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1answer
12 views

Conformal Mapping with homeomorphic extension

Suppose that $$D=\{z:0<x<a,0<y<b\}$$ and that $$D'=\{w:0<u<c,0,v<d\}$$ Then there is a conformal mapping $f$ of $D$ onto $D'$ whose homeomorphic extension $\tilde{f}$ to ...
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0answers
12 views

Reflecting function over a line

I need to derive a function that is the reflection of the function $y=(0.0056x)/(1+0.006x)$ across the line $y=(1-z)x/1000$ where $z$ is just a pre-determined variable. Thanks so much!!
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2answers
45 views

Why does reflecting a point (x,y) about y=x result in point (y,x)?

I noticed that whenever reflecting a point (x,y) about the line y=x the x and y coordinates become swapped in order to give (y,x). However, I do not know why this is the case. Is there any way to ...
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1answer
39 views

Hermitian matrix the only diagonizable

During the last lecture one of my professors claimed that the hermitian matrix is the ONLY complex matrix which was diagonizable. This seems strange to mee (not to say a very very strong claim to ...
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1answer
18 views

Common elements Job - Groups and Algebras Lie

I need some help to prepare my final master thesis. It is difficult for me, because I want to use things/informations/notes/concepts which I usually use in my job. The maths concepts which I need ...
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2answers
67 views

How to find plane of reflection from transformation matrix

If you have an orthogonal matrix with a determinant of -1, how do you determine the plane of reflection? Thanks
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1answer
57 views

Composition of orthogonal projection

Given $\gamma: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (rotation around $o$) and $\sigma: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (reflection in one of the lines through the origin), I have to show that ...
2
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1answer
63 views

Harmonic function reflection

I'm learning some harmonic function theory by reviewing some problems. I came across two: 1) Prove that a real harmonic function $u$ from $\mathbb{R}^n$ to $\mathbb{R}$ such that $u(x, 0) = 0$ for ...
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3answers
109 views

Why are orthogonal matrices generalizations of rotations and reflections?

I recently took linear algebra course, all the I learned about orthogonal matrix is that matrices is that Q transposed is Q inverse, and therefore it has a nice computational property. Recently, to my ...
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2answers
58 views

Reflection question

This is a practice question to a test I will be taking soon. My conjecture is that it's none of the choices given. I tried reflections about y=x, y-axis, x-axis and it doesn't work. Does anyone ...
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2answers
73 views

Rotations/Reflections of a point

It's been $10$ years since I see some kind of geometry and I'm preparing for a test of these sort of questions. I need help figuring out the following problems: 1) A point $P(x,y)$ is rotated $180$ ...
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0answers
61 views

A light beam enters a closed room. What is the maximal number of reflections?

I have the following problem: a light beam enters a mirror room with integer coordinates in the plane (consider it as a polygon). One of the walls of the room is removed and a light beam enters the ...
0
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1answer
43 views

point deflecting off of a circle

I know that this is a very simple question, but I am stuck at the very last part of this process and can't find the solution elsewhere (I figured I'd find it on this site, but I didn't see it). I ...
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1answer
39 views

Show that every element on $O(\mathbb{R} ^2)$ is either a rotation or reflection

Where $O(\mathbb{R} ^2)$ is the orthogonal group of $\mathbb{R} ^2$ or; The set of all linear maps $g: \mathbb{R} ^2 \rightarrow \mathbb{R} ^2$ represented by an $n \times n$ matrix $M$ w.r.t. the ...
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3answers
82 views

Show that $(A',B',C')$ form the vertices of an equilateral triangle.

Let $ABC$ be a triangle with $AB = AC $ and $angle BAC = 30.$ Let $(A')$ be the reflection of A in the line BC $(B')$ be the reflection of $B$ in the line CA $(C')$ be the reflection of C in the line ...
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0answers
41 views

virtual cell mirror reflections

Please note that this is an extension of this question: Mirror reflection Questions Mirrors A and B with the space between them, including the bird, are part of the real world. Those are all a part ...
2
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1answer
196 views

Reflection Matrix linear algebra

I am practicing some linear algebra question to prepare for my test. I have come across one question that has given me much trouble. It states: If $\lVert u\rVert = 1$, then $Q = I - 2uu^T$ is a ...
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1answer
18 views

Why does knowing where two adjacent vertices of regular $n$-gon move under rigid motion determines the motion?

I am reading the book Abstract Algebra by Dummit and Foote. In the section about the group $D_{2n}$ (of order $2n$) the authors claim that knowing where two adjacent vertices move to, completely ...
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3answers
39 views

Why $Hx=x-(\rho u^Tx)u$?

A Householder reflection is a matrix of the form $$H=I-\rho uu^T$$ with $\rho=2/\|u\|^2$. Obviously, $Hx=x-\rho uu^Tx$. Textbook http://www.mathworks.se/moler/leastsquares.pdf says that ...
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1answer
24 views

Linear Algebra, reflected linear image

If I have a linear image of the room where v1 and v2 is an image of theirselves and v3 is an image of the null vector. If that gives me the matrix A=(a, b, c; d, e, f; g, h, i;) then A^n = A because ...
3
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2answers
78 views

Prove linear operator is a reflection

Prove that a linear operator on $\mathbb{R}^2$ is a reflection if and only if its eigenvalues are $1$ and $-1$ and the eigenvectors with these eigenvalues are orthogonal. $\Rightarrow$: Let $r: ...
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1answer
38 views

Reflection along subspace

A symmetric space is a Riemannian manifold M with the following property: For every point $p \in M$ there is an isometry $\phi: M \rightarrow M$ such that $\phi(p) = p$ and $\phi_*(v_p) = -v_p \in ...
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1answer
243 views

About the use of Schwarz reflection principle in the proof of the mapping formular between the upper half plane to a given polygon

In Chap 8 of Stein's complex analysis, he proved that all the conformal maps $f$ from the upper half plane $\mathbb{H}$ to a given polygon $P$ is of the form $c_1S(z)+c_2$, where $c_1, c_2$ are ...
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3answers
145 views

Two Parabolic Mirrors Opposite of Each other

Suppose we have two parabolic mirrors opposite of each other (e.g. $x=y^2$ and $x= -y^2+10$). Also suppose the first mirror is smaller than the second mirror. If a light ray enters into the opening ...
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2answers
578 views

Finding reflection transformation matrix

I have two 3 dimensional points. $A [x_1, y_1, z_1]$ and $B [x_2, y_2, z_2]$. I need to find a transformation matrix which when multiplied to $A$ will give me $B$ and when multiplied by $B$ give me ...
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2answers
488 views

The Schwarz Reflection Principle for a circle

I'm working on the following exercise (not homework) from Ahlfors' text: " If $f(z)$ is analytic in $|z| \leq 1$ and satisfies $|f| = 1$ on $|z| = 1$, show that $f(z)$ is rational." I already know ...
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2answers
57 views

How to find what point a wave is reflected off

If a wave is reflected off a surface, the angle of reflection is equal to the angle of incidence. But, how can we use this to find the actual path of the incident and reflected waves if we only know ...
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3answers
127 views

Reflecting a golfball off a wall to a hole and compensating for the balls radius

Problem: I'm struggling to compensate for the radius of a ball when reflecting it off a wall towards a target. (sorry I cannot yet post images) What I want is to do this: golf reflections but this ...
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1answer
392 views

Using the three reflection theorem

There is this sample question from my book that I dont know how to go about. please help out Use the three reflections theorem to show that the only transformations of the Euclidean plane are ...
1
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1answer
107 views

Householder reflections

Let $x=\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$ I want to use a Householder reflector U to keep only first element in vector x, and make everything else zero but I'm doing something wrong... ...
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2answers
217 views

Reflecting a point by a line in $\mathbb R^3$

I would like to know if it's possible, given the vector equation of a line and the coordinates of a point, whether it's possible to reflect the point by the line.
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0answers
36 views

Invariants of parabolic subgroups

Let $G$ be a finite reflection group acting on $R^n$. Then for each point $x\in R^n$ we can look at its stabilizer $G_x$. Since $G$ is a finite reflection group, its ring of invariants is a polynomial ...
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1answer
52 views

Solve for $x$ (three equations)

I've been having trouble with these equations. The three equations have no relation to one and other. I've been trying to solve for $x$: $$y = 1.15^{x+2}$$ $$y = \frac{1}{1.15^{x+2}}$$ $$y = 1 - ...
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2answers
94 views

Raster scan with a rotating mirror

I'm trying to figure out a way to scan a 2D raster (like a TV or CRT monitor) with a beam using two rotating mirrors for the X and Y positions. I'm looking for a shape that will, when spinning, ...
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1answer
76 views

Question about matrix relfections

The question is: Determine the matrix of the following reflection in $\Bbb R^2$: $R$ is a relection in the line $x_1-5x_2 = 0$.
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1answer
146 views

Reflecting an exponential function over a y = 3 line.

How would you write the equation of $f(x) = 4^x$ that reflects over the line $y = 3$? I've put in $f(x) = 3 + 4^{-x}$ which I thought was the right answer, but it isn't. Thanks in advance!
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1answer
278 views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook ...
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1answer
363 views

Reflect a complex number about an arbitrary axis

This should be really obvious, but I can't quite get my head round it: I have a complex number $z$. I want to reflect it about an axis specified by an angle $\theta$. I thought, this should simply ...
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2answers
981 views

Angle of reflection off of a circle?

I've made simple 2D games in the past using mostly just squares. If an object collided with another object (all squares/rectangles) it would just change the slope to the opposite based on what side ...
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2answers
6k views

How do I reflect a function about a specific line?

Starting with the graph of $f(x) = 3^x$, write the equation of the graph that results from reflecting $f(x)$ about the line $x=3$. I thought that it would be $f(x) = 3^{-x-3}$ (aka shift it three ...
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1answer
65 views

Question about reflection, eight queens problem.

When I am to reflect a current chessboard around say the horizontal line Y=4, would that imply that I need to reflect the bottom part to the top part, and then the top part to the bottom part?
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1answer
2k views

Why does the formula for calculating a reflection vector work?

The formula for calculating a reflection vector is as follows: $$ R = V - 2N(V\cdot N) $$ Where V is the incident vector and N is the normal vector on the plane in question. Why does this formula ...
1
vote
1answer
303 views

Question about generalizing Schwarz Reflection

You can use the proof of the Schwarz reflection principle to show that if $D$ is a domain in the complex plane, where $D^+$ denotes the subset of D above the real line, and $D^-$ the subset below, if ...
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2answers
1k views

Geometric intuition for the Householder transformation

I am studying QR decomposition. Could you explain the geometric intuition for what the Householder transformation does in that context, and why it's sometimes referred to as the Householder ...
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2answers
2k views

Point reflection across a line

Let's say that we have three points: $p = (x_p,y_p)$, $q = (x_q,y_q)$ and $a = (x_a,y_a)$. How can i find point $b$ which is reflection of $a$ across a line drawn through $p$ and $q$? I know it's ...
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2answers
192 views

do Householder reflections describe all reflections?

I'm familiar with Householder reflections; they are a simple transformation that, given a normal vector, describes reflection in the hyperplane perpendicular to that vector. But do Householder ...