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We present a general Riemannian framework for shape analysis in which metrics and other analyses are invariant to certain transformations beyond that of the similarity transformation of rigid motion and scale. We apply this framework to three specific analyses: 1) point sets invariant to affine transformation, 2) planar contours invariant to affine transformation and re-parameterization, and 3) point sets invariant to projective transformation. For each scenario we construct a pre-shape manifold of canonical, or "standardized," representations of orbits under the appropriate group transformation and develop a path-straightening technique for computing geodesics on the resulting nonlinear manifold. The removal of the rotation and, in the case of the case of planar curves, the re-parameterization groups results in a quotient space, which intersects each orbit uniquely. Geodesics on this quotient space are used for shape comparison, retrieval, and statistical modeling of given point sets or curves.