The average of a set of compatible curves and surfaces
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The proposed thesis focuses on providing definitions, closed-form expressions and algorithmic solutions for computing averages of sets of shapes and associated standard deviation fields. By shapes, we mean lines, planes, curves, and surfaces. Note that these shapes are embedded in a common ambient space -- by which we mean that their scales, orientations, and positions are fixed and important. We establish sufficient conditions on sets of shapes for which our proposed solutions yield topologically valid results, and call such sets compatible. We propose, study and apply suitable generalizations of the statistical formulations of the average and standard deviation of sets of numbers to sets of compatible shapes. In particular, the solutions proposed extend naturally the popular concept of the medial axis to two and more curves in the plane or in three-dimensions and to two or more surfaces. We demonstrate the usefulness of our developed algorithms for tasks such as: morphing between shapes, sketch specification by overdrawing, and, analyzing the path of the cutting tool of a CNC machine.