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Analyzing shape manifolds as Riemannian manifolds has been shown to be an effective technique for understanding their geometry. Riemannian metrics of the types H^0 and H^1 on the space of planar curves have already been investigated in detail. Since in many applications, the basic shape of an object is understood to be independent of its scale, orientation or placement, we consider here an H^2-metric on the space of planar curves modulo similitudes. The metric depends purely on the bending and stretching of the curve. Equations of the geodesic for parametrized curves as well as un-parametrized curves and bounds for the sectional curvature are derived. Equations of gradient descent are given for constructing the geodesics between two given curves numerically. |
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