Approximate continuity in geomeric modeling

科学和工程应用通常需要将现有物体复制为具有指定连续性的补丁状表面。目前的解决方案不能满足许多工业和科学应用的平滑标准。我的研究目标是找到可接受的表面复制技术。以前的研究者使用G1连续性作为光滑的定义。不幸的是，G1方案产生的表面在视觉上可能不光滑。在早期的工作中，我定义了两个补丁之间的eG1连续性，这意味着相邻的补丁共享一个共同的边界，但是两个补丁在边界上一点的法线之间的夹角由用户定义的epsilon限定。通过将G1连续性条件放宽到eG1，我已经能够改善斑块形状并创建视觉上更平滑的斑块网络。对于工业应用，这种小的不连续性是可以接受的，因为制造过程的精度有限。****我之前在近似连续性方面的工作是测试不连续的最大角度，改进表面以减少不连续，并通过实验发现正常不连续可以有多大，而不会在阴影图像和等影线中创建视觉伪影。这是重要的概念验证工作，现在我将采取下一步措施。特别地，贝塞尔曲面和b样条曲面是用控制点构造的。执行精确的连续性约束对这些控制点的位置施加了限制。我将研究的第一个主题是确定我们在控制点的位置上有多少额外的自由，以便相邻补丁的正常不连续小于用户指定的公差。作为第一步，我将研究函数域的边界，然后将这些结果推广到参数补丁。****我要研究的第二个问题是如何在表面贴片的构造中使用这种额外的自由来改善它们的形状，以满足表面上的工业要求。****我要研究的第三个问题是研究邻近斑块之间有间隙的分段多项式曲面；即，斑块不满足C0，尽管在不连续上会有一个界。这里的数学问题更难，因为C0连续性提供了相邻补丁之间的关联，例如允许我们测量补丁之间共享点法线之间的不连续。虽然Hausdorff度量可以在两个补丁的边界之间提供某种程度的关联，但Hausdorff距离无法区分间隙和重叠，这对制造有非常不同的影响。这里的工作将包括确定一个度量（或多个度量）来区分这两种情况和其他情况，并确定控制点上的条件，以保证工业可接受的大约0连接

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.**