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Item Fast Updates for Least-Squares Rotational Alignment(The Eurographics Association and John Wiley & Sons Ltd., 2021) Zhang, Jiayi Eris; Jacobson, Alec; Alexa, Marc; Mitra, Niloy and Viola, IvanAcross computer graphics, vision, robotics and simulation, many applications rely on determining the 3D rotation that aligns two objects or sets of points. The standard solution is to use singular value decomposition (SVD), where the optimal rotation is recovered as the product of the singular vectors. Faster computation of only the rotation is possible using suitable parameterizations of the rotations and iterative optimization. We propose such a method based on the Cayley transformations. The resulting optimization problem allows better local quadratic approximation compared to the Taylor approximation of the exponential map. This results in both faster convergence as well as more stable approximation compared to other iterative approaches. It also maps well to AVX vectorization. We compare our implementation with a wide range of alternatives on real and synthetic data. The results demonstrate up to two orders of magnitude of speedup compared to a straightforward SVD implementation and a 1.5-6 times speedup over popular optimized code.Item Normal-Driven Spherical Shape Analogies(The Eurographics Association and John Wiley & Sons Ltd., 2021) Liu, Hsueh-Ti Derek; Jacobson, Alec; Digne, Julie and Crane, KeenanThis paper introduces a new method to stylize 3D geometry. The key observation is that the surface normal is an effective instrument to capture different geometric styles. Centered around this observation, we cast stylization as a shape analogy problem, where the analogy relationship is defined on the surface normal. This formulation can deform a 3D shape into different styles within a single framework. One can plug-and-play different target styles by providing an exemplar shape or an energy-based style description (e.g., developable surfaces). Our surface stylization methodology enables Normal Captures as a geometric counterpart to material captures (MatCaps) used in rendering, and the prototypical concept of Spherical Shape Analogies as a geometric counterpart to image analogies in image processing.Item Levitating Rigid Objects with Hidden Rods and Wires(The Eurographics Association and John Wiley & Sons Ltd., 2021) Kushner, Sarah; Ulinski, Risa; Singh, Karan; Levin, David I. W.; Jacobson, Alec; Mitra, Niloy and Viola, IvanWe propose a novel algorithm to efficiently generate hidden structures to support arrangements of floating rigid objects. Our optimization finds a small set of rods and wires between objects and each other or a supporting surface (e.g., wall or ceiling) that hold all objects in force and torque equilibrium. Our objective function includes a sparsity inducing total volume term and a linear visibility term based on efficiently pre-computed Monte-Carlo integration, to encourage solutions that are as-hiddenas- possible. The resulting optimization is convex and the global optimum can be efficiently recovered via a linear program. Our representation allows for a user-controllable mixture of tension-, compression-, and shear-resistant rods or tension-only wires. We explore applications to theatre set design, museum exhibit curation, and other artistic endeavours.