# 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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### How to prove that the is A and a product of reflections R is an element of SO(V)?

Let $V$ be a finite-dimensional vector space $A$ be an ortogonal matrix and let $R:=R_1R_2 … R_n \in \mathrm{O}(V)$ be the product of $n$ reflections (it doesn’t matter if $n$ is even or odd). We know ...
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### What 'type' of reflection principle is used in ill-founed set theories, and how does it work?

Recently I have been studying the reflection principle and non well founded set theories. As of lately I have wondered what is there relation? and how does a reflection principle fit in a non-well ...
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### Does anyone know the Reflection method in solving PDEs?

I am looking at solutions for a diffusion equation, where we have: \frac{1}{4}u_{xx}=u_t , \ \ \ \ \ \ \, 0<x<\infty, t>0 \\ u_x(0,t)=0 \\ u(x,0)=\begin{cases} 6 \ \ \ \ 0 \...
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### explicit formula for hyperbolic translation

Let $S(a,r)$ be a sphere around $a \in \mathbb{R}^n$ of radius $r \in \mathbb{R}_+$. Assume further that this sphere is orthogonal to $S^{n-1}$, i.e. $r^2 = |a|^2-1$. Let $\sigma_a$ be the reflection ...
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### Reflection principle: can any property of a set (model) be reflected?

Can any property of a set be reflected? If no, why? Is there some sort of requirement that must be fulfilled? I've read on the Wikipedia page https://en.m.wikipedia.org/wiki/Reflection_principle ,that ...
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### Eigenvectors of linear transformations: Reflections vs Rotations

I'm curious why reflections can have real eigenvectors/eigenvalues whereas rotations always have imaginary numbers. The two linear transformations seem similar to me in spirit so this difference is ...
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### For the reflection $\phi$ in the sphere $S(a,r)$ it holds that $\phi(B^n) = B^n$ if and only if $\phi(a*) = 0$

This is a question about Theorem 3.4.2 in "Geometry of discrete groups" from Beardon. Let $\phi$ be the reflection in a sphere $S(a,r)$ for $a \in \hat{\mathbb{R}^n}$, $r \in \mathbb{R}$. ...
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### A matrix with inner product that respects norm is a reflection matrix

Let $\langle \cdot | \cdot\rangle$ be the standard inner product over $\mathbb{R}^{2}$ and let $A \in M_{2}(\mathbb{R})$ be a matrix such that $\|Av\| = \|v\|$ for every $v \in \mathbb{R^{2}}$ and ...
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### Order of Reflection Group of Type $A_n, B_n, C_n$

The following exercise comes from section 2.11 in Humphreys' book Reflection groups and Coxeter Groups: Use the method of this section to derive again the orders of the groups of types $A_n$, $B_n$, ...
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### Simple Reflections on Simple Roots

I have two related questions concerning simple reflections and simple roots. Let $\Phi$ be a root system for a reflection group $W$, let $\Pi \subset \Phi$ be a positive system, and let $\Delta$ be a ...
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### Using the linear algebra formula to prove homotopic through reflections.

I am trying to understand the solution of the following problem: Show that any two reflections of $S^n$ across different $n$-dimensional hyperplanes are homotopic, in fact homotopic through ...
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### When the graph of $y = f(x)$ is reflected in the line $y = x$, the number of invariant points is

What are the invariants? Would the invariant points be where the points of reflected graph and original intersect?
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### Issue to plot reflection coefficient and transmission coefficient in matlab

I want to plot the reflection coefficient and transmission coefficient of an incident EM wave to a Dielectric layer. I wrote the below code but it does not work properly. I highly appreciate it if ...
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### Any two reflections of $\mathbb{R}^n$ are conjugate in $I(\mathbb{R}^n)$.

Let $H_1,H_2$ be arbitrary hyperplanes in $\mathbb{R}^n$ where $H_1 \cap H_2 \neq \{ \}$ Then, reflections $R_{H_1},R_{H_2}$ satisfies $R_{H_1}R_{H_2}=R(h,\alpha)$ where $h$ is some $n-2$ ...
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### Finite reflection group: Reflections with the same reflecting hyperplane are equal?

Let $V$ be a finite-dimensional vector space over a field of characteristic zero. We call a linear map $s:V \rightarrow V$ a reflection if $s^2=id_V$ and the set of its fixed points $V^s$ is a ...
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### Representing reflection matrix as rotation matrix

I have a reflection matrix that represents the orientation of a particular domain. The determinant of this matrix is (-1). Is it possible to rewrite this reflection matrix in the form of a rotation ...
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### Conditions for finiteness of a reflection group

Given a subgroup $G$ of $O(V)$ that is generated by reflections ($V$ a Euclidean space), under what conditions can we determine that $G$ is finite? I realize that if the angles between any two roots ...
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### An artist needs help from mathematicians! Angles of reflections: should I paint these distant trees in the water's reflections?

So forgive the unfinished work(the first rule of being an artist is to NEVER show unfinished work...) But, I find myself wondering if I would see the reflections in the little pond I'm getting ready ...
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### Coordinates of the reflection of a point across a line of specific angle

I am going through an exercise to derive the point $P_1$ which is a reflection of of $P_0$ across line $D$. The following diagram (I couldn't make it accurate for the post, I am sorry) shows the idea: ...
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### Informally constructing the reflection numbers

Motivation The complex numbers allow us to model rotations in the plane as multiplication of the elements in it. A more fundamental operation in the plane is that of reflection, since we can write ...
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### If two linear reflections $s, t$ has the same $(+1)$-eigenspace, then their duals have the same $(-1)$-eigenspace

Let $V$ be a two-dimensional real vector space and let $s, t$ be two linear reflections on $V$ with the same $(+1)$-eigenspace. I would like to show that the dual maps $s^*$ and $t^*$ in the dual ...
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### How to get the reflections, what's the technique for that?

So i have this question, and they have solved it like this, now i don't get the reflections like how they got those, can anyone help me understand that: Q) Determine the number of nonequivalent ...
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### what happens in each stage of the render pipeline in the Phong lighting model? [closed]

I asked this question to understand the specular reflection. I read from this website to understand phong shading. But didn't understand Problems with Phong shading from that sites which are given ...
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### Optics of convex mirrors

I recently got in an argument with a friend while I was driving. When I moved to a different lane, I did not turn my head to check for any vehicles in my blind spot. Instead, how I checked was by ...
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