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.
Questions tagged [reflection]
600 questions
39
votes
3 answers
Why are orthogonal matrices generalizations of rotations and reflections?
I recently took linear algebra course, all that I learned about orthogonal matrix is that Q transposed is Q inverse, and therefore it has a nice computational property.
Recently, to my surprise, I learned that transformations by orthogonal matrices…
Alby
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20
votes
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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 about the reflection principle for the case of a…
user1337
- 24,859
15
votes
4 answers
Reflect a ray off a circle so it hits another point
my problem is the following:
I have two points ($e$ and $p$) in a 2D space and I am trying to figure out where on the circle is the reflection of $p$ as seen from $e$.
$$$$
$$$$
So the way I approached this is by looking for the vector from the…
Орлин Митев
- 151
14
votes
4 answers
An artist needs help from mathematicians! Angles of reflections: should I paint these distant trees in the water's reflections?
So forgive the unfinished work(the first rule of being an artist is to NEVER show unfinished work...)
But, I find myself wondering if I would see the reflections in the little pond I'm getting ready to paint.
Now some of them seem obvious, the…
jackwarner
- 143
13
votes
1 answer
Why does the formula for calculating a reflection vector work?
The formula for calculating a reflection vector is as follows:
$$
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 work? I haven't seen any good explanations of it. I…
jakecard
- 231
12
votes
1 answer
What is the reflection across a 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 the distance $R^2/d$, where $R$ is the radius.…
Conifold
- 12,093
11
votes
6 answers
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 the surface of the sphere. For this, I want to find…
ButterDog
- 161
9
votes
3 answers
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 simple to calculate, when we have $p$, $q$ etc. But I…
guram
- 347
8
votes
1 answer
Prove that for an orthogonal matrix $A$ and a certain reflection $R$, the $\mathbb C$-linear part of $RA$ is invertible
Hello: I need help with this problem:
Let $V = (V,b)$ be a finite-dimensional vector space equipped with a symmetric and positive definite bilinear form $b$. And let $\{e_1,…,e_n\}$ be a orthonormal basis for the subspace $\ker((P_A)^t)$ ($P_A$ is…
James Garrett
- 2,687
8
votes
1 answer
What is a Coxeter Group?
I've recently started investigating abstract algebra and have now stumbled upon "Coxeter Groups", which are a mystery to me.
I've read that Coxeter Groups
have something to do with reflections (in which way is entirely unclear)
are related to…
schuelermine
- 611
8
votes
2 answers
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 reflection.
NPE
- 320
7
votes
2 answers
How many non-infinite plane curves with infinite reflectional symmetry?
If we consider a curve with infinitely many distinct lines of reflectional symmetry, we generally think of a circle. However, can we prove that the circle is the only finite plane curve that has infinitely many lines of symmetry, or do more such…
gz839918
- 195
7
votes
1 answer
Relation between reflection group and coxeter group
Reflection group is defined see https://en.wikipedia.org/wiki/Reflection_group.
An abstract Coxter group is defined to have generators $s_1$, $s_2$, ..., $s_n$ and relations $s^2_i=e$, $(s_is_j)^{m_{ij}}=e$ for some $2\leq m_{ij}\leq \infty$.
I…
bing
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6
votes
2 answers
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 units to the right and reflect it), but it's…
kubasub
- 307
6
votes
0 answers
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 the light beam enters the room. The initial (not reflected) beam is defined…
Marin Shalamanov
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