Surface design and rationalization for robotic hot-blade cutting

In this thesis, we are motivated by robotic hot-blade cutting. Hot-blade cutting is a technology where a robot moves a heated flexible rod through a block of expanded polystyrene (EPS). By controlling the endpoints and end tangents, the robot controls the shape of the rod. The outcome of the cutting procedure is a cut in the EPS that can be used as a mold for concrete casting. Because the rod takes the shape of an elastic curve, we can cut curved and interesting designs that otherwise would be expensive to fabricate.
To fully utilize hot-blade cutting, architects and designers need design tools for defining the shape of the rod, i.e., computational tools for designing elastic curves, elastic splines, and surfaces foliated by elastic splines in a CAD (Computer Aided Design) environment. In this thesis, we present algorithms for accommodating this need. First, we introduce algorithms for solving the boundary value problem. By using these methods, one is able to design a single elastic curve via specification of endpoints, end tangents, and curve length. We compare the algorithms and describe their advantages and disadvantages. One of the disadvantages of solving the boundary value problem is the non-uniqueness of the solution. Consequently, the methods lack shape control. We, therefore, propose to design the elastic curves through a cubic curve interface. We present a projection tool that projects a cubic Bézier curve to a cubic curve visually close to an elastic curve. The projection keeps the endpoints and end tangent angles fixed. Along with the design of a single elastic curve, we also introduce algorithms for designing elastic splines, i.e., concatenated elastic curve segments. Among others, we present a data-driven tool for interactively designing an elastic spline. Finally, we extend the spline algorithms to the design of surfaces foliated by elastic splines.