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Jan
7
comment Is this (self intersecting) surface considered one sided?
@Alex If we treat the given surface as parameterized by $f:[0,1]\times[0,1]\to\mathbb R^3:(s,t)\mapsto sc(t)$ "glued" along $f(s,0)=f(s,1)$, where $c:[0,1]\to\mathbb R^3$ is the input closed curve, then we can define the normal at any point $(s,t)$ on the parameterization by normalizing the vector $c(t)\times c'(t)$. This normal vector field is "consistent" in that the normals point in the same direction on both sides of the gluing (unlike the Möbius strip, where the normals necessarily point in opposite directions).
Jan
7
comment Is this (self intersecting) surface considered one sided?
I think this still counts as a two-sided surface, because you can define a consistently oriented normal field. In fact because you are connecting a closed curve to a point, this is just an immersion of a disk into 3-space (with the center of the disk mapped to the origin and the circumference mapped to the curve).
Jan
7
revised Is this (self intersecting) surface considered one sided?
inserted images
Jan
7
comment Any other Caltrops?
Are you sure all the faces of your polyhedron are planar? I believe the bottom-left two and bottom-right two in the leftmost image (i.e. the ones a knight's move away from the triangle) are not.
Jan
6
comment How to mathematically and/or programmatically find if an integer can't be written as a sum of defined integers?
Going by your own answer it sounds like you were asking about the Frobenius coin problem.
Jan
5
comment Why is the eigenvector centrality considered a generalized version of degree centrality?
This sentence from the Wikipedia article seems relevant: "Restricting consideration to this group [based on counting the number of walks starting from a given vertex] allows for a soft characterization which places centralities on a spectrum from walks of length one (degree centrality) to infinite walks (eigenvalue centrality).[2][5]"
Jan
5
comment How to find the equidistant middle point for 3+ points on an arbitrary polygon?
According to your edit, you're looking for the geometric median. However, the centroid as mentioned by Frentos is much easier to compute and may be sufficient for your application. P.S. the geometric median minimizes the total distance, while the centroid minimizes the total squared distance; minimizing the maximum distance is the smallest enclosing circle problem.
Jan
5
comment How to find the sum of the infinite series whose general term is not easy to visualize
@JMoravitz: The first term does fit the pattern: it's $1/6$ times the empty product.
Jan
5
comment What method can I use to determine continuity of squares on a 2d grid?
en.wikipedia.org/wiki/…
Jan
5
comment is this function an ill-shape convex function?
What is the sum over? Do you mean $\log\sum_i\exp\alpha_i$ instead? // Also if $D$ and $n_i$ are constant then the whole $N\exp(\cdots)$ thing is just a constant factor.
Jan
4
comment What does the semicolon ; mean in a function definition
"nestlings"?? Who approved that edit?
Jan
4
comment What is the probability density function of pairwise distances of random points in a ball?
In mathematics it is conventional to use "sphere" to refer to the surface and "ball" to refer to the enclosed region. You mean the points are selected from the latter, yes?
Jan
4
comment Equation of a 3D curve shaped like a logarithmic spiral
Your image $z = \sin(\theta + r)$ corresponds to the Archimedean spiral $\theta + r = 0$. So for the logarithmic spiral $\theta + \log r = 0$, try $z = \sin(\theta + \log r)$.
Jan
4
comment Is the point of a shape with the greatest average ray length also the “centroid”?
How is that an $n$-gon? It's just a Riemann sum.
Jan
4
comment Is the point of a shape with the greatest average ray length also the “centroid”?
Wait, I think I see the problem. You keep talking about measures over the curve. Why do we need a measure over the curve? We just need a measure over the space of directions emanating from $\mathbf p$, which is essentially the uniform measure on $[0,2\pi)$.
Jan
4
comment Is the point of a shape with the greatest average ray length also the “centroid”?
The fact that the harmonic mean of $\{0.1, 1, 2, 3, 4, 5\}$ is $0.49$ also conflicts with the idea that most of the numbers in the set are no less than $1$. Anyway, I think I'm done with this conversation. Have a nice day/night.
Jan
4
comment Is the point of a shape with the greatest average ray length also the “centroid”?
But the angular average from $\mathbf p$ is the entirety of the question; if you remove it there is nothing left. You are now speculating about the OP's intent after rejecting all of their express statements.
Jan
4
comment Is the point of a shape with the greatest average ray length also the “centroid”?
We don't want the mean of the distances to all vertices of the polygon. We want the mean of the distances over rays in different directions emanating from $\mathbf p$. That's the point.
Jan
4
comment Is the point of a shape with the greatest average ray length also the “centroid”?
You'll have to walk me through that logic. What exactly is wrong with $\mathrm d\theta$ here?
Jan
4
comment Does the constraint $\mathbf{w'1}=1$ make $\mathbf{w}$ an eigenvector of any square symmetric matrix?
From $\mathbf 1^T\mathbf M\mathbf w = k$ you can't get $\mathbf M\mathbf w = k\mathbf w$.