Data for studies of biologicalshape oftenconsist of the locationsof individually named points, landmarks considered to be homologous(to correspond biologically) from form to form. In 1917 D'Arcy Thompson introduced an elegant modelofhomology as deformation: the configuration of landmarklocations for any one form is viewed as a finite sample from a smooth mapping representing its biological relationship to any other dirrlemlioX),s, m,ulti"l.ariate stali~ti.cal analysis oflamlmark:locati()ns m~y

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... C·A)J()3.. A) for sets of landmark trianglesABC. •These ofone vertex/landmark after scalingso that the remainingtwo vertices are at (0,0) and (1,0). Expressed in this fashion, the biological interpretation of the statistical analysis as a homologymappingwould appear to dependon the triangulation. This paper introduces an analysisof landmark data and homology mappings using a hierarchyof geometric components of shape differenceor shape change. Each componentis a smooth deformation takingthe form of a bivariatepolynomial in the shape coordinates and The simplest component model expresses homogeneous shape change over a form and is represented algebraically by a linear function of the shape coordinates. Modelsof increasinggeometriccomplexityinclude, forexample,growtllgradients; they are derivedfrom higher.. ()rderpolynomials.,For studies of samPles varYing whether individualcomponents are significantas features ofthe homologymapping, and whether the residuals from a fitted model continue to include significant features. These methods are demonstrated data on ph(J~tograpllS of 36 children, to