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.
432
questions
0
votes
1answer
18 views
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 ...
0
votes
1answer
38 views
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 ...
1
vote
0answers
29 views
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 ...
2
votes
1answer
38 views
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\, \_ \,,\, ...
1
vote
1answer
20 views
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(...
1
vote
1answer
15 views
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....
0
votes
0answers
28 views
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 ...
0
votes
1answer
28 views
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 ...
1
vote
0answers
30 views
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 ...
1
vote
0answers
33 views
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 ...
0
votes
0answers
72 views
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 (...
0
votes
1answer
33 views
Commutative reflections in the plane
How can i show formally that two reflections $S_v ,S_w$ in the plane commute iff $v=Ā±w$ or v and w are orthogonal.
I tried and searched a lot, but did not manage to prove this.
0
votes
1answer
59 views
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 $AD\cdot AF$. ...
0
votes
0answers
12 views
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 ...
0
votes
0answers
32 views
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\...
1
vote
1answer
34 views
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\...
0
votes
1answer
29 views
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 $$r_u(v) = v - 2 \langle v, u \rangle u = v - 2proj_{span(u)}(...
4
votes
1answer
86 views
2012 EGMO P7: Show that the lines $KH$, $EM$ and $BC$ are concurrent [ Proof Verification needed]
Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$. Let $K$ be a point of $\Gamma$ on the other side of $BC$ from $A$. Let $L$ be the reflection of $K$ in the line $...
0
votes
3answers
56 views
Let $A$ be $2\times 2$ Orthogonal matrix such that det$A= -1$. Show that $A$ represents reflection about the line in $R^2$.
Every matrix is corresponding to a linear transformation. Reflection is a linear transformation called $T$. So $T^2=I$. Also modulus of eigen values of $A$ is $1$.Since it has determinant $-1$ so odd ...
0
votes
0answers
37 views
Orthogonal matrix decomposition into rotations and reflection
Given an n-dimensional orthogonal matrix, how can one decompose it as a product of elementary rotations (rotations in orthogonal planes of rotation) and / or reflections, and then retrieve the ...
4
votes
3answers
52 views
Axis of reflection
Is there any way to get the equation of axis of reflection given two intersecting lines without sketching?
Example question: The image of the line $p:\; y-2x=3$ is the line $q:\;2y-x=9$. Find the ...
0
votes
1answer
43 views
Set of possible reflections of a vector.
How many vectors can one construct by by reflecting a vector $b\in\mathbb{R}^d$ for $b\neq 0$?
Reflections can be described by Householder matrices $H=I-2vv^T/||v||_2^2$.
In other words, I'm ...
6
votes
1answer
82 views
Does this method work for reflecting over $x^2$?
I am investigating reflecting over any quadratic. In the graph, I have the simplest scenario (reflecting $y=0$ over $y=x^2$).
My method was to use the tangent (red dotted line) find the intersect that ...
0
votes
0answers
22 views
Reflection of polar functions
For whatever reason, I am having a complete brain block; I would like to reflect the arithmetic spiral
$$
r=\frac{\theta}{2\pi}
$$
about the polar axis. This should be possible? But I can't quite ...
2
votes
1answer
54 views
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 ...
1
vote
2answers
139 views
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. ...
2
votes
0answers
35 views
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 ...
0
votes
0answers
29 views
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$?
0
votes
0answers
12 views
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).
...
1
vote
1answer
87 views
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 $(...
0
votes
1answer
62 views
Determine the matrix of the reflection over the plane $2 x_1 + x_2 ā 2 x_3 = 0$ in $\Bbb R^3$
Determine the matrix of the reflection over the plane $2x_1 +x_2 ā2x_3 = 0 \in \mathbb{ R^3}$ .
3
votes
0answers
61 views
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 120Ā° opening angle with infinitely extended sides. The surface of the reflector has homogeneous Dirichlet boundary conditions imposed,...
0
votes
0answers
26 views
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 ...
0
votes
1answer
39 views
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/...
1
vote
0answers
19 views
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 ...
0
votes
1answer
42 views
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 ...
1
vote
1answer
37 views
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 ...
1
vote
0answers
44 views
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 &...
0
votes
0answers
21 views
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\...
1
vote
2answers
39 views
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=\...
0
votes
0answers
19 views
$f$ 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'...
0
votes
0answers
32 views
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 ...
0
votes
0answers
8 views
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 \...
0
votes
1answer
32 views
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 ...
1
vote
0answers
50 views
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 ...
0
votes
0answers
19 views
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^{ā...
0
votes
0answers
19 views
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 ...
0
votes
0answers
14 views
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
-1
votes
1answer
33 views
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?
0
votes
0answers
17 views
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 ...
