Title: Interactive, Computation Assisted Design Tools
Description: Realistic modeling, rendering, and animation of physical and virtual shapes have matured significantly over the last few decades. Yet, the creation and subsequent modeling of three-dimensional shapes remains a tedious task which requires not only artistic and creative talent, but also significant technical skill. The perfection witnessed in computer-generated feature films requires extensive manual processing and touch-ups. Every researcher working in graphics and related fields has likely experienced the difficulty of creating even a moderate-quality 3D model, whether based on a mental concept, a hand sketch, or inspirations from one or more photographs or existing 3D designs. This situation, frequently referred to as the content creation bottleneck, is arguably the major obstacle to making computer graphics as ubiquitous as it could be. Classical modeling techniques have primarily dealt with local or low-level geometric entities (e.g., points or triangles) and criteria (e.g., smoothness or detail preservation), lacking the freedom necessary to produce novel and creative content.

A major unresolved challenge towards a new unhindered design paradigm is how to support the design process to create visually pleasing and yet functional objects by users who lack specialized skills and training. Most of the existing geometric modeling tools are intended either for use by experts (e.g., computer-aided design [CAD] systems) or for modeling objects whose visual aspects are the only consideration (e.g., computer graphics modeling systems). Furthermore, rapid prototyping, brought on by technological advances 3D printing has drastically altered production and consumption practices. These technologies empower individuals to design and produce original objects, customized according to their own needs. Thus, a new generation of design tools is needed to support both the creation of designs within the domain's constraints, that not only allows capturing the novice user's design intent but also meets the fabrication constraints such that the designs can be realized with minimal tweaking by experts.

To fill this void, the premise of this thesis relies on the following two tenets: 1. users benefit from an interactive design environment that allows novice users to continuously explore a design space and immediately see the tradeoffs of their design choices. 2. the machine's processing power is used to assist and guide the user to maintain constraints imposed by the problem domain (e.g., fabrication/material constraints) as well as help the user in exploring feasible solutions close to their design intent. Finding the appropriate balance between interactive design tools and the computation needed for productive workflows is the problem addressed by this thesis. This thesis makes the following contributions: 1. We take a close look at thin shells--materials that have a thickness significantly smaller than other dimensions. Towards the goal of achieving interactive and controllable simulations we realize a particular geometric insight to develop an efficient bending model for the simulation of thin shells.

Under isometric deformations (deformations that undergo little to no stretching), we can reduce the nonlinear bending energy into a cubic polynomial that has a linear Hessian. This linear Hessian can be further approximated with a constant one, providing significant speedups during simulation. We also build upon this simple bending model and show how orthotropic materials can be modeled and simulated efficiently. 2. We study the theory of Chebyshev nets--a geometric model of woven materials using a two-dimensional net composed of inextensible yarns. The theory of Chebyshev nets sheds some light on their limitations in globally covering a target surface. As it turns out, Chebyshev nets are a good geometric model for wire meshes, free-form surfaces composed of woven wires arranged in a regular grid. In the context of designing sculptures with wire mesh, we rely on the mathematical theory laid out by Hazzidakis~\cite{Hazzidakis1879} to determine an artistically driven workflow for approximately covering a target surface with a wire mesh, while globally maintaining material and fabrication constraints.

This alleviates the user from worrying about feasibility and allows focus on design. 3. Finally, we present a practical design tool for the design and exploration of reconfigurables, defined as an object or collection of objects whose transformation between various states defines its functionality or aesthetic appeal (e.g., a mechanical assembly composed of interlocking pieces, a transforming folding bicycle, or a space-saving arrangement of apartment furniture). A novel space-time collision detection and response technique is presented that can be used to create an interactive workflow for managing and designing objects with various states. This work also considers a graph-based timeline during the design process instead of the traditional linear timeline and shows its many benefits as well as challenges for the design of reconfigurables.