Abstract: ic is the construction of a detailed parametric average of three-dimensional surface form from a sample of specimens labelled in accordance with the features of that average. The labelling, which takes the conceptual and topological form of a smooth wire mesh, is a hybrid of biological information and linearized geometric schemes applying to a variety of geometric elements: points, curves, and surface patches. We do not 'smooth' the individual specimens of a sample; instead, we carefully restrict the scope of differentiations of empirical geometric elements so as to apply only after specimens have been averaged. Likewise, while we carry out all averaging in a single Cartesian coordinate system, we carefully 'unwarp' the largest- scale aspects of biological variability within this system before proceeding with the averaging of features at smaller scales. The resulting algorithms underlie useful visualizations of 'typical' or 'normative' anatomy and its variability for consideration in formal computations of statistical atypicality and optimization of interventions and should be crucial to future approaches to image analysis via the automatic deformation of geometrically extended templates.!32
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