Approximate continuity in geomeric modeling

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Scientific and engineering applications often require an existing object be reproduced as a patchwise surface of designated continuity. Current solutions fail to meet the smoothness criteria of many industrial and scientific applications. The goal of my research is to find acceptable surface reproduction techniques. Previous researchers used G1 continuity as the definition of smooth. Unfortunately, surfaces produced by the G1 schemes can fail to appear visually smooth. In earlier work, I defined eG1 continuity between two patches to mean that adjacent patches share a common boundary, but the angle between the normals of the two patches at a point on the boundary is bounded by a user defined epsilon. By relaxing the G1 continuity conditions to eG1, I have been able to improve patch shape and create visually smoother patch networks. For industrial applications, such small discontinuities are acceptable since the manufacturing process has limited precision.****My previous work on approximate continuity was on testing the maximum angle of discontinuity, refining surfaces to reduce the discontinuity, and finding through experiments how large the normal discontinuity can be without creating visual artifacts in shaded images and isophote lines. This was important proof of concept work, and I will now take the next steps. In particular, Bezier and B-spline surfaces are constructed with control points. Enforcing exact continuity constraints puts restrictions on the locations of these control points. The first topic I will investigate will be to determine how much extra freedom we have in the locations of the control points so that neighboring patches have normal discontinuities less than a user specified tolerance. As a first step, I will investigate bounds in the functional domain, and later generalize these results to parametric patches.****The second question I will investigate is how to use this additional freedom in the construction of surface patches to improve their shape to meet industrial requirements on the surfaces.****The third question I will investigate is to study piecewise polynomial surfaces with gaps between the neighboring patches; i.e., the patches do not meet C0, although there will be a bound on the discontinuity. The mathematical questions here are harder, since C0 continuity provides an association between neighboring patches, allowing us for example to measure discontinuity between normals at points shared between patches. While a Hausdorff metric can provide some level of association between the boundaries of two patches, the Hausdorff distance fails to distinguish between a gap and an overlap, which have very different impacts on manufacturing. The work here will involve determining a metric (or metrics) to distinguish between these two and other cases, and determine conditions on control points that will guarantee approximately C0 joins acceptable to industry.**