# Questions tagged [reflection]

Reflection is a transformation that fixes a line or plane or a more general subset. Reflections appear in geometry, linear algebra, complex analysis, differential equations, etc -- therefore, this tag must be used with a tag describing the area of mathematics.

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### Find Reflection of A point with respect to a line mirror in 3D

I need to find the reflection of point $P(1,2,3)$ w.r.t line mirror $(x-1)/2 =(y-1)/3 = (z+1)/1$ I know one method to do it i.e by first finding the foot of perpendicular of P on the line by using ...
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### what is the slope of a line that 1) can rotate around a point, and 2) its reflection from a circle has a specific direction

I have a question. The figure of the problem: I have a line that intercepts a circle. the line (in vector form) has equation i + td , where i is the direction of the line, d is on point of the line ...
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### Why Two line coincide in the product of reflections

Example. Let $l$ and $m$ be two different lines intersecting at a point $P$. Show that $R_m R_l =R_{p, \theta}$ where $B$ is twice the directed angle $\alpha$ from $l$ to $m$. Solution. Let $h$ be ...
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### invariant space, complex reflexion group

I am reading an article written by Pavel Etingof: "Symplectic reflection algebra, Calogero-Moser-Space and deformed Harish Chandra homomorphism". I am trying to figure out the isomorphism (4....
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### Are all the elements of order $2$ in a symmetric group reflections of some object with respect to some axis of reflection?

Are all the elements of order $2$ in a symmetric group reflections of some object with respect to some axis of reflection? I know that reflections are of order $2.$ Is the converse true? Any help ...
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### Is there a particular order to doing reflections when there is an inverse reflection (about the line y=x) involved?

For example, the original function would be y = f(x) and the transformed would be x = -f(y) If an x-intercept on the original graph was (-2,0) depending on which function was done first, the resulting ...
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### Mirror reflection and orthogonal transformation

I know a orthogonal transformation can be represented by the product of a series of mirror reflections. But I meet a question which requires one to show that the number of the mirror reflections in ...
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### Number of ways to arrange N objects of type A and N objects of type B

There are N objects of type A. There also N objects of type B. Consider random permutations of these 2N objects in to an array of size 2N. Let's call this array $A$. What is the number of permutations ...
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### On the reflection of a point $\vec{r}$ with respect to either the point $\vec{p}$ or to the line $\vec{l}=\vec{p}+t\vec{u}$: book recommendation

The reflection of a point $\vec{r}$ with respect to the point $\vec{p}$ is $\vec{r'}=2\vec{p}-\vec{r}$, satisfying $\frac{\vec{r}+\vec{r'}}{2}=\vec{p}$. Would you suggest any elementary book (...
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### Commutative reflections in the plane

How can i show formally that two reflections $S_v ,S_w$ in the plane commute iff $v=±w$ or v and w are orthogonal. I tried and searched a lot, but did not manage to prove this.
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### Find $AD\cdot AF$ when point $C$ is reflected across $AD$

$ABC$ is a triangle with side lengths $AB=AC=20$ and $BC=18$. $D$ is a point on $BC$ such that $BD<CD$. $C$ is reflected across $AD$ and lands on $E$. $EB$ and $AD$ meet at $F$. Find $AD\cdot AF$. ...
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### Basic derivations of reflection groups in positive characteristic

Orlik and Terao found the basic derivations for the irreducible unitary reflection groups in their book Arrangements of Hyperplanes with the statement: Since a finite group of unitary transformations ...
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### $f$ is analytic in $|z|\leq 1$ and $|f|=1$ when $|z|=1$ then $f$ is rational [duplicate]

This is a problem from Ahlfors' Complex Analysis "$f$ is analytic in $|z|\leq 1$ and $|f|=1$ when $|z|=1$ then $f$ is rational" This is in the section of Reflection Principle, but I don't know what'...
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### How is SVD computing unitary square matrices for rank-1 matrices (Matlab)

Consider matrix $\mathbf X=[\mathbf x ~\mathbf x] \in \mathbb R^{D \times 2}$. Of course, $\mathbf X$ has rank-1. Background: $\bullet$The full Singular Value Decomposition (SVD) of $\mathbf X$ is ...