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#### Team 4: Loft-free unlofting methods for geometric design

August 6, 2008
Keywords:
• geometric
MSC:
• 78A05
##### Abstract
Project Description: The process of laying out the curves and surfaces needed to describe free form shapes in mechanical design is called lofting. Examples of lofting include shapes such as ship hulls, airplane wings and bodies, automobile exteriors, and so on. The best lofting procedures take a vector of inputs, which can contain items like wing span, wing sweep angle, aspect ratios, wing leading edge curvatures, etc., and produce a mathematical model of the geometric shape. Good lofting procedures necessarily have to process the input data nonlinearly in order to produce acceptable shapes. Additionally, it is frequently important to solve the inverse problem. Specifically, one is given a mathematical model of a geometric shape and, with any luck, a lofting code and wants to know what vector of inputs to the lofting code will produce the given shape. This problem has been called the unlofting problem, and it can usually be solved with with standard techniques in non-linear least squares and non-linear parameter estimation. Just as frequently, though, the unlofting problem arises in contexts where no lofting code exists, requiring such a code to be produced as part of the solution. So far, the requirement to produce a lofting code as part of the solution to the unlofting problem has ruined all attempts to produce a fully automatic solution. This project will attempt to construct a prototype unlofting code given only a final geometric shape with no accompanying lofting code. Some recent developments in multiresolution modeling have suggested a promising approach to this problem that we will explore during the workshop, focusing initially on 2D curves and then migrating to simple 3D shapes if time permits. References: "Multiresolution morphing for planar curves," by S. Hahmann, G.-P. Bonneau, M. Cornillac, and B. Caramiaux. Computing 79 (2-4), pp. 197-209 (2007) Prerequisites: Required: 1 semester of numerical analysis and computing skills. Desired: Knowledge of non-linear least squares, splines, and Python programming. Keywords: Lofting, geometric morphing, inverse problems, multiresolution modeling, nonlinear parameter estimation.