User ramsay - MathOverflow

Answer by Ramsay for Delaunay triangulations and convex hulls

2012-01-23T11:40:43Z

This paper by Bobenko and Izmestiev "Alexandrov's Theorem, Weighted Delaunay Triangulations, and Mixed Volumes" is based upon a generalization of this observation.

One reason that this convex-hull characterization may seem obscure is that it really only applies to points on the sphere. The paragraph in the OP that begins "There is no reason ..." is misleading because it implies that the planar Delaunay triangulation can be characterized in this way. The lower envelope of the convex hull of the traditional lifting (to a paraboloid) gives exactly the combinatorics of the planar DT. However, going to a sphere, by stereographic projection, say, is only going to work if the radius of the sphere is "big enough". For a fixed number of points n, there will be no upper bound on the radius that is big enough for all point sets of size n.

The Delaunay triangulation is brittle, and unless unusual demands on the vertex configurations are made, the combinatorial structure can change with an arbitrarily small perturbation of the metric. I.e., the point set can be arbitrarily close to being degenerate.

Answer by Ramsay for Gaussian curvature radius

2011-02-17T14:00:01Z

Deane's answer is similar to what I would have tried to say if I'd got here on time. I don't recall seeing the "Gaussian curvature radius" defined before, so I can't point you to other references. The definition is natural. On the one hand the bound on the distance to a conjugate point (Morse-Schönberg lemma) is given in terms of a bound on the Gaussian curvature radius, and on the other hand the Gaussian curvature radius provides an upper bound to the "maximal curvature radius"(reciprocal of the maximum of the absolute values of the principal curvatures). As Deane pointed out, these two curvature radii coincide on the sphere.

Since we are only using it as an upper bound, we just define it to be infinite if the curvature is non-positive. In flat or negatively curved spaces, conjugate points are not an issue; geodesics diverge.

As to your alternative, $\rho_K(x) = 1/\sqrt{|K(x)|}$, I guess it depends on what you want to do. You are making a smaller sizing function, but why? The spirit of the Morse-Schönberg lemma is better captured without the absolute value signs. If the infinite values disturb you, you have not avoided them when the Gaussian curvature vanishes.

Answer by Ramsay for Is this formulation of the Singular Value Decomposition standard?

2010-10-24T20:37:23Z

The book "Numerical Linear Algebra" by Trefethen and Bau, also introduces the SVD in this way. The relevant chapters (4 and 5) seem to be complete in Google books. The exposition is not tainted by the 'numerical' in the title.

I don't understand why the SVN isn't given more emphasis in standard introductions to linear algebra. It seems to give a good intuitive decomposition of a linear operator. I would expect it to be a useful theoretical tool even if numerical issues are not being considered.

Answer by Ramsay for finding cutting edge papers and books

2010-10-22T15:59:56Z

There are some good resources mentioned already on this site. In particular see this question: http://mathoverflow.net/questions/2147/most-helpful-math-resources-on-the-web

Ryan Budney makes a comment under Thanos D. Papaïoannou's response about MathSciNet. Given any relevant paper on the topic of interest, you can find papers that cite it.

This question may also be worth looking at: http://mathoverflow.net/questions/1722/free-high-quality-mathematical-writing-online

Answer by Ramsay for Shortest morphing between shapes embedded in $\mathbb{R}^3$

2010-10-08T14:18:52Z

There is another solution in the computer graphics literature that you didn't mention:

Greg Turk and James O'Brien, "Shape Transformation Using Variational Implicit Functions," SIGGRAPH 99, August 1999, pp. 335-342.

http://www.cc.gatech.edu/~turk/my_papers/schange.pdf

This uses the same idea that you outline, except instead of taking the convex hull they create a 'best fit' shape in the ambient space using radial basis functions. In this way they don't need the shapes to be isotopic or even homeomorphic.

Maybe people familiar with cobordism would be able to provide insight into what might be desired? Anyway, I like the O'Brien-Turk solution.

Geodesic confluence as witness to a local extrinsic bound

2010-09-22T14:53:18Z

This problem seeks a relationship between the lengths of geodesics which emanate from a point on a surface in $\mathbb{R}^3$ and then come together again, and the bounds on the lengths of centred normal fibres through them in a maximal tubular neighbourhood of the surface.

Let $S \subset \mathbb{R}^3$ be a smooth closed (compact without boundary) surface. The local reach is a function $\rho: S \rightarrow \mathbb{R}$ such that $\rho(x)$ is the radius of the smaller of the two maximal empty tangential balls on either side of $S$ at $x$. An empty tangential ball at $x$ is an open Euclidean ball in $B \subset \mathbb{R}^3$ that does not intersect $S$, and whose boundary sphere is tangent to $S$ at $x$. It is maximal if it is not contained in any other such ball.

Let $\alpha$ and $\gamma$ be two geodesics which emanate from $p \in S$ in distinct directions and then meet again at $q \in S$.

I wish to show that there exists some point $z$ on either $\alpha$ or $\gamma$ such that $$ \rho(z) \leq \frac{\ell(\alpha) + \ell(\gamma)}{2\pi}, $$ where $\ell(\alpha)$ is the length of $\alpha$.

If for example $\alpha$ were the shorter geodesic and it contained a point conjugate to $p$, then our bound would follow from a standard result that relates the length of $\alpha$ to a bound on the square root of the sectional curvature it sees (the local reach is bounded by the radii of the osculating balls).

Also, if $\alpha$ and $\gamma$ together form a geodesic loop, then the result follows by considering the curvature of that loop as a space curve ( ... I asked for help with that too).

These two observations allow us to show that at least somewhere within a geodesic disk centred at $p$ and of radius bounded by the injectivity radius at $p$, we can find a $z$ satisfying the inequality, but I want more. I want to find such a $z$ on one of these two arbitrary geodesics.

This question came up within the context of surface sampling, where a popular means of defining the sampling density is through a function that is bounded by the local reach. We worked around it, but I'm still curious, and hope it might provide insight.

Comment by Ramsay

2012-07-25T07:53:57Z

If you have the Voronoi diagram then farthest point sampling (Delaunay refinement) seems like the obvious choice. The farthest point from a station will always be a Voronoi vertex, (or a point where a Voronoi edge intersects your quadrilateral boundary, or a corner of your boundary). Insert a new "station" at that point, and continue until the farthest point is close enough to a station.

Comment by Ramsay

2012-07-07T15:03:12Z

Could we not graft a cylinder (with $z$-axis) to the $xy$-plane (with disk removed) in such a way that no geodesic can self-intersect?

Comment by Ramsay

2012-04-09T11:13:32Z

An update to jc's comment above: Ken Baker made a subsequent post that describes a deformation retraction: sketchesoftopology.wordpress.com/2010/06/23/…

Comment by Ramsay

2012-02-25T14:19:35Z

Another relevant blog post: scottaaronson.com/writings/bignumbers.html

Comment by Ramsay

2012-02-02T14:05:37Z

I guess in the definition of a cone you meant to say that $r\alpha$ is the distance from $\xi$ to $\zeta$ in $\Sigma$?

Comment by Ramsay

2011-11-02T19:07:11Z

The rigidity of the punctured torus is listed as an open problem (number 13) by Ghomi here: people.math.gatech.edu/~ghomi/Papers/op.pdf . I guess either the list is wrong, or I am missing a distinction between things I don't understand.

Comment by Ramsay

2011-05-22T15:45:30Z

D. Glickenstein has done some work that might interest you, "Combinatorial Yamabe flow in three dimensions" arxiv.org/abs/math/0506182

Comment by Ramsay

2011-05-05T11:41:55Z

de Boor's "An empty exercise" on vacuity in linear algebra: ftp.cs.wisc.edu/Approx/empty.pdf or here: dx.doi.org/10.1145%2F122272.122273

Comment by Ramsay

2011-05-05T11:38:33Z

There is a neat article by de Boor [An empty exercise](ftp.cs.wisc.edu/Approx/empty.pdf) ([doi](dx.doi.org/10.1145%2F122272.122273)), where he argues for the inclusion of $0\times n$ and $m \times 0$ matrices in Matlab. He discusses definitions of span and determinant etc. I found this from a link on the Wikipedia page for determinant.

Comment by Ramsay

2011-04-23T13:10:42Z

A thoughtful answer. There is an arithmetical error on the first page of the short article linked at the end of the post. It is quite amusing in its context, so I am not entirely sure it isn't intentional.

Comment by Ramsay

2011-04-23T10:03:49Z

I heard this very anecdote for the first time two days ago. Strange coincidence. Even if it is apocryphal, it does provide an amusing caricature of the Bourbaki philosophy.

Comment by Ramsay

2011-04-17T01:04:41Z

This metric is sometimes used in constructing an approximating inscribed polyhedron for the surface. There are some optimality results, e.g. K. Clarkson (2006) "Building triangulations using epsilon nets" and the works of P. M. Gruber cited therin.

Comment by Ramsay

2011-02-18T17:04:30Z

By smaller I mean that $\rho_K(x) \leq \rho_G(x)$ for all $x$, where $\rho_G$ is the original definition. They agree everywhere except when the curvature is negative and $\rho_G$ provides no bound. I would be surprised if you found this to be a useful way to capture the geometry of negatively curved surfaces. Since it is purely intrinsic, you will never be able to control triangle normals this way, for example. It is not clear to me what is represented by the bound you're proposing to introduce. I don't recall seeing it anywhere previously.

Comment by Ramsay

2011-02-18T09:15:52Z

In a response to his own question here: mathoverflow.net/questions/31222/… Joseph O'Rourke linked to a paper by Zalgaller: springerlink.com/content/hu76g212137g2864 which describes a PL embedding of a flat torus in $R^3$. I am curious to know where that paper sits in the line of successive rediscoveries.

Comment by Ramsay

2010-12-07T19:33:19Z

A topical preprint has been posted on ArXiv (asserting that $S^6$ has a complex structure): front.math.ucdavis.edu/0505.5634