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0
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2answers
32 views

Show that a curve which is perpendicular to a vector exactly twice is the unknot

If I know that the tangent vector $T(s)$ of $\gamma(s)$, a smooth closed curve in $\mathbb{R}^3$, is perpendicular to some vector $\vec{v}$ exactly twice, say at $s_0$ and $s_1$, how can I show that ...
2
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1answer
39 views

Proving that a proposed function is a Borel measure

Suppose $K$ is a fixed compact convex subset of $\mathbb{R}^n$. I wish to define a measure $M(K,\cdot):\{Borel subsets of \mathbb{R}^n\} \to \mathbb{R}$ where intuitively $M(K,A)$ (where $A$ is a ...
1
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1answer
42 views

Radon Transformation

I have been tinkering over the line segmentation of images. I found that it is very well implemented in matlab with the Hough algorithm. Now the Hough-transformation is just a special form of the ...
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0answers
26 views

Center of mass in a straight rod [duplicate]

I got an assignment to prove that in a straight homogeneous rod, you can always choose a coordinate system in such a way that $\int_S x_1dx_1dx_2=0 $ $\int_S x_2dx_1dx_2=0 $ $\int_S x_1x_2 ...
3
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1answer
66 views

A question from Selected Topics in Integral Geometry

I'm referring to Gel'fand, Gindikin, and Graev's Selected Topics in Integral Geometry, pages 4-5, section 1.4 (see here and here). Now in page 5 they write that: $$dx_1 dx_2 = d(\xi_1 x_1 +\xi_2 x_2 ...
3
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0answers
146 views

Integral Geometry Reference Request

I am looking for a good introductory reference (book, lecture notes, survey article) on integral geometry. I am especially interested in the Crofton formula in $\mathbb{R}^n$ and its extensions to ...
4
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1answer
607 views

The Cauchy-Crofton Formula

I am trying to understand the most basic formula from integral geometry. I have been looking at this website. The problem is things aren't working out for very simple examples. The circle works ...