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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Range of a standard brownian motion, using reflection principle

With a standard brownian motion $B_t$, I'm trying to find the distribution of the "range": $$R_{t} = \sup_{0 \leq s \leq t} B_s - \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The ...
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How to determine the reflection point on an ellipse

Here is my problem. There are two points P and Q outside an ellipse, where the coordinates of the P and Q are known. The shape of the ellipse is also known. A ray comming from point A is reflected by ...
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How to reflect a complex point A through complex point B?

I'm doing a math project and I was wondering if anyone could tell me the formula for reflecting a complex point B through point A? I know that across the x axis its the complex conjugate, but I wanted ...
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Reflecting coordinates over the line $x = -1$

I know how to reflect a coordinate over the $y$ and $x$ axis, but is there a rule I could use to help me find the reflected point over $x = -1$? This is what I know already: Over the $x$-axis: ...
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30 views

Finding the glide reflection using a compass or straightedge

Given two congruent triangles that are not a rotation, translation or reflection of each other; how can I find the glide reflection (the last remaining option) using only compass and straightedge. ...
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Possible eigenvalues of projection and reflection operators

What are the eigenvalues of an (orthogonal) projection operator that projects vectors onto some hyperplane passing through the origin? Well, for vectors $v$ orthogonal to this hyperplane, the ...
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Given $a,b \in S^n$, then there exists an isometry $f: S^n \rightarrow S^n$ such that $f(a) = b$

$S^n = \{x\in \mathbb{R}^{n+1} : \|x\| = 1\}.$ I am using this definition: isometry is a surjective function $f:M \rightarrow N$ between two metric spaces $(M,d)$ and $(N,\rho)$ such that $$\rho ...
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Isometries of reflections

Suppose $R_\rho \in M_2$ denotes a reflection across a line which is through the origin and at a anticlockwise angle of $\rho$ with the x-axis. The question asks that for any $n \in \mathbb{N}$, does ...
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Find Normal from Reflection

I have a vector A with direction $\vec D$. I also have another vector B with direction $\vec R$. (think of $\vec R$ to be the reflection of $\vec D$ at point B.) From the reflection formula:$$ \vec ...
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Linear algebra reflection or rotation

I have a matrix $$\begin{bmatrix}2&-\frac12\\1&1\end{bmatrix}.$$ How to define matrix is rotation or reflection?
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$s$ is a reflection of the plane about $x=1$. $r$ is a reflection across the x-axis.

$s$ is a reflection of the plane about the vertical line $x=1$. $r$ is a reflection across the x-axis. $g$ is an isometry and $grg^{-1}=s$. What is $g$?
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Express rotation of the plane about a point as the product of a translation and a rotation about the origin

I need help with the following problem: a) Let $s$ be a rotation of the plane with angle $\frac{\pi}{2}$ about the point $(1,1)^t$. Write the formula for $s$ as a product $t_a \rho _\theta $ Edit: ...
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Order of affine reflections (described with complex numbers operations)

Let be the affine reflection described as an operation with complex numbers : $$s_\beta,_\nu : z \mapsto \nu + \overline{\beta z},$$  where $z, \nu \in \Bbb C$ and $\beta \in \Bbb C^1 = \{x+iy \ ...
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35 views

Vector Reflection Matrix over line

I need to find a matrix that reflects vectors over any line y= mx + b. Furthermore, I need to be able to find the components of both of these vectors. Thanks!
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Regarding the Schwarz Reflection Principle, is getting the analyticity of f(z) on the real axis a consequence of the theorem itself,

or a consequence of Morera's Theorem? Basically, I want to be able to cite it correctly, e.g., can I say we have not only continuity of f(z) along $R$ (by assumption) but also it turns out that ...
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let X has a finite dimensional show that X is reflexive.

We know If a normed space X is reflexive, then $X'$ is reflexive.and also Reflexive normed spaces are Banach. but can you proof if X has a finite dimensional Then X is reflexive.
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Householder QR problem

Can somebody give me a hint or help me to solve this problem. Let V be a p×q matrix with orthonormal columns (p > q), and $M = I−2VV^T$ , with I being the p × p identity matrix. The matrix M can ...
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Find the Inverse Matrix of a Transformation

Let $ f: \mathbb R^3 \rightarrow \mathbb R^3$ be a linear mapping which reflects $\bar{x}$ over the plane $x_1+x_2+x_3 = 0$ . You are given the standard matrix for $f$ is: $$ ...
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Roots of height 1 are necessarily simple.

Suppose $\Phi$ is a root system, $\Pi \subset \Phi$ is a fundamental system (let $\Pi = \{r_1,...,r_l\}$). Now any root $r \in \Phi$ is a linear combination of the elements of $\Pi$ with all the ...
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Reflection through a hyperplane, orthogonality propoerty

While I was studying for my exam, I found Stanford's EE263 course's old homework questions and this particular one attracted my attention, however couldn't solve it. If you can help me, I will really ...
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105 views

Find a 3x3 Matrix that reflects in an Arbitrary Line y=mx+c

I want to find a 3x3 Matrix that reflects in an Arbitrary Line y=mx+c I have a matrix with me but it doesnt take into account the '$c$' This is the matrix \begin{pmatrix} \frac{1-m^2}{m^2+1}& ...
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1answer
86 views

Finding the reflection that reflects in an arbitrary line y=mx+b

How can i find the reflection that reflects in an arbitrary line , $y=mx+b$ I've examples where its $y=mx$ without taking in the factor of b But i want to know how you can take in the factor of $b$ ...
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1answer
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$PQRS$ is a convex quadrilateral.Find the area of $PQRS.$

Let $P$ be the point $(3,2).$Let $Q$ be the reflection of $P$ about the $x-axis.$Let $R$ be the reflection of $Q$ about the line $y=-x$ and let $S$ be the reflection of $R$ through the origin.$PQRS$ ...
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A subset of roots whose mutual angles agree with those of a simple system

I would appreciate help/hints solving the following exercise from Humphreys book "Reflection Groups and Coxeter Groups", page 11, exercise 1. Let $\Phi$ be a root system of rank $n$ of unit ...
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Can reflection across a line segment be done using the rational field?

Assume that I have a point and a line segment, all specified using rational coordinates. Can I compute the reflection of the point across the line segment using only rational numbers? This previous ...
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Every positive system of roots contains a unique simple system.

The following question is in effort to understand a proof to a theorem appearing in "Reflection Groups and Coxeter Groups" by Humphreys on page 8. Let $\Phi$ be a root system in the euclidean space ...
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Describing the symmetries of a $2n$-gon in $\Bbb R^2$ with matrices.

Problem: Consider a regular $2n$-gon in the Euclidean plane $\Bbb R^2$ centered at the origin $(0, 0)$ and with its $2n$ vertices equally distributed on the unit circle. Label the vertices from ...
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Proof of conformal mappings onto polygons (Stein)

Recently I am reading Stein and Shakarchi's Complex Analysis and I find a great difficulty in understanding the proof of theorem 4.6 in Chapter 8 (p.242-244), which talks about conformal mappings onto ...
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1answer
40 views

Linearity of Certain Reflections in $\Bbb R^n$

Let $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ be a reflection about a hyperplane passing through $\vec 0$. Is $f$ always a linear transformation? If so, how can the matrix of the reflection be ...
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$\mathbb{R}^3 \to \mathbb{R}^3$ transformation: reflection across a plane

Notation: $v$// is $v$ parallel symbol, $v\bot$ is $v$ perpendicular, and both are relative to plane $\sqcap$ Let $\sqcap \subseteq$ $\mathbb{R}^3$ be the plane whose equation is $x + y + z = 0$. ...
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How to reflect an object without changing its geometry

I have an object in which has its orientation defined by 3 vectors and a point and I need to reflect it across a plane without changing the geometry of the object (For example if there was text on the ...
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Schwarz reflection principle, understanding the conjugated function:

Given a symmetric region $\Omega$, say, symmetric w.r.t. the real line, and f(z) defined and analytic only on $\Omega^{+}$, we can analytically continue the function to $\Omega^{-}$ with the analytic ...
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How can i reflect position and direction vectors from a plane

I'm now working on a project that has mirrors. I'd like to reflect a virtual camera and the way which i can do this is to reflect two vectors - position and normalized direction vectors of the camera. ...
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Product of a Householder transformation and reflection through the origin in 3 dimensions

This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$ $v^T\cdot w=0$, and the Householder transformation ...
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Interesting property of reflection matrix

By inspection, is is evident that the reflection matrix is an orthogonal matrix, while its transpose is equivalent to its non-transpose. The reflection matrix can be represented by the square matrix: ...
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Is it possible to reflect a linear equation across a curved equation?

I understand how to reflect equations across the x and y axises as well as the line y=x. But how do you reflect an equation across more complex equations such as other slant lines and higher order ...
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find out the point of reflection

Given a point $(1, -5, 6)$ is reflected about the plane having equation $-2x+7y+9z=4$. What will the point of reflection about the plane? a.) $\left(\frac{98}{67}, \frac{-389}{67}, ...
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Schwarz Reflection Principle for the four quadrants in the plane and for two intersecting circles,

I'm looking at an old exam problem that shows a picture of what the function f does to the plane. On the upper right quadrant, there is a + sign, which indicates that f maps this quadrant one-to-one ...
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What is reflection across parabola?

Reflection across a line is well familiar, reflection across a circle is the inversion, the point at a distance $d$ from the center is reflected into a point on the same ray through the center, but at ...
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almu and gemu of orthogonal projections and reflections?

Let Vbe an m-dimensional subspace of R^n. I have already found the eigenvalues of the nxn orthogonal projection matrix A onto V as 0 and 1 with respective eigenspaces V_perp. and V, and dimensions of ...
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transforming $(A,B,C)$ to $(0, 0, 1)$ by rotations

I'm trying to reflect the "world" through a specified plane $p:Ax+By+Cz=0$. I know how to reflect the "world" through the $xy$-plane, so I want to rotate $p$ in the $3$ axes ($x,y,z$-axes) so it will ...
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Identify the combination formed by first applying the glide reflection $γ_{A,B}$ and then applying $γ_{C,D}$

Question: Identify the combination formed by first applying the glide reflection $γ_{A,B}$ and then applying $γ_{C,D}$ where $A = (0, 0), B = (2, 0), C = (1, −2), D = (1, 0)$. Before, I jump in ...
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Reflection combined with a glide reflection

Suppose that $A, B, C,$ and $D$ are the vertices $(0, 0), (2, 0), (2, 2),$ and $(0, 2)$ of a square. The transformation $ρ_{AC} ◦ γ_{DA}$ can be decomposed as the combination of reflections across ...
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Translation or rotation? Identify $R_{C,-120} \circ R_{B,-60} \circ R_{A,-180}$

Let $ABC$ be a right triangle that is oriented clockwise and has angles of $90, 30, 60$ at vertices $A,B,C$. Identify $R_{C,-120} \circ R_{B,-60} \circ R_{A,-180}$ I started out with: $R_{B,-60} ...
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Reflection matrix in $ \mathbb{R}^{3} $.

I need help in understanding how they got the transformation matrix $ Q_{L} $ from Theorem 2 and $ P_{M} $ at the bottom of the page. They skipped some steps and I find it confusing. Any help would ...
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Identifying translations and rotations as compositions.

I am having trouble understanding the below which are the ones underline in red and blue. For the red: Why is that $R_{A,90}(A)=A$ and that $\tau_{AB}(A)=B$ As for the blue: Why is that ...
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Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections I am having trouble showing $P \circ R$ is a glide reflection, I manage to get $R \circ P$, ...
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Identifying compositions of reflections, and rotations in a hexagon

Let $ABCDEF$ be a regular hexagon that is oriented clockwise (so that a rotation from $A$ to $B$ to $C$ to $D$ to $E$ to $F$ is clockwise). i) Identify $R_{D,120} \circ R_{A,60}$ which are two ...
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The Composition of Two rotations

So far I rewrote the halfturns of d,c,b,a to halfturn (p,n)(m,l) where n=m because lines c and d are parallel so I can make ambiguous lines n and p parallel too. I also know that lines c,d can be ...
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Equation of one of the two lines whose angle bisector is given.

A ray of light falling along the line $ lx+my+n=0 $ strikes a plane mirror at point $P$. Find the equation of reflected ray if $px+qy+r=0$ is the equation of normal to the plane at point $P$.