# Tagged Questions

Questions about understanding a problem geometrically.

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### Interpreting results of matrix transform

This is my first post on Mathematics Exchange, so I hope you'll be easy on me! I'm trying to project points in one 2-d coordinate space into another 2-d coordinate space using a simple matrix ...
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### Interpretation of this linear application

If $r$ is a unit vector, the reflection with respect to the hyperplane of normal $r$ corresponds to the matrix $I-2rr^\top$ (known as a Householder matrix). Now, let $r_1,r_2$ be two unit (column) ...
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### Is $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$?

I was reading a little about how to imagine the projective plane and I have some weird intuition that says $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$. Is this true, and if ...
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### Adding two functions represented by a table of values with a different step size?

Let $f(t)$ be some numerically obtained $T$-periodic function represented by a table of values over one period or a set of points $(t, y)$ with a time step $\Delta t.$ Now let's change the frequency/...
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### How to find the closest point to three vector lines?

So this is the question here I know the angles $A$ and $B$ for each individual, and their positions in longitude and latitude (assuming height of person $z =0$), am I correct in thinking that for any ...
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### What is the interpretation of homogeneous line intersection?

I understand homogeneous coordinate systems. I read the intersection of lines in homogeneous coordinate can be computed by taking a cross products of lines $l_1(a_1,b_1,c_1)$ and $l_2(a_2,b_2,c_2)$. ...
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### Covering maps as bundles

One geometric way to see a continuous map (or any set function really) is as a "fiber bundle" with the usual picture of a comb - the base space indexes the fibers of the map and there's a nice picture ...
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### Pulling back along surjective étale maps vs being “locally in $\mathcal M$” vs being “locally in $\Sigma \mathcal M$”

(Closely related) This question centers around section 6.5 of Borceux and Janelidze's Galois Theories. Definition 1. Let $\mathcal M$ be a class of arrows in a category (in our case $\mathsf{Top}$). ...
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### What is geometrical interptetation of a set being measurable.

What is geometrical interptetation of a set being measurable. I mean what does it mean geometrically by a set is measurable...
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### Is there a geometrical interpretation of this equality $2\cdot 4\cdot 6\cdot\ldots\cdot(2n)=2^nn!$?

$$2\cdot 4\cdot 6\cdot\ldots\cdot(2n)=2^nn!$$ How it can be seen in a plane? I have found many proofs with by induction but I wish to understand it geometrically.
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### Relationship between gradient and velocity?

I recently learned that when you have a real-valued function $f(x,y) = c$, you create a plane parallel to the $xy$-plane. Inside that plane we have a level set, such that the derivative of its ...
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### Geometrical representation of speed?

I've been learning about position vectors, and how their derivatives show the velocity (first derivative), and acceleration (second derivative) of a moving body. From Mechanics I learned that, the ...
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### Question about geometric interpretation of modules

I would like to understand the accepted answer to this MO question about the geometric interpretation of modules. In particular, I would like clarification on the following excerpt. Let $R$ be the ...
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### Geometric meaning of vanishing of higher cohomology of quasi-coherent modules over affine schemes

One of the basic vanishing results about quasicoherent (sheaves of) modules over affine schemes is that their non-zero cohomology vanishes. My only geometric intuition for sheaf cohomology is via ...
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### What is the geometric implication of subtracting two Matrices representing linear transformations?

If we have two linear transformations denoted by matrices $A, B$ operating on an arbitrary vector $v \in \mathbb R^n$, then how does $Av$ and $Bv$ differ geometrically from $(A-B)v$ ? Does the ...
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### Whats the terminology for defining a point on a triangle?

There are 2 common methods of representing a point on a 2d/3d triangle. 2 numbers (often called "UV coordinates" in 3D graphics):Where 2 edges of the triangle are axes, which the point is translated ...
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### How to resample translated grid to ensure consistent interpolation?

I have a grid of values, sampled at certain locations. I'd like to translate grid by some offset and resample it. The questions is: what should be the resampling and interpolation formulas such that ...
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### Interpretation of $a+b \ | \ a^n + b^n$ for odd $n$

It is not hard to show that $a+b \ | \ a^n + b^n$ for odd $n$. (because $f(x) = x^n - b^n = (x-b)h(x)$ we have $a - b \ | \ a^n - b^n$, so $a - (-b) \ | \ a^n - (-1)^n b^n$) Is there a nice ...
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### How to mathematically model a realistic aperture illumination?

I want to know a mathematical expression that i can use to model a realistic aperture illumination to produce the primary beam of an antenna so that the radial distribution of this aperture ...
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