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I hereby authorize whoever it may concern to post my comments as answers if they wish, whether as community wiki or otherwise if you like imaginary internet points.


Jul
9
comment Why isn't the volume formula for a cone $\pi r^2h$?
Only one vertex of the triangle travels a distance of $2\pi r$. The other two travel $0$ distance. Surely it's more reasonable to take the average, $(2\pi r+0+0)/3$, which gives the volume to be...
Jul
8
revised Two-way matrix optimization
added 255 characters in body
Jul
8
comment How can I define a measure of similarity between two line segments in $\mathbb{R}^2$?
@This: That's a great suggestion, but your formula can't be right. The Hausdorff distance between the two diagonals of a unit square is $1/\sqrt2$ but your formula gives $1$. I'd believe that it's only off by a bounded factor though.
Jul
8
answered Two-way matrix optimization
Jul
8
comment Maximum number of vertices in intersection of triangle with box
A box-plane intersection forming a hexagon is shown halfway down this page, and one can easily truncate three of its vertices with a triangle.
Jul
8
comment Is chess Turing-complete?
The question is about a generalization of chess played on an infinite board as stated in the very first sentence.
Jul
8
comment Characterization of sphere.
@user1963: It's not. If you squash the plane down in the $y$ direction, lines parallel to the $x$-axis are now much more likely than lines parallel to the $y$-axis.
Jul
8
comment filling an occluded plane with the smallest number of rectangles
Thanks, that is indeed a productive search query. As for 3D, I'm familiar with BSP trees, but I don't see how one could use them to find the smallest partition rather than just any partition. If the answer is too long for a comment, I can ask a new question.
Jul
8
comment Number of binary trees with N nodes
A binary tree with $k$ internal nodes must have $k+1$ leaves and therefore $2k+1$ total nodes. Ergo, any binary tree with $n$ total nodes is a binary tree with $(n-1)/2$ internal nodes, and you already know how many of those there are. Edit: At least if you're talking about full binary trees where each node has either two or no children.
Jul
8
comment filling an occluded plane with the smallest number of rectangles
+1: I wish this answer had existed five years ago when I was dealing with a related problem! Two questions, if you happen to know the answer offhand: (i) Is the problem easier if the polygon is without holes? (ii) Do the algorithms generalize to higher dimensions?
Jul
8
comment How to slice an area in rectangles optimally?
Joseph O'Rourke has now posted a $O(n^{5/2})$ time solution (where $n$ is the number of vertices) at this duplicate question.
Jul
8
comment How to find the “average” direction of a set of vectors?
It's exactly the "direction of sum of vectors" approach which you said you didn't want. I provided the reference to show that it is a standard approach so you may want to reconsider not wanting it.
Jul
8
comment How to find the “average” direction of a set of vectors?
What about four vectors equally spaced at 90 degrees? What do you expect the average direction to be there? Further reading: mean of circular quantities
Jul
7
comment Non-uniform sampling of N-sphere
Replace $(x_1,x_2,x_3,\ldots,x_n)$ with $(x_1,\lambda x_2,\lambda x_3,\ldots,\lambda x_n)$, then normalize. When $\lambda=1$ you get the uniform distribution on the sphere. When $\lambda=0$ you get all the points at the poles.
Jul
7
comment Two-way matrix optimization
$U\mapsto A-UW$ and $U\mapsto RU-H$ are both linear operators on matrices, and the Frobenius norm is just the $L^2$ norm on the entries of the matrix. So you just have a quadratic minimization problem on your hands.
Jul
7
comment Length of a curve without function?
+1, but the triangles are a little superfluous. You could just say "pick a number of points on the curve, and add up the distances between them".
Jul
7
revised Distinguishing sets according to more fine-grained notions than cardinality.
spelling
Jul
7
comment Positive semi-definite of a matrix composed of semi-definite blocks
@Gal: If you are satisfied with the answer, you can accept it by clicking the check mark on the left.
Jul
7
answered Prove or disprove a statement about testing the convexity of a set using the vertices
Jul
6
revised Extrema homework — maximizing the viewing angle of a picture on a wall
edited title