Approximate continuity in geomeric modeling

科学和工程应用通常要求现有对象被复制为具有指定连续性的面片状表面。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。以前的研究人员使用G1连续性作为平滑的定义。不幸的是，由G1方案产生的表面可能在视觉上看起来不光滑。在以前的工作中，我定义了两个面片之间的eG1连续性，意味着相邻的面片共享一个公共边界，但边界上一点上两个面片的法线之间的角度由用户定义的epsilon界定。通过将G1连续性条件放宽为eG1，我已经能够改善面片形状并创建视觉上更平滑的面片网络。对于工业应用，这种小的不连续是可以接受的，因为制造过程的精度是有限的。我之前在近似连续性方面的工作是测试最大不连续角度，细化曲面以减少不连续，并通过实验发现法向不连续可以达到多大，而不会在阴影图像和等光线中产生视觉伪影。这是重要的概念验证工作，我现在将采取下一步行动。特别地，Bezier曲面和B-Spline曲面是用控制点构造的。强制实施精确连续性约束会对这些控制点的位置进行限制。我要研究的第一个主题是确定我们在控制点的位置上有多大的额外自由度，以便相邻面片的正常不连续性小于用户指定的公差。作为第一步，我将研究函数域中的边界，并将这些结果推广到参数曲面片。第二个问题是如何在构造曲面片时利用这种额外的自由度来改善其形状，以满足工业对曲面的要求。第三个问题是研究相邻曲面片之间有间隙的分段多项式曲面；即，曲面片不满足C0，尽管在不连续性上会有界。这里的数学问题比较难，因为C0连续性提供了相邻面片之间的关联，例如，允许我们测量面片之间共享的点处的法线之间的不连续性。虽然Hausdorff度量可以在两个补丁的边界之间提供一定程度的关联，但Hausdorff距离无法区分缺口和重叠，这对制造业有非常不同的影响。这里的工作将包括确定一个(或多个)指标以区分这两种情况和其他情况，并确定控制点上的条件，以确保行业可接受的大约C0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.