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 the signature of a reflection.

I am doing some self-study and I have this task: Show that the signature of the following mapping, the reflection about the hyperplane $a^{\perp}$, given by $S_a(v) = v - 2\frac{<v,a>}{<a,a&...
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Bath towel on the spheric rope: minimize the area of self-intersection of a 'folded' spheric rectangle

Some time ago I was curious about a question related to my bath towel, which I hang on a rope to have fun (you can use your own towel to do this experiment in bath-o if you want): 'There is this ...
Mikhail Gaichenkov's user avatar
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Hyperbolic Reflection of polygon

I'm working on visualizing the reflections of a polygon in the Poincaré disk along each side of it using SageMath. The figures below show the reflections of a polygon (a 4-gon and a 3-gon, the ...
Rowing0914's user avatar
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What are All the Reflections in Minkowski Space $\mathbb{R}^{1,n}$?

All the literature on reflections in minkowski space, that I have found, have defined ways to reflect about an arbitrary planes or lines and they always add the disclaimer eventually that the plane or ...
intravertig0's user avatar
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How can we distinguish elements of $D_n$ that include reflections versus those that don't?

For $n \geq 3$, the dihedral group $D_n =\langle r, s \rangle$, where $r ^n = s^2 = e$. Within this group, we can distinguish two types of elements: Those of the form $r^i$, where $i$ is any integer ...
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Formula for the equation of the reflection of a line on another line, and application to convergence of reflections of vertical lines on a parabola

Is there a well-known formula to find $p, q, r$ in the equation $px+qy+r=0$ of the line which is the reflection of a line $ax+by+c=0$ on an axis $dx+ey+f=0$ ? I vaguely see a way by computing angles ...
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Reflection of a continuous differentiable curve about a line

Let $g(x)$ be the reflection of the continuous and differentiable curve $f(x)$ about the line $x\cos(\theta)+y\sin(\theta)=r$. Find $g(x)$. Also, examine the case in which a curve $f(x)$ is reflected ...
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Generalize a reflection of a function over a line

I want to write a parametric function that can take any function $f(x)$ and reflect it over a given line in the form $y=mx+b$ (excluding the vertical line). I know the reflection matrix is given by: $...
SebtheSong's user avatar
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Finite cyclic groups are reflection groups.

I am following a note on complex reflections. There the notion of pseudoreflection groups are introduced. A pseuforeflection $\sigma : \mathbb C^n \longrightarrow \mathbb C^n$ is a linear isomorphism ...
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Proving two circles are concentric given some conditions involving an isosceles trapezium and reflections.

In an isosceles trapezoid, $ABCD$ ($AB ∥ CD$) points $E$ and $F$ lie on the segment $CD$ in such a way that $D$, $E$, $F$, and $C$ are in that order, and $DE = CF$. Let $X$ and $Y$ be the reflection ...
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Finding the plane curve resulting from a reflection onto another plane curve

Question: What is the best way to describe the set of reflected points of one plane curve along the perpendicular lines at points of another plane curve? I am trying to find a way to find the ...
vallev's user avatar
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Can I extend the solution of my wave equation to satisfy one more zero boundary condition?

Using d'Alembert's formula, I found the solution to the following wave equation on a half-line problem: $$u_{xx} = u_{tt} \tag 1$$ $$u_x(0, t)=0 \tag 2$$ $$u(x, 0)=0 \tag 3$$ $$u_t(x, 0)=(1-x)e^{-kx} ...
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Express reflection w.r.t. combination of vectors as combinations of reflection w.r.t. vectors

Let $\alpha\in\mathbb{R}^m$, then for every $x\in\mathbb{R}^m$ we call reflection of $x$ w.r.t. $\alpha$ the reflection of $x$ w.r.t. the hyperplane $\alpha^\perp=\{y\in\mathbb{R}^m\mid <\alpha,y&...
Giulio Binosi's user avatar
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Unitary operators and a product of reflections

The book Advanced Linear Algebra by Steven Roman states that Reflections or Housholder transformations are self-adjoint and unitary. Moreover, Theorem 10.17 of this book states every unitary $\tau \in ...
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Questions About Generalized Reflections in Linear and Curved Lines

A few weeks ago, I was taking the transformations unit in my precalculus class. I remember being intrigued by the reflections you could perform on a graph, and eventually derived this generalized ...
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Finding the Mirror Plane Mapping One Plane to Another and Understanding the Relationship between Normal Vectors

How can I find the mirror plane that maps the plane $ax+by+c = 0$ to $a'x+b'y+c' = 0$? My approach involves finding the normal vector of the mirror plane. I believe that if $(a, b, c)$ and $(a', b', c'...
martina p's user avatar
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Billiard Shot Angles for Circular Table: Return to Starting Point

A circular billiard table is given with a cue ball at the circumference. It is shot at an angle of θ to the line from the ball to the center of the table. For what angles θ will the ball return to ...
Nithin Kamalahasan's user avatar
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Existence of orthogonal transform that switches a linear subspace and a complement of it

Suppose $V$ is an $m$ dimensional linear subspace of $\mathbb{R}^{2m}$, $W$ is a complement of $V$, ($V$ and $W$ might not be orthogonal). Does there exist a orthogonal transform $Q \in \mathbb{R}^{2m ...
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Bilinear form associated to Coxeter group

Let me just set the scene before asking my question. Let V be a real vector space with basis $u_1,...,u_n $. Let $W$ be a Coxeter group with generating set $S=\{s_1, ..., s_n \}$ and let $m_{ij}=|...
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Vector reflection relative to a unit vector

I am reading a book by Eric Lengyel, Foundations of Game Development - Mathematics, and I am stumped on a particular statement: "Suppose the component v⊥a represents everything that is ...
Sole Core's user avatar
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Reflecting a point over a line

I am working with my son on a math question: Triangle $DUC$ with coordinates $D(-3,-1), U(-1,8)$, and $C(8,6)$. After reflecting $U$ over $DC$ prove that the resulting shape $DUCU'$ is a square. In ...
gunygoogoo's user avatar
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Reflection matrix such that $Ax = y$ and $Ay = x$

I am working on the following problem: Let us say that a real symmetric $n \times n$ matrix $A$ is a reflection if $A^2 = I_n$ and $rank(A − I_n) = 1$. Given distinct unit vectors $x, y \in \mathbb{R}^...
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Rotation matrix as product of reflection matrices [duplicate]

I would like to prove that any rotation matrix can be represented as finite product of reflection matrices. Almost everything I found earlier is that product of 2 reflections is a rotation, but this ...
nezudem's user avatar
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Non-straight lines of symmetry

This is a less concrete question I was just curious about. We always talk about straight lines of symmetry and reflections but can non-straight lines of symmetry exist when reflecting something, and ...
d0uble_a_b4ttery's user avatar
4 votes
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223 views

Trajectory of light rays in a mirror polygon

Given a general polygon and we are given a ray of light bouncing between the sides of the polygon where each side is a mirror. they hit at points $P_1,P_2...$, we define $\alpha_i$ to be the smaller ...
razivo's user avatar
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Find the matrices $S_1$ and $S_2$ of the reflections in $\mathbb{R}^3$ according to the following lanes

Find the matrices $S_1$ and $S_2$ of the reflections in $\mathbb{R}^3$ according to the following planes defined by the following subspaces: $V_1=\{\vec{x}:x_1+x_2-x_3=0\}$ \ $ V_2=\{\vec{x}:x_1-x_2+...
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Reflection lines in cube tilings

When considering tilings of the plane by the unit cube. That is, a tiling by the unit square $I^2=[0,1]^2$ which covers $\mathbb{R}^2$. Usually one says that the unit cube tiles the plane using ...
DaveGoneRogue's user avatar
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Eikonal equation reflection to solve for boundaries.

Given a point source of radially expanding wave (2D) I need to change its wavefront shape (partially) to planar by reflection. A simple parabolic reflector will not work since the circular wave ...
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Proving consequence of BM reflection principle [duplicate]

I'm currently working on the maximum of Brownian Motion and have found that the mirror principle can be really helpful with this. I have found and understand a proof of the following theorem: Given a ...
Jord van Eldik's user avatar
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Function with same symmetry as planar hexagonal lattice

I'm currently trying to define a function which has the same symmetry as a planar hexagonal lattice. For a quadratic lattice the solution $f(x,y) = \cos(x) + \cos(y)$ is quite simple but I wasn't able ...
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Reflections of Graphs?

When I reflect a function over the $y$ axis: $$f(-x)$$ Consider the function $f(x+3)$. When reflecting this over the y axis: $$f(-x+3)$$ I cannot intuitively understand why the reflection is not $f(-x-...
James Chadwick's user avatar
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Condition for a group generated by reflections of the triangle's sides to be discrete

Consider a triangle in the plane with angles $\alpha,\beta,\gamma$ and let $G$ be the group generated by the reflections across the triangle's sides. I need to find a condition on the triangle's ...
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Relation between tangent circles and side of a triangle

Consider a triangle $ABC,$ and let $D$ the foot of the bisector of angle $\angle{BAC},$ $M$ the midpoint of $BC,$ $D'$ the symmetric of $D$ with respect to $M.$ Consider the circle $\omega_1$ tangent ...
PS48725's user avatar
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Reflections of a point about n lines returns point to its original position

Here's a very interesting problem that I made up with a friend this morning: For which even $n$ does there exist a permutation $\pi$ of $\{1,2,\cdots,n\}$ such that when we reflect any point $P$ in ...
TheBestMagician's user avatar
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Diffusion Equation with inhomogeneous Dirichlet boundary condition

How to solve: \begin{cases} u_t - ku_{xx} = 0 \hspace{2cm} k>0, x>0, t>0 \\ u(x,0) = 0 \hspace{2.8cm} x>0\\ u(0,t) = h(t) \hspace{3cm} t>0 \end{cases} I am trying to solve diffusion ...
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Linear algebra - reflection matrix of vectors of the same length

I would appreciate a hint for a question I have been struggling with for the last couple of days. Let $v, w \in R^2$, and assume $||v||=||w||$. Prove that there is only one reflection matrix that ...
GreekMustard's user avatar
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Properties of a reflection matrix

If you are given a transformation matrix, how do you check if it represents a reflection quickly? Or if it represents rotation? What is uniq about them that I can easily spot(it doesn't need to be ...
Need_MathHelp's user avatar
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Find reflection vector of 2 lines in 2D with General form

I am having problems computing the reflection of 2 lines using the General form (Ax + By + C = 0). I am currently using the following method: We have lines a and b; ...
Sasho Trubata's user avatar
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3 answers
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Reflecting a triangle over a line $y=2x+1$

A triangle with vertices at $A(-2,2), B(-8,2)$ and $C(-8,-1)$ is reflected about the line $y=2x+1$. Express the coordinates of the reflection of A as an ordered pair. (A) $(-2,0)$ (B) $(0,-2)$ (C) $(...
Ben Liang's user avatar
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Linear algebra — similarity of reflection matrices

The question is as follows: Let $A,B$ be $2\times2$ reflection matrices. Are $A$ and $B$ similar? What I’ve tried: It did seem like a proof to me: I have calculated the characteristic polynomial and ...
GreekMustard's user avatar
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Can this linear algebra / geometry lemma in Humphreys be proven by induction?

Many questions have been asked on this site about the proof of Lemma 9.1 on page 42 of Humphreys's Lie algebra book. The full proof is posted here. I understand the proof, but I don't like it and ...
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Reflection in three concurrent lines

i have some problems in my proof of the following task: Let $l_1, l_2, l_3\in L(\mathbb{R²})$ be three concurrent lines with the common point $q\in\mathbb{R²}$. Show that there exist lines $h_1,h_2\in ...
BananaHead's user avatar
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Inner product is invariant under reflections in the real hyperplane?

Let $V$ be an (even) finite-dimensional vector space. I’m trying to prove that $$d(r_v(u), r_v(w))=d(u,w),$$ where $d:V \times V \to \mathbb{R}$ is a positive definite symmetric bilinear form (i.e. a ...
James Garrett's user avatar
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How I can find direction vector of line if I know only two angles (azimuth and polar) and one start point(x0,y0,x0)?

I have one start point x0,y0,z0, and two angles. Theta(polar(0-180)) and Phi(azimuth (0-360)). And now I need to find the direction of the vector of the reflected line 3d space box with parameters (4,...
Aknur Sarsenbay's user avatar
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Find $\det(\mathrm{I} - 2\mathrm{P})$ where $\mathrm{P}\mathrm{P} = \mathrm{P}$ is a projection matrix

Let $\mathrm{P}$ be a well-defined $n\times n$ projection matrix and $\mathrm{I}$ be the identity matrix. What is the determinant of the following reflection matrix? $$ \mathrm{R} = \mathrm{I} - 2\...
Euler_Salter's user avatar
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Find two ratios given an inequality about a triangle with integer coordinates

Let $A(0,0)$, $B(p,q)$, $C(r,s)$ where $p,q,r,s \in \mathbb{Z}$. Also, it is known that $(|AB|+|BC|)^2<8\cdot \textbf{Area}(\triangle{ABC})+1$ $1.$ If $\triangle{ABC}$ is reflected across side $AC$...
Vanessa's user avatar
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Given two vectors $A\in\mathbb{R}^{3}$ and $B\in\mathbb{R}^{3}$, does it hold that $A\cdot B = (-A)\cdot(-B)$ and $A\times B = (-A)\times(-B)?$

Following a space inversion, vector $\bf{A}$ goes to $\bf{A}^{'} = -\bf{A}$ and $\bf{B}$ to $\bf{B}^{'} = -\bf{B}$. How does $\bf{A}^{'} \cdot \bf{B}^{'}$ compare to $\bf{A} \cdot \bf{B}$ and $\bf{A}^{...
Austin's user avatar
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There is at most one orthogonal reflection for a given reflection line

$ \def\char{\operatorname{char}} \def\id{\operatorname{id}} \def\rank{\operatorname{rank}} \def\Hom{\operatorname{Hom}} $ Let $k$ be a field with $\char k=0$ and let $V$ be a finite-dimensional $k$-...
Elías Guisado Villalgordo's user avatar
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386 views

Proving that a product of $k$ reflections and an orthogonal matrix has positive determinant

I’m studying Linear Algebra and I stumbled into an interesting problem: Let $V=(V,b)$ be a finite-dimensional vector space with a symmetric and positive-definite bilinear form $b$. Let $M \in \mathrm{...
James Garrett's user avatar
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1 answer
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Multiple reflections and determinant

If a linear transformation swaps two axes is said to perform a reflection and the determinant will be negative. Testing for the sign of the determinant can tell me whether a reflection has happened. ...
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