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We present a new algorithm for fairing of space curves with respect spatial constraints based on a vector valued curvature function. Smoothing with the vector valued curvature function is superior to standard Frenet techniques since the individual scalar components can be modeled similar to curvature-based curve smoothing techniques in 2d. This paper describes a curve smoothing flow that satisfies strict spatial constraints and allows simultaneous control of both curvature functions.

A new method for noise removal of arbitrary surfaces meshes is presented which focuses on the preservation and sharpening of non-linear geometric features such as curved surface regions and feature lines. Our method uses a prescribed mean curvature flow (PMC) for simplicial surfaces which is based on three new contributions: 1. the definition and efficient calculation of a discrete shape operator and principal curvature properties on simplicial surfaces that is fully consistent with the well-known discrete mean curvature formula, 2. an anisotropic discrete mean curvature vector that combines the advantages of the mean curvature normal with the special anisotropic behaviour along feature lines of a surface, and 3. an anisotropic prescribed mean curvature flow which converges to surfaces with an estimated mean curvature distribution and with preserved non-linear features. Additionally, the PMC flow prevents boundary shrinkage at constrained and free boundary segments.

We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in Euclidian $3$-space. The notion of totally normal convergence is shown to be equivalent to the convergence of either one of the following: surface area, intrinsic metric, and Laplace-Beltrami operators. We further s how that totally normal convergence implies convergence results for shortest geodesics, mean curvature, and solutions to the Dirichlet problem. This work provides the justification for a discrete theory of differential geometric operators defined on polyhedral surfaces based on a variational formulation.

We introduce a scheme for real-time nonlinear interpolation of a set of shapes. The scheme exploits the structure of the shape interpolation problem, in particular, the fact that the set of all possible interpolated shapes is a low-dimensional object in a high-dimensional shape space. The interpolated shapes are defined as the minimizers of a nonlinear objective functional on the shape space. Our approach is to construct a reduced optimization problem that approximates its unreduced counterpart and can be solved in milliseconds. To achieve this, we restrict the optimization to a low-dimensional subspace that is specifically designed for the shape interpolation problem. The construction of the subspace is based on two components: a formula for the calculation of derivatives of the interpolated shapes and a Krylov-type sequence that combines the derivatives and the Hessian of the objective functional. To make the computational cost for solving the reduced optimization problem independent of the resolution of the example shapes, we combine the dimensional reduction with schemes for the efficient approximation of the reduced nonlinear objective functional and its gradient. In our experiments, we obtain rates of 20-100 interpolated shapes per second even for the largest examples which have 500k vertices per example shape.

We propose a new framework for spacetime optimization that can generate artistic motion with a long planning horizon for complex virtual characters. The scheme can be used for generating general types of motion and neither requires motion capture data nor an initial motion that satisfies the constraints. Our modeling of the spacetime optimization combines linearized dynamics and a novel warping scheme for articulated characters. We show that the optimal motions can be described using a combination of vibration modes, wiggly splines, and our warping scheme. This enables us to restrict the optimization to low-dimensional spaces of explicitly parametrized motions. Thereby the computation of an optimal motion is reduced to a low-dimensional non-linear least squares problem, which can be solved with standard solvers. We show examples of motions created by specifying only a few constraints for positions and velocities.

Many efficient computational methods for physical simulation are based on model reduction. We propose new model reduction techniques for the approximation of reduced forces and for the construction of reduced shape spaces of deformable objects that accelerate the construction of a reduced dynamical system, increase the accuracy of the approximation, and simplify the implementation of model reduction. Based on the techniques, we introduce schemes for real-time simulation of deformable objects and interactive deformation-based editing of triangle or tet meshes. We demonstrate the effectiveness of the new techniques in different experiments with elastic solids and shells and compare them to alternative approaches.

Creating motions of objects or characters that are physically plausible and follow an animator’s intent is a key task in computer animation. The spacetime constraints paradigm is a valuable approach to this problem, but it suffers from high computational costs. Based on spacetime constraints, we propose a technique that controls the motion of deformable objects and offers an interactive response. This is achieved by a model reduction of the underlying variational problem, which combines dimension reduction, multipoint linearization, and decoupling of ODEs. After a preprocess, the cost for creating or editing a motion is reduced to solving a number of one-dimensional spacetime problems, whose solutions are the wiggly splines introduced by Kass and Anderson [2008]. We achieve interactive response using a new fast and robust numerical scheme for solving a set of one-dimensional problems based on an explicit representation of the wiggly splines.