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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Higher-dimensional reflections

I would like to know if it is possible to find a hierarchy on reflections in the following sense: Let $V$ be a finite-dimensional euclidean vector-space with standard inner product $\langle\, \_ \,,\, ...
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Graph Transformation - reflection and right shift

If I have a function $f(x)$ and I then want to move it to the left by 2, I could represent this as $f(x + 2)$. Similarly, if I want to move the function to the right by 2, I could represent this as $f(...
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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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The exact number of Givens rotations required to describe an orthogonal matrix.

Let $U\in \mathbb{R}^{d\times d}$ be an orthogonal matrix, i.e., $U^T=U^{-1}$. Any orthogonal matrix can be written as a product of ${n \choose 2 }= \frac{n(n-1)}{2}$ Givens rotations, i.e., $U=R_1\...
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The Householder “Rank” of Orthogonal Matrix

Any vector $v\in \mathbb{R}^d$ defines a Householder matrix $H=I-2\frac{vv^T}{v^Tv}\in\mathbb{R}^{d\times d}$. Any orthogonal matrix $U\in \mathbb{R}^{d\times d}$ can be written as a product $U=H_1\...
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Questions about the reflection operator on an inner product space

Let $V$ be an $n$-dimensional real inner product space. Suppose $u \in V$ has norm 1, and define the reflection in the direction of $u$ as $$r_u(v) = v - 2 \langle v, u \rangle u = v - 2proj_{span(u)}(...
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2012 EGMO P7: Show that the lines $KH$, $EM$ and $BC$ are concurrent [ Proof Verification needed]

Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$. Let $K$ be a point of $\Gamma$ on the other side of $BC$ from $A$. Let $L$ be the reflection of $K$ in the line $...
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Let $A$ be $2\times 2$ Orthogonal matrix such that det$A= -1$. Show that $A$ represents reflection about the line in $R^2$.

Every matrix is corresponding to a linear transformation. Reflection is a linear transformation called $T$. So $T^2=I$. Also modulus of eigen values of $A$ is $1$.Since it has determinant $-1$ so odd ...
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Orthogonal matrix decomposition into rotations and reflection

Given an n-dimensional orthogonal matrix, how can one decompose it as a product of elementary rotations (rotations in orthogonal planes of rotation) and / or reflections, and then retrieve the ...
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Axis of reflection

Is there any way to get the equation of axis of reflection given two intersecting lines without sketching? Example question: The image of the line $p:\; y-2x=3$ is the line $q:\;2y-x=9$. Find the ...
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Set of possible reflections of a vector.

How many vectors can one construct by by reflecting a vector $b\in\mathbb{R}^d$ for $b\neq 0$? Reflections can be described by Householder matrices $H=I-2vv^T/||v||_2^2$. In other words, I'm ...
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Does this method work for reflecting over $x^2$?

I am investigating reflecting over any quadratic. In the graph, I have the simplest scenario (reflecting $y=0$ over $y=x^2$). My method was to use the tangent (red dotted line) find the intersect that ...
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Reflection of polar functions

For whatever reason, I am having a complete brain block; I would like to reflect the arithmetic spiral $$ r=\frac{\theta}{2\pi} $$ about the polar axis. This should be possible? But I can't quite ...
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Is it possible to reflect over non-linear functions like quadratics or cubics?

This is a problem I have been investing for a while now and I have come up with several ideas. For the purpose of simplicity I am taking $y=x^2$ as my example and reflecting $y=0$ over it. Take the ...
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How to find the coordinates of the reflection of the point $(0,1)$ in the line $y=mx$?

So I understand the perpendicular line is $y=-\frac{1}mx+c$ and I think the point of intersection is $(\frac{-1+m}{m^2}),(\frac{-1+m}{m^3}+c)$ based on my calculation but I think I have made an error. ...
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Reflecting over non-linear functions/curves

Currently I am writing an IA (Internal Assessment) for the IB. It is a build up of deriving general rules to reflect over any line with equation $y=mx+b$. One step further that my teacher told me to ...
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Can we derive an equation to reflect any point/line over any quadratic function?

This is a derivation for reflecting any point over any function with the equation $y=mx+b$. Is it possible to further this to apply to any quadratic function with the equation $y=Ax^2+Bx+C$?
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How to shift and compress a parameterized log curve

Suppose I have some nice way to create a log curve that I need for a certain task (here in Python). ...
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Reflection of a curve/point using graph transformations.

The original question is to reflect the curve $y^2=4ax$ about the line $y+x=a$. The general method to solve such a question is to consider the parametric coordinates of the given curve (in this case $(...
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Determine the matrix of the reflection over the plane $2 x_1 + x_2 − 2 x_3 = 0$ in $\Bbb R^3$

Determine the matrix of the reflection over the plane $2x_1 +x_2 −2x_3 = 0 \in \mathbb{ R^3}$ .
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What is the analytical form of the cylindrical wave appearing on reflection of a plane wave from a corner?

Consider a plane 2D wavelet moving towards a corner reflector with 120° opening angle with infinitely extended sides. The surface of the reflector has homogeneous Dirichlet boundary conditions imposed,...
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Reflect an angle around an arbitrary axis.

What is the formula to reflect an angle around an arbitrary axis? Let D be an arbitrary angle rotated counter-clockwise from the x-axis. Let ...
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Combination of reflection symmetries in $\mathbb{E^4}$

Is the combination between point reflection (https://en.wikipedia.org/wiki/Point_reflection) symmetry and hyperplane(or axial using Hodge duality) reflection symmetry (https://en.wikipedia.org/wiki/...
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Reflection by a curve

Consider point $A(x_1, y_1)$ and a line $y = x.$ After reflecting point $A$ over $y = x$ we get new point, $A'(y_1, x_1).$ Is there general approach to this? Which point we get after refleting it by ...
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Derivative of $e^x$ after geometric transformation

Inverse function of $f(x) = e^x$ is of course $f^{-1}(x) = \ln{x}.$ We have, by definition, $\frac{d}{dx}e^x = e^x$. In other words, $e^x$ in some sense describes slopes of tangent lines on a curve ...
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On higher dimension, does rotation matrix product some vector match reflection matrix product the vector? [closed]

When you reflect a vector with reflection matrix on 2 dimensional space, and 3 dimensional space, intuitively we know there's rotation matrix can make same result. But is it possible on higher ...
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General form of an orthogonal matrix involving angles, sines, and cosines

In $\mathbb{R}^2$, all orthogonal matrices are one of two forms: $$\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \text{ or } \begin{pmatrix} \cos\theta &...
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Extension of $f\in\mathcal{H}(\mathbb{D})$ that maps a circular arc to an analytic curve

Let $f\in\mathcal{H}(\mathbb{D})$ (i.e. it is a holomorphic function on the open unit disc) and $\gamma$ a (open) circular arc $\in \partial\mathbb{D}$ such that $f\in\mathcal{C}^0(\mathbb{D}\cup\...
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Given what a reflection matrix does on a subspace, find the subspace - Can't solve

I can't really solve this exercise I've been trying to solve for some time now. It goes like that: Matrix $R$ ($\in \mathbb R^{3\times3}$) is a reflection matrix, in relation to subspace $U$, $U=\...
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$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 ...
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Evaluating pool ball sidespin deflection vector

I'm working on a pool simulation and I want to predict the deflection vector after a bounce on the wall with side spin. I'm using $V_f = Reflect(V_i, wall\_normal) + \underbrace{(||V_i|| \times 0.3 \...
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Isometries: Find the vectors and matrices

We consider the isometries of $\Bbb R^2$. Let $\varphi$ be the rotation of $90^\circ$ (counterclockwise) around the point $\begin{pmatrix}3 \\ 5\end{pmatrix}$ and let $\psi$ be the reflection about ...
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Extension (or “counterexample”) of the Schwarz reflection principle

The usual statement for Schwarz reflection principle on a general analytic arc is something along these lines: Given a holomorphic function $f$ on a region $\Omega$, assume that The function has a ...
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How to reflect the matrix Y = WX

I'm trying to reflect this matrix in order to find a value in X. $Y=WX$ means $y' = w_0, + w_1 * x$ And I would like to find the vector form for $ x' = \frac{y -w_0}{w_1}$ I'm trying with $X = YW^{−...
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Classification of reflection operators

How do you call a reflection operator whose matrix $R\in\mathbb R^{n\times n}$ is a diagonal of $\pm1$, i.e. the operator flips the signs of some coordinates? For example in dimension 3: $R = I;\quad ...
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How to find reflection matrix about a line x=a(ex reflections matrix of x=1)

I want to know how to make reflection matrix The reflection about line x=a for example i want to know the reflection matrix about line x=1, in homogeneous coordinate
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Dihedral subgroup of a infinite Coxeter group

I have seen a conclusion that every infinite Coxeter group contain an infinite dihedral subgroup, but I have no idea how to prove it. Could anyone give me some hint?
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The rank of a reflection subgroup of a finite Coxeter

Suppose $W'$ is a reflection subgroup of a Coxeter group $W$.I have seen some examples that the rank of $W'$ can be larger than that of $W$. But in those examples $W$ is an infinite group. I guess ...

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