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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 would one classify point groups?

By point group I mean a finit subgroup of $\mathrm O(\Bbb R^n)$. Lists of point groups for some small dimensions are found on Wikipedia, but I am not certain about their completeness. As there seem ...
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Euclidean geometry problem from Euclidean and Non Euclidean Geometry: An Analytic Approach by Patrick J Ryan

Let $l=P+[v]$ be a line. Let $m=Q+[v]$. Show that if $|v|=1$, then $Ω_{l}Ω_{m}=τ_{w}$, where $w=2<P-Q,V^{⟂}>V^{⟂}$ and $Ω_{m}Ω_{l}=τ_{-w}$ I honestly have no idea how to start this. I know the ...
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The condition for the existence of a symmetric form for the reflection formula $f(1-x)= \chi (x) f(x)$

Suppose we have a functional equation in the form $$f(1-x)=\chi (x) f(x)$$ with given function $\chi (x)$. What is the condition on the function $\chi (x)$ so that we can write this reflection ...
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Reflecting x-axis over a line of tangent of a function

I learned about linear transformations of vectors and shapes during the linear algebra class. Amongst all the transformations, I saw the reflections over x and y axis by multiplying certain matrices. ...
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2 vector of equal length given how to find vector about which one vector reflected to that other?

Let $ w\in \mathbb R^n$ be vector of length $1$. $U$ is orthogonal space $w^\perp $ The reflection $r_w $ about $U$ is defined as follows if $v=cw+u$ , $u\in U$ then $r_w(v)=-cw+u$ Let $ u ,v$ be ...
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Reflecting/Rotating a 2 dimension manifold in a 3 dimensional space

I have a convex tilted hexagonal figure (6 corners) in 3-D space (imagine a tilted hexagon floating in a cube). I acquire this object by applying a projection $Q$ onto a cuboid. So a point of the ...
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Understand reflections in the plane

A plane is spanned by the vectors $\vec{a}$ and $\vec{b}$. The angle between the mirror axes $\vec{a^\perp} $ and $\vec{b^\perp}$ is equal to the angle of the two vectors $\vec{a}$ and $\vec{b}$. Let'...
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Orthogonal reflection in the plane

Question I'm solving for part (f). I get (1,0,0) (0,1,0) (0,0,1) as my ordered basis for part (e). But how do I find the matrix that represents the orthogonal reflection in the plane for the ordered ...
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3answers
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how to find the point obtained by reflecting over a line?

Given the point $A=(-2,6)$ and the line $y=2x$,what are the coordinates of the point B obtained by reflecting A over the line $y=2x$ ? Can someone teach me how to solve this question please?
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1answer
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reflect a point over another point using matrix transformation

We know that if we want to reflect any point over an origin, i.e. $ O\left(0, 0\right) $, we can use matrix transformation like this $$ \left(\begin{matrix}x' \\ y'\end{matrix}\right) = \left(\begin{...
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Find the matrix A of the reflection in the line $\mathbb{R}^2$ that consists of all scalar multiples of the vector $(6,5)$.

My professor didn't go over this in class at all. I was looking online and found some solutions, but was not able to get a $2\times 2$ matrix in any of the examples. The formula I was using was $v (V^...
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Why the restriction to "elements of $V$' in the output of formulas used in Reflection axiom schema of Ackermann class theory?

Can we upgrade the Reflection axiom schema in Ackermann to the following: Modified Reflection axiom schema: if $\psi(y)$ is a formula that doesn't use the symbol $V$, in which only symbols $y,x_1,..,...
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2answers
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Matrices in $\operatorname{O}(n) \setminus \operatorname{SO}(n)$

Can any matrix $M \in \operatorname{O}(n) \setminus \operatorname{SO}(n)$ be written as $I_n - uu^T$ where $I_n \in \mathbb{R}^{n \times n}$ is the identity matrix and $u \in \mathbb{R}^n$, $||u||_2 = ...
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Fundamental domains in Reflection groups and Coxeter groups - by Humphreys

In this thm I do not understand how he is using the induction in item d). The step t=1 is clear. enter image description here enter image description here
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1answer
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A geometry problem from Malaysia Math Olympic 2018 [closed]

Let ABC be an acute triangle. Let D be the reflection of point B with respect to the line AC. Let E be the reflection of point C with respect to the line AB. Let T1 be the circle that passes through A,...
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3answers
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Find the $\angle ACB$ of $\triangle ABC$.

If $PC=2BP$, $\angle ABC= 45^\circ$, and $\angle APC=60^\circ$, find $\angle ACB$. All solutions are acceptable but please try solving using reflection of point $C$ through the line segment $AP$. I ...
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Maintain right handedness of a rotation matrix

I have a rotation matrix represented by $$\left[ \begin{matrix} xx & xy & xz \\ yx & yy & yy \\ zx & zy & zz \\ \end{matrix} \right]$$ which is also an ...
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3answers
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geometry - find the path a light ray must take to reach a destination with one bounce off a mirror

this seems like it should be simple, but i've run out of leads with a similar-triangles approach, and the algebraic approaches seem pretty daunting, so i'm asking for help. i'm working on a ...
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Good material on reflecting boundaries for stochastic processes

restricting attention to continuous time, continuous state space ( say $\mathbb{R}$) stochastic processes. Can someone point me in the right direction of how to one imposes reflecting boundary ...
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3answers
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Variation of the sum of distances

Let $l$ be a line and $A$ and $B$ two points on the same side of $l$. To find the point $P$ for which $AP+PB$ is minimum we take the intersection of $l$ and the line joining $B$ and the symmetric $A'$ ...
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1answer
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High school geometry problem: Reflect a vertex about opposite side.

Let $ABC$ be a triangle and $A'$ be the reflection of $A$ about $BC$. Let $M$ be the mid-point of $BC$ and $I$ be the point of intersection of $A'M$ with the circumcircle of triangle $ABC$ (See figure)...
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Why is this matrix not a reflection in a plane matrix?

$$ \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix} $$ This matrix clearly has det -1 and is orthogonal. Why is it not reflection in ...
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1answer
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How to represent reflection of a particle inside a standard simplex?

I am trying to simulate the trajectory of an evolutionary system represented by a vector of probabilities $\vec p = [p_1, p_2,...] $. Values are restricted between 0 and 1. As a result we can think ...
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1answer
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Linear algebra: Analytic geometry problem.

Let $p$ be a line given by the equation $x-1=\frac{y}{2}=\frac{z+3}{2}$, and $q$ a line with the equation $\frac{x}{2}-1=y-2=\frac{z+1}{2}$. If we reflect the $p$ over the plane $\Pi$ we get the line $...
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Higher dimensional version of “a product of two reflections is a rotation”

In higher dimensions, which orthogonal matrices are a product of two reflections? Is it all of $SO(n)$? In the complex case is it $U(n)$?
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Conjugacy classes of stabilizer subgroups

Let $G\subseteq GL(V)$ be a complex finite reflection group. I would like to understand the stabilizer subgroups of $G$, and their normalizers. By this I mean (the conjugacy classes of) subgroups $H&...
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2answers
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Matrix of reflection in $R^3$

Please, can you explain me how do we get this formula $$ A = I - 2nn^{T} $$ in $$ R^{3} $$? This should be matrix of reflection, but I don't know how to prove that.
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How do you reflect a graph about an arbitrary line for a precalculus student?

Here's a question one of my precalculus students asked me, paraphrased for clarity: You know how if you have the line $y=x$, and you want to reflect the graph of a function $f(x)$ across it, you ...
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1answer
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Convolution Notation: The difference between (f*g)(x) and f(x)*g(x)

What is the difference between (f*g)(x) and f(x)*g(x) [1] for convolutions? Are they the same? I ask this because I have been asked to prove the Reflection of Convolution property for my course in the ...
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1answer
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Hermitian Property of a Householder Transform on a Complex Field

Let $\mathbf{H}$ be a Householder (i.e. elementary reflector), such that $\mathbf{Hx} = \mathbf{e}_1$, for an $\mathbf{x} \in \Bbb{C}^n$, having $\|\mathbf{x}\|_2 = 1$. For this I have defined $\...
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Restriction of Bruhat order to stabilizer of a vector

Let $W$ be a Weyl group (maybe better a Coxeter group, i.e. a group with action on vector space $V$ generated by reflections with some conditions). Consider $v \in V$ be a vector. Let $\text{Stab}_v \...
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Using quaternion algebra, determine matrix of reflection given a plane

The question asked us to find the matrix of the reflection on the plane with equation $x+2y-2z=0$. From what I have learnt, the (unit) normal to the plane is $[\frac{1}{3},\frac{2}{3},-\frac{2}{3}]$. ...
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Pixel coloring problem

We have an images $m$ pixels length and $n$ pixels width. There are at least $k$ pixels with different colors. Image and its reverse, reflect and rotation are considered to be equivalent. Permuting ...
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1answer
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What does “Reflection along the subspace generated by v” means?

I got a problem which includes "Reflection along the subspace generated by $v$ in $\mathbf{R}^{n+1}$". I need some clarification, what does it mean? Does it mean reflection about the hyperplane $v^{\...
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Reflection through a line

I need to find the matrix of reflection through line $y=- \frac 23 x$ . I'm trying to visualise a vector satisfying this. The standard algorithm states that we need to find the angle this line makes ...
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1answer
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Linear Algebra with Reflection Across Axis [closed]

Point $P(-2,5)$ is reflected on the line $y=\frac{2}{5}x$. How do you find the coordinates of the reflection using linear algebra and not the distance formula?
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How to characterize rotation and reflection in linear algebra

How can I prove this fact using Linear Algebra: A rotation is formed by the composition of two reflections in which the lines of reflection intersect. The composition of reflections over two ...
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How to show the product of two reflections is a translation?

If $\ l = P+ [v]$ and $\ m = Q+ [v]$ are lines. And $\ |v|=1 $, then $$\Omega_l\Omega_m= \tau_w$$ where $\ w= 2<P-Q,V^\perp>V^\perp $ and also $$\Omega_m\Omega_l=\tau_{-w}$$ My work: If $\ l$ ...
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If T is a reflection then $Ker (T-Id) = Im (T-Id)^{\perp}$

Let V be a finite dimensional space with inner product . If T is an orthogonal transformation and a reflection then $ Ker (T-Id) = Im (T-Id)^{\perp}$, where $Id$ denotes the identity matrix I know ...
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1answer
264 views

Normal of a point on the surface of an ellipsoid

Given an ellipsoid in the form: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$ and an arbitrary point $p$ on the surface of the ellipsoid, how can I compute the normal vector of the ...
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Finding a mapping from a segment of the real line to an ellipse that satisfies the given property.

Here's the exact problem statement as it was given: Consider the ellipse $x^2 + a^2y^2 = 1$ Find a holomorphic isomorphism from a neighborhood of 0 onto some neighborhood of $a^{-1}i$ which transforms ...
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How to construct ray reflection from convex curved surface

I'm trying to understand how to construct the reflection path of a ray from a curved surface. Here's the basic setup: In a 2D space, assume a point S is the source of a ray and point R is the ...
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1answer
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Find mirror point in unusual plane

I was working on an assignment and I ran into this weird curve-ball situation where part of it is to find the mirrored point of $P=(3,10,9)$ in the plane $x-z=0$. I can see the plane is basically the ...
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How to the reflection of a point about a line?

I've been doing some problems on the topic Reflection.And struggling with it when I am doing it graphically. I have solved it using ...
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1answer
29 views

Reflection operator and plane

Let $R:\mathbb R^3\to\mathbb R^3$ is reflection linear transformation on plane $\pi: x+2x+3z=0$: a)Find a matrix of linear transformation $R$ using base $B=\{v1,v2,v3\}$ where $v_1=(1,1,-1), v_2=(-1,...
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2answers
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Reflection vector doesn't match book solution

Let L be the line in R3 that consists of all scalar multiples of [2 1 2]. Find the reflection of the vector [1 1 1] in line L. I'm calculating this using ...
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1answer
60 views

Does proof of the fact that a line only intersects a conic section once imply that it is either a tangent to the conic section?

I found this proof of the reflection property of a hyperbola which is short and uses no algebra (https://www.geogebra.org/m/m6cz5fqR). However, the author says that since the line only intersects the ...
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1answer
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What is a Coxeter Group?

I've recently started investigating abstract algebra and have now stumbled upon "Coxeter Groups", which are a mystery to me. I've read that Coxeter Groups have something to do with reflections (in ...
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spectral theorem - why does it only apply to a symmetric matrix?

The real spectral theorem asserts that any symmetric matrix can be decomposed into a composition of rotations, reflections and scaling. Why can't a non-symmetric matrix be represented as such? Are ...
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2answers
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Reflection matrix - are these two definitions equivalent?

Are these two definitions of a reflection matrix Q equivalent (if and only if)? Definition 1: $Q^TQ = I$ and $det(Q) = -1$ Definition 2: $Q = I-2nn^T$ where $n$ is a unit normal vector to the ...