Robust modeling of constant mean curvature surfaces
ACM Transactions on Graphics (TOG) - SIGGRAPH 2012 Conference Proceedings
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This thesis addresses the application of nonlinear optimization to three different problems in computer graphics: the generation of simple motions for legged creatures, the generation of models of truss structures, and the generation of models of constant mean curvature structures. Our technique for generating motion for legged creatures is a reformulation of spacetime optimization, which poses the task of motion synthesis as the process of solving a large, constrained nonlinear optimization problem. Traditionally, the objective function of these problems is a measure of consumed energy, and the constraints are a combination of the laws of physics and a high-level description of the motion we wish to see. Our technique replaces the Newtonian constraints that previous techniques have used to enforce physical realism with a dynamic simulation, which makes the spacetime constraints framework more flexible. We then present a method for using nonlinear optimization to design truss structures, a common and complex category of buildings. Truss structures are ubiquitous in the industrialized world, appearing as bridges, towers, roof supports and building exoskeletons, yet are complex enough that modeling them by hand is difficult and time consuming. We represent trusses as a set of rigid bars connected by pin joints, which may change location during optimization. By including the location of the joints as well as the strength of individual beams in our design variables, we can simultaneously optimize the geometry and the mass of structures. The third application we examine is the task of generating models of surface area minimizing, constant mean curvature objects. Constant mean curvature objects, which include such diverse natural and man-made structures as thin film membranes, sails, pneumatic structures, and soap bubbles and films, are both common and often difficult to create by hand. Using the technique of constrained nonlinear optimization, we can automatically generate models of these structures by minimizing surface area while maintaining a constant mean curvature on each surface in addition to volume and other geometric constraints. We conclude this thesis with a discussion of the advantages and shortcomings of optimization as a technique for solving modeling and animation problems in computer graphics.