Interpolation from Landmark Locations and Arcs The measure of roughness  The vector space and its associated functions  Interpolation from boundary values  Interpolation from boundary values and interior points  Note on computation.- Technical Note 3. Construction of Integral Curves.- Technical Note 4. On Homologous Points.- Seven. Examples of Bl£§