Cynosure对《Generalized Curvatures (Geometry and Computing)》的笔记(1)

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搞图形的人，学ddg的话未必能从本书中得到想要的。
这本书讲什么？可以从《On the angular defect of triangulations andthe pointwise approximation of curvatures》窥一斑：
 a significant number of papers advocate the use of normalized angular defects to estimate the Gauss curvature of smooth surfaces. ....But none of these contributions address the question of the accuracy of these estimates or that of their convergence when the mesh is refined....We show that the statements made in these papers are erroneous in general, although they may be true pointwise for very specific meshes.
In particular, Meek and Walton observe on a counterexample that the angular defect does not estimate the Gauss curvature, but no analysis is carried out. The missing analysis is presented in this paper.
We show that vertices of valence 4 and 6 are the only ones where kG can be inferred from the angular defect. .... n = 4 is the only value of n such that angular defect depends upon the principal directions, and that n = 6 is the only value such that angular defect provides an exact estimate for kG.
作为应用，ddg可以选择研究regular mesh，这无可厚非。但在数学上，ddg还需要研究更一般的情况。既然“angular defect does not estimate the Gauss curvature”，那就尝试去估计Gauss curvature的范围，然后发现这是可行的。于是便有了书里的n多不等式。
https://www.rapidtables.com/math/symbols/Basic_Math_Symbols.html
geometric quantity：
We say that aquantity Q(S) associated to S is geometric with respect to G0 if the correspondingquantity Q[g(S)] associated to g(S) equals Q(S), for all g ∈ G0。
it is important to point out that theproperty of being geometric depends on the chosen group。In this book, we only consider the group of rigid motions,
a geometric quantity is “interesting” if it possesses “fundamental” properties：continuity condition, inclusion–exclusion principle。
Different Possible Classifications：
the fundamental difference between a convergence result and an approximation one: when one deals withapplications (like medical imaging, structural geology, or computer graphics for instance), a convergence result of geometric invariants is often elegant and reassuring.But how to apply it? Conversely, an approximation result gives a bound on the error.However, in both cases, we are often dealing with a “real-world object,” extremelydifficult to define. We must have permanently in mind the difference between a“real” physical object, the perception of this object, and its mathematical modeling.
In our context (of approximation), these fundamental results appear as negativeones. In fact, this means that the shape of a smooth surface or a polyhedron cannotbe detected by integrating or summing the pointwise Gauss curvature.
The aim of this book is now clear: to construct a theory which allows us to define geometric invariants (which will be called curvature measures) on a large classof objects including smooth submanifolds, piecewise linear spaces, convex subsets(and more generally, singular objects). By using a coherent topology, our goal is toderive approximation and convergence results.
normal cycle generalizes the unit normal bundle.
从22.6至最后一章是前面理论在2d、3d上的特化，相对容易理解