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: \begin{equation} \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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Reflect a point about a given plane $Ax + By + Cz = D$

Write a program to find the reflection of an polygonal object (for example, input a triangle or a rectangle) in $\mathbb{R}^3$ with the standard inner product about a given plane $ax + by + cz = d$. $(...
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Angles of the Fundamental Alcove (Chamber?)

I am trying to calculate the angles of the fundamental alcove (chamber?) for the root systems of type $B_2$, $C_2$, and $G_2$; the fundamental alcove (chamber?) forms a triangle in these cases, so I ...
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Reflection Group of Type $D_n$

Here is the description of the reflection group of type $D_n$ in Humphreys' book Reflection Groups and Coxeter Groups: ($D_n$, $n \ge 4)$ Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of ...
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Reflection Group of Type $C_n$

In Humphreys' book Reflection Groups and Coexeter groups, he defines a group of type $B_n$ (for $n \ge 2$) in the following way: Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of all vectors of ...
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How to find a formula of a reflection in hyperbolic geometry.

I'm looking for a hyperbolic geometry formula for a reflection. However, I only know how to find a reflection point and not a formula, so I'd appreciate some assistance. Let $B$ be the quadratic form $...
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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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Eigenvalue of Householder reflector in $\mathbb{C}^{m \times n}$

I know that the eigenvalues of the Householder reflector in $\mathbb{R^{m\times n}}$, $H=I-2qq^T$, is $\pm 1$. But I have no idea whether this statement is true for the Householder reflector in $\...
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Prove that $``ff^{-1}(x)$ $=$ $x$ $=$ $f^{-1}f(x)"$ $\implies$ $``$the graph of $f$ and $f^{-1}$ are reflections of each other in the line $y = x"$.

According to the Cambridge International AS & A Level Pure Mathematics $1$ book $(2019$ edition, page $48)$, The graph of $f$ and $f^{-1}$ are reflections of each other in the line $y = x$. This ...
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How to construct the centroid of a quadrilateral

I know how to construct the centroid of a quadrilateral as mentioned here. But my question is different from that. We know that if points B,C,G are given in geometry plane and for locating point A ...
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About another proof of Witt's theorem

This is a proof of Witt's cancellation theorem from Uzi Vishne's book (I wrote it in my words (in english) so if there is anything that does not seem accurate please tell me). Witt's cancellation ...
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Are reflection subgroups corresponding to closed root subsystems always parabolic?

In the third paragraph of this reference, the following is stated: let $W$ be a Coxeter group with set of roots $R$, and let $H$ be a subgroup of $W$ generated by reflections (i.e. by conjugates of ...
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Is this converse results of Varignon's theorem known?

The Varignon's Theorem on Quadrilateral is very well known results of Plane Geometry and we have find the Converse of this theorem on Quadrilateral and generalise this for 2n-sided convex irregular ...
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Analogues of circle inversion for general conics

I saw a picture of a small object near the edge of a circle along with its circle inversion, and it looked a lot like a reflection. That’s when I remembered that they’re both anti conformal ...
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Reflection across an Ellipsoid

A reflective ellipsoid is given by $$ \dfrac{x^2}{16} + \dfrac{y^2}{9} + \dfrac{(z - 2)^2}{4} = 1 $$ A light source, emitting rays in all directions, is placed at $A=(10,4,3)$. Find the point $C=(x,y,...
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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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What's the consistency strength of these term allowing reflective theories?

Working in mono-sorted first order logic with equality and membership: Define: $set(x) \equiv_{df} \exists y \, (x \in y)$ Axiomatize: Extensionality: $(\forall x \, (x \in a \leftrightarrow x \in ...
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Analytic solution to differential equations with reflected arguments

I am curious about whether or not there are any techniques to find solutions to functional differential equations where the arguments of $y,y',...$ are only either of $\pm x$ (which could be ...
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How to prove r = n * 2(l·n) - l in specular reflection?

I asked this question where I understand all basics concepts of specular reflection. From that question I read the reflection of the vector (r) across a normal...
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Can Bernays reflection scheme be formalized in mono-sorted FOL?

Working in mono-sorted first order logic with equality $``="$ and membership $``\in"$: Define: $set(x) \equiv_{df} \exists y \, (x \in y)$ Axiomatize: Extensionality: $( a \subseteq b \land b \...
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Show that there exists a matrix $A$ such that $R_a(x)=Ax$ for all $x\in \mathbb R^n$.

Let $a(\neq 0)\in \mathbb R^n$. Let us define the reflection operator $R_a=x-2\frac{x.a}{a.a}a$. Show that there exists a matrix $A$ such that $R_a(x)=Ax$ for all $x\in \mathbb R^n$ and prove that $\...
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Is this simple reflective separative theory inconsistent?

Working in mono-sorted first order logic with membership: Define: $set(x) \equiv_{df} \exists y \, (x \in y)$ Define: $a \approx b \equiv_{df} \forall x \, (a \in x \leftrightarrow b \in x)$ ...
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Is separation + subset.reflection on transitive sets strong enough to go beyond ZF?

This question is related to this comment, and to this follow up posting. Working in mono-sorted first order logic with equality and membership: Define: $set(x) \equiv_{df} \exists y \, (x \in y)$ ...
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Can ZFC prove reflection on stages of the cumulative hierarchy whose size is equal to their height?

Working in first order logic with equality and membership, add the following axiom schemata: Specification:$(\exists! i:i=\{x \in A \mid \phi\})$; if $``i"$ not used by $\phi$. Reflection:$(\phi \...
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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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Equidecomposability of arbitrary subsets of the plane

I'm currently teaching a segment on the Bolyai-Gerwein theorem, and I was asked about the following question: Suppose I have measurable sets $P,Q\subset\mathbb R^2$ such that there exists sets $P_1,\...
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Reflect a curve about a point

What is the equation of reflection of the curve $$ y=5x^{2}-7x+2 $$ about the pont (3,-3)? The answer is explained as follows; To get the reflection about origin , the x and y co-ordinates have to be ...
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Reflection in 3 Dimensions

Say you have a set of $N$ points in $\mathbb{R}^3$ with the centroid at origin with fixed distance between the points (assume a rigid body constraint). Assuming the centroid to be fixed at the origin, ...
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Derived functor of a reflection (BGP) functor

I'm studying the book "quiver representations and quiver varieties" of Kirillov and I'm in Theorem 3.10: The functor $\Phi_i^+$ is left exact. Moreover, $R^n\Phi_i^+(V)=0$ for all $n>1$, ...
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Three lines and three rotations using reflections.

The questions are from Naive Lie Theory by John Stillwell, problem 1.5.2.-1.5.3.: My main question is about 1.5.3., but I believe I should explain how I understood 1.5.2. For example, if we want to ...
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