Computing efficiently the singularities of surfaces embedded in R~3 is a difficult problem, and most state-of-the-art approaches only handle the case of surfaces defined by polynomial equations. Let F and G be C~∞ functions from R～4 to R and M = {(x, y, z, t) ∈ R～4 | F(x, y, z, t) = G{x, y, 2, t) = 0} be the surface they define. Generically, the surface M is smooth and its projection Ω in R~3 is singular. After describing the types of singularities that appear generically in Ω, we design a numerically well-posed system that encodes them. This can be used to return a set of boxes that enclose the singularities of Ω as tightly as required. As opposed to state-of-the art approaches, our approach is not restricted to polynomial mapping, and can handle trigonometric or exponential functions for example.
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机译：有效地计算嵌入R〜3中的曲面的奇点是一个难题，大多数最新技术仅处理由多项式方程式定义的曲面的情况。令F和G为从R〜4到R的C〜∞函数，并且M = {（x，y，z，t）∈R〜4 | F（x，y，z，t）= G {x，y，2，t）= 0}是它们定义的表面。通常，表面M是光滑的，并且其在R〜3中的投影Ω是奇异的。在描述了一般以Ω出现的奇异点的类型之后，我们设计了一个数字良好定位的系统，对它们进行编码。这可用于返回一组框，这些框根据需要将Ω的奇数紧紧地包围起来。与最新技术相反，我们的方法不限于多项式映射，并且可以处理例如三角函数或指数函数。
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