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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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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1answer
13 views

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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15 views

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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Let A,B,C be the vertices of the triangle, find the center of the following rotations

Let A,B,C be the vertices of the triangle, find the center of the following rotations: a) $R_{A,\frac{\pi}{2}} \circ R_{B,\frac{\pi}{2}}$ Two rotations that are composed together is another ...
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1answer
21 views

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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Intersection of the composition of two glide reflections

i am taking a geometry course and we are learning about isometries. I am having a hard time with glide reflections and this problem is giving me some issue, mainly because my professor usually tells ...
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22 views

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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22 views

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 $P$.
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40 views

Prove: $G = HN$, and $H \cap N =$ {$e$}, (Isometries in $\mathbb{R^2}$)

I'm trying to solve this problem: Let $G = E(\mathbb{R^2})$ be the group of all isometries of $\mathbb{R^2}$, so $G$ consists of translations, rotations about the origin and reflections in a line ...
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1answer
68 views

Reflection of a curve around a slant line

a fifth-degree function: y = 80*x^5-225*x^4+350*x^3-300*x^2+150*x-20 (the green curve in the image) needs to be reflected/mirrored around the line ...
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When is a stabilizer group a reflection group?

Let $G$ be a compact, connected Lie group and $K$ a closed, connected subgroup. If $K = T$ is a maximal torus, it is well known that $W := N_G(T)/Z_G(T) = N_G(T)/T$ is a finite reflection group, the ...
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37 views

Weak* topology on Hilbert space

I am a little confused about the weak* topology on Hilbert space $H$. Beyond doubt, the weak* topology on $H^{**}$ is $\sigma(H^{**},H^*)$. Suppose $\tau$ is the natural embedding from $H$ onto ...
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30 views

Rotations around the origin not a normal subgroup of the group of all isometries

I have to prove this exercise for my math study: With a rotation I mean {$\rho_{x}| x \in \mathbb{R^{2}}$} and $x$ is the angle of rotation. Let $H$ be the group of all rotations around the origin, ...
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Formula for reflection across a line in $\mathbb{R}^2$? [duplicate]

$\newcommand{\Reals}{\mathbb{R}}$I have an equation of a line: $4x - 3y = 0$. Let $S : \Reals^2 \to \Reals^2$ be reflection through that line, and let $P : \Reals^2 \to \Reals^2$ be projection onto ...
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Need help with linear transformations (with projection and reflection)?

Let $L$ be the line given by the equation $4x − 3y = 0$. Let $S : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be reflection through that line, and let $P : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be ...
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110 views

Ray\curve mirror problem

I have an idea for a space station, but there is the following problem. I have a patch of grass on a space station. If the sun (yellow rays) shines from below it, what is the best shape of mirror ...
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102 views

How do I deal with reflections inside an ellipse?

Suppose I have an ellipse with foci $F_1$ and $F_2$. How do I show that any ray of light which intersects the segment connecting the foci will have subsequent reflections that always are tangent to ...
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20 views

Vector reflection with limited angle

I am struggling with following challenge in my free time programming project $-$ how is it possible to make reflection vector that reflects along normal with angle that is not larger than some ...
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1answer
28 views

Form of Matrix for Reflection about a Line

I've seen a bunch of variations on the wonderful properties of this specific matrix. My textbook gives one algebraic form in particular that I'm having a bit of trouble verifying: Any help here? I ...
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3answers
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Should a reflection matrix of a vector have the same form as a rotation matrix?

According to the book: I know that it is not possible to write a reflection as a rotation, but from the text it seems that the matrix of the form $$ A=\left[ \begin{array}{ c c } a & ...
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Group Duality with respect to Generators and Relations

Although the following question is not phrased in the most accurate way, I would like to ask it in the same way it rushed to my mind: "Looking at some basic examples of group theory with geometrical ...
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1answer
85 views

Reflection of a vector across a line

For homework, I need to find the reflection of the vector <1,1,1> over the line defined by all the scalar multiples of <2,1,2>. I tried looking at the other questions about similar topics here, ...
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3answers
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Every reflection is an isometry proof

The theorem is that every reflection $R_{S}$ in an affine subspace $S$ of $\mathbb{E}^{n}$ is an isometry: $R_S:\ \mathbb{E}^{n} \rightarrow \mathbb{E}^{n}:\ x \mapsto R_{S}(x) = x + 2 ...
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Reflecting a line from plane

I have a plane given by equation $0=10x+2y-3z$ and a line described by vector with variable $t\ge0$ $(1-t, 1+2t, 1+t)$ How to calculate reflected vector of this line from plane? We treat line as ...
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115 views

Constructing a 2x2 matrix R which represents reflection in the x-y plane,

Construct a 2x2 matrix R which represents reflection in the x-y plane through the line $$(cos(\theta)x+(sin(\theta)y=0$$, where $\theta$ is any real number. (Let's call this line "L".) Write an ...
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Reflections on a sphere

There is a sphere located in a point s with radius r. The Sphere is a perfect mirror. If i'm sitting in the point c, I want to cast a ray to the sphere such that I hit the point p after bouncing in ...
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3answers
49 views

Matrix with linear transformation with reflection

Find the matrix of the linear transformation A which is the reflection in the line $y = \sqrt{2}x$ with respect to the standard basis in $\mathbb{R^2}$. I Have no idea how to approach this problem... ...
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32 views

Householder reflector sign error

I am studying Householder reflectors from Trefethen and Bau but am having trouble creating a simple example for it. I am given the equation for the vector v that the Householder reflector H is based ...
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Proving two reflections across a line are commutative if and only if the two lines are perpendicular

Please note brackets () denote inner product - for some reason the traditional symbols for inner product do not show up when I type them in! Hi, I am trying to prove that the two reflections in ...
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ODE with Reflection

Some Background: I come from a computer science background and differential equations was not part of my course work. I'm still in contact with my professor that I ...
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Reflection to get within convex polygon

Let $P$ be a convex polygon, and let $A_1$ be a point on the same plane as $P$. Prove that we can find an integer $n$, and points $A_2,A_3,\ldots,A_n$, such that $A_{i+1}$ is a reflection of $A_i$ ...
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Why isn't $f(x)=\sqrt{2-x}$ reflected across the y-axis?

If I try to graph this function, it does not appear to reflect across the y-axis when it comes time to do the reflection. Rather, it is reflected around the point where the function begins on the ...
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Simple systems are the “smallest”

Let $\Phi$ be a root system of a finite reflection group $W$. Let $\Delta$ be a simple system in $\Phi$. I want to prove that $\Delta$ is "the smallest" set which generates $W$, more precisely: There ...
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reflection(reflection) = rotation

Lel $\alpha$ and $\beta$ be two distinct simple roots in a root system $\Phi$. How to prove that i) $S_{\alpha} S_{\beta}$ is a rotation in $\mathbb{R}\Phi$ ii) Composition of two reflection is a ...
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Do involutions suffice to find reflected vectors in a reflection group representation?

Consider a reflection group $W$ acting by isometries on a Euclidean space $V$. I want to understand the union of $(-1)$-eigenspaces for this action, the set $$\{v \in V : \exists w \in W\ (w\cdot v = ...
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60 views

Reflections - Understanding

I have been asked Does the set of all reflections in the plane form a group? Explain. I thought that the answer was that it wouldn't as if I reflected the parabola $y=x^2$ through the y axis, I ...
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Complete Triangle Given 3 Parallel Planes and 2 Points

I have a problem where a point B connects to a point C at a known angle and distance. Both point B and C are on two separate parallel axis, GH and JK respectively. I need to find a third point, A, on ...
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Reflecting and Translating Points

Point R (-4,5) is reflected about the x-axis onto point R'. It is then reflected about the line y=-x to find R''. It is then translated using the vector <2,-2> to find R'''. Find R'''
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Tangent and angle bisectors [closed]

The tangent to the incircle of a triangle ABC is reflected about the external angle bisectors. Show that the triangle formed by the resulting 3 lines is congruent to ABC .
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Fixed point set of orthogonal Transformation

I need some help with this problem. Let $g$ be an element of the orthogonal group and $s$ a reflection. Then the dimension of the fixed point set of $g$ and $gs$ differ by $\pm 1$. Since that ...
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Prove that this transformation is a reflection

Let $ v \in V$ be a unit vector in a Euclidean vector space. Prove that the endomorphism $$\phi: V \to V, \qquad \phi(x)=x - 2<x,v> v$$ is a reflection. I know that a reflection is an ...
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1answer
90 views

The composition of two different glide reflections is a rotation

Denote by $G_{XY}$ a glide reflection which reflects around the $XY$-axis and then takes the point $X$ to $Y$. I would like to prove that the composition of two different glide reflections $G_{XY} ...
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26 views

Order of rotation

For $\alpha \in \mathbb{R}^n$, let $s_\alpha$ be the reflection in $\alpha$, i.e. $s_\alpha$ sends $\alpha$ to $-\alpha$ and fixes pointwise the hyperplane orthogonal to $\alpha$. Then if ...
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angles between simple roots are obtuse, problem with proof

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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71 views

Angle between roots in a root system

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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185 views

Reflection in a plane.

What is the exact definition of a reflection through the plane $a.r=0$ for a given vector a and $r=(x,y,z)$. Of course I know what it is but I don't know what's part of its definition and what's part ...
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59 views

Rotation-Reflection-Symmetries

I have the following exercise: Do a rotation of $2 \cdot 72 ^\circ$. Then do a reflection of the axis $d4$. Then do a reflection of the axis $d3$. Then do a rotation of $2 \cdot 72 ^\circ$. ...
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What is this and why is it so important $(x,y) \to (y,x)$?

When doing a $y=x$ reflection the notation is $(x,y) \to (y,x)$. My teacher told us to find out what it is, what it is called, and why it is important? Please help ?
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Reflect the plane in the $x$-axis, and then in the line $y = \frac12$. Show that the resulting isometry sends $(x,y)$ to $(x,y+1)$

I have a hard time proving this without using any numbers. How do I show that the point $(x,y)$ reflected across $y=\frac12$ is $(x, 1-y)$ ?
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Image of a point reflected over $y=mx+b$ using dot product

So, I know that the image for a generic point is $$\small((1-m^2/1+m^2)x + (2m/1+m^2)(y-b), (2m/1+m^2)x - (1-m^2/1+m^2)(y-b)+b))$$ when you reflect it over the line $y=mx+b$. It's straightforward ...