Technical Report MSC-2006-20

Title: Piecewise Developable Surface Approximation of General NURBS Surfaces, with Global Error Bounds
Authors: Jacob Subag
Supervisors: Gershon Elber
Abstract: Developable surfaces possess qualities that are desirable in the manufacturing processes of CAD/CAM models. Specifically, models formed out of developable surfaces can be manufactured from planar sheets of material without distortion. This quality proves most useful when dealing with materials such as paper, leather or sheet metal, which cannot be easily stretched or deformed during production.

In this work, we present a semi-automatic algorithm to form a piecewise developable surface approximation of a general NURBS surface. Each such developable surface is constructed from the envelope of tangent planes along a curve on the input surface. Along this envelope surface, individual lines are sampled and analyzed to determine a finite interval along each line and the region it approximates on the input surface. Then, a set of intervals is used to generate a finite developable surface that is contained in the envelope surface and interpolates the intervals. Next, the Hausdorff distance between the finite developable surface and a region on the input surface is bounded and the finite developable surface is refined until this bound is smaller than or equal to the user provided threshold. The refined version of the finite developable surface is modified further by a smoothing algorithm that is intended to remove jagged features from the surface's boundaries. Lastly, more curves on the input surface are used to generate additional approximating developable surfaces in order to obtain a complete approximation of the input surface. Seeking an exclusive coverage of the input surface, these additional surfaces are trimmed.

We also present two intermediate results of our research, which are utilized by the approximation algorithm: an algorithm for generating a conservative approximation of implicit curves and an algorithm for bounding the distance between two general NURBS surfaces. The latter uses the Hausdorff metric (also known as Hausdorff distance), which is a widely acceptable metric for measuring distances and similarities between objects. The conservative approximation of implicit curves algorithm is used to determine the region of the input surface, approximated by a single line on the developable surface and the distance bounding algorithm is used the measure the quality of the generated approximating developable surface.

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