0
votes
1answer
68 views

How to define an “anisotropic vector” for a given object?

Dear experts, I am looking for a way to define an "alignment vector" (or anisotropy or orientation vector?) for a given geometrical object. I am not sure how to put this into correct technical terms, ...
1
vote
1answer
418 views

Is this min not less than a min

Let $\mathbf{D}$ be the unit disk, is $$\inf_{\begin{array}{c} v_{1},v_{2},v_{3},v_{4}\in\mathbf{D},\\ v_{0}\in\mbox{convexhull}\left(v_{1},v_{2},v_{3},v_{4}\right) \end{array}}\max_{0\le ...
8
votes
1answer
974 views

Naive definition of surface area doesn't work?

A first stab at a definition of surface area might go like this: Let S be a surface. Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the ...
0
votes
1answer
287 views

Minimum distance between two data sets

Suppose we have two sets of data, $X$ and $Y$, each of which contains $10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$, $x_{1}\ge\cdots\ge x_{10}>0$ and ...
6
votes
2answers
477 views

Computational complexity of integration in two dimensions

I have an algorithm for solving a certain problem that requires that I compute two-dimensional integrals as a subroutine, and I'd like to make some kind of statement about its running time. Suppose ...
0
votes
0answers
433 views

A product sum inequality question

For any $x_{1},x_{2},\cdots x_{6}$ with $\sum_{i=1}^{6}x_{i}^{2}=1$ and $y_{1},y_{2},\cdots y_{6}$ in $\mathbb{R}$ with $\sum_{i=1}^{6}y_{i}^{2}=1$, do there always exist $z_{1},z_{2},\cdots z_{6}$ in ...
1
vote
3answers
1k views

How do you calculate the solid angle of a rectangular, axis aligned section of a surface defined by a two dimensional function?

I have $f(x,y) = \frac{1}{2} (1 - x^2 - y^2)$, which is a paraboloid centered around the origin (plot). Now I want to calculate the solid angle (with the origin as the viewpoint) of the surface area ...
0
votes
1answer
206 views

What is the definition of product of ideal sheaves?

Each book on algebraic geometry write I^2 when it deal with nongsingular varieties, here I is a ideal sheaf. But no one give the definition. I guess it's the sheafification. It's right? Thanks.
7
votes
1answer
964 views

Burnside's Lemma and Geometry

I think one of the most interesting results in Elementary Group Theory is the so-called "Burnside's Lemma", counting the numbers of orbits of a (finite) group action. I wonder if there is any ...
0
votes
1answer
387 views

Generic coordinate system representations

Please excuse the verboseness which follows, as the question is rather basic, so I would like to state it carefully so that it will not be accidentally neglected as automatically trivial. If, after my ...
6
votes
2answers
1k views

Generalizing square wheels rolling on inverted catenaries

It is not uncommon to see in a science museum a bicycle with square wheels that rides smoothly over a washboard-like surface made from inverted catenary curves (e.g., at the Münich museum). The ...
11
votes
9answers
4k views

Geometric imagination of differential forms

In order to explain to non-experts what is a vectorfield, one usually describes an assignemnt of an arrow to each point of space, and this works quite well, also when moving to manifolds (where a ...
26
votes
2answers
2k views

Drawing of the eight Thurston geometries?

Do you know of a picture, drawing, or other concise visual representation of the eight three-dimensional Thurston geometries? I am imagining something akin to the standard picture (of a sphere, ...
5
votes
2answers
371 views

Triangles, squares, and discontinuous complex functions

Is there some onto function $f:$ $\mathbb{C}$ $\rightarrow$ $\mathbb{C}$ such that for each triangle $T$ (with its interior), $f(T)$ is a square (with interior, too) ? I would have the same question ...
10
votes
2answers
732 views

Geometrically interpreting the answer to a vector calculus question involving tangent line segments to ellipses.

Let E be an ellipse centered at the origin on the x, y plane with major radius b and minor radius a. The length of the shortest line segment tangent to E that begins on the x-axis and ends on the ...
5
votes
4answers
966 views

Coloring Points in the Plane

Suppose one wants to color the points in the plane so any two points at distance one apart are different colors. How many colors are needed? I heard this problem when I was a kid. Back then the most ...
5
votes
6answers
856 views

Database of polyhedra

As part of many hobbies (origami, sculpting, construction toys) I often find myself building polyhedra from regular polygons. I am intimately familiar with all of the Archimedean and Platonic solids, ...
7
votes
5answers
725 views

How can I sample uniformly from a surface?

Given an equation of a parametric surface, is there a general way to sample of points uniformly distributed on that surface? I'm interested in this problem for purposes of visualisation - rather than ...
2
votes
1answer
892 views

How to find the Fermat Point using the construction of the tangent to ellipse?

Be done the triangle ABC, it is known the method to finding the point Q that minimises the sum QA+QB+QC among all points Q in the plane (The Fermat point). I want a hint for solving this problem using ...
3
votes
4answers
870 views

How to partition R^3 into pairwise non-parallel lines?

Problem. How to partition R^3 into pairwise non-parallel lines? A possible solution is to stack infinitely many ``concentric'' hyperboloids, by increasing radius and decreasing slope. And don't ...