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Figure 1: Geodesic interpolation and extrapolation. The blue input poses of the elephant are geodesically interpolated in an as-isometricas-possible fashion (shown in green), and the resulting path is geodesically continued (shown in purple) to naturally extend the sequence. No semantic information, segmentation, or knowledge of articulated components is used. AbstractWe present a novel framework to treat shapes in the setting of Riemannian geometry. Shapes -triangular meshes or more generally straight line graphs in Euclidean space -are treated as points in a shape space. We introduce useful Riemannian metrics in this space to aid the user in design and modeling tasks, especially to explore the space of (approximately) isometric deformations of a given shape. Much of the work relies on an efficient algorithm to compute geodesics in shape spaces; to this end, we present a multiresolution framework to solve the interpolation problem -which amounts to solving a boundary value problem -as well as the extrapolation problem -an initial value problem -in shape space. Based on these two operations, several classical concepts like parallel transport and the exponential map can be used in shape space to solve various geometric modeling and geometry processing tasks. Applications include shape morphing, shape deformation, deformation transfer, and intuitive shape exploration.

Fascinating and elegant shapes may be folded from a single planar sheet of material without stretching, tearing or cutting, if one incorporates curved folds into the design. We present an optimization-based computational framework for design and digital reconstruction of surfaces which can be produced by curved folding. Our work not only contributes to applications in architecture and industrial design, but it also provides a new way to study the complex and largely unexplored phenomena arising in curved folding.

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Surface texturing has proved to be very efficient in full and mixed lubrication, reducing the friction coefficient and the wear rate of mating surfaces. By partially texturing the inlet zone of a thrust washer pad load carrying capacity is generated in the system. In the present paper, a partially textured thrust bearing with square dimples is analysed theoretically using a thermohydrodynamic model. The equations are solved numerically by the finite-difference method. The bearing was realized by the photolithographic method and the theoretical results (fluid film thickness and friction torque) were compared with the experimental data obtained on the test rig. It is found that an optimal number of 12 sectors maximize the load carrying capacity of the bearing. The optimal textured fraction, which maximizes the load carrying capacity is 0.5 on the circumferential direction and 0.9-1 on the radial direction. A good correlation was found between the theoretical and experimental results for the two measured parameters (fluid film thickness and friction torque).