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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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…
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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
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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…
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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
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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…
jakeboxer
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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
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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 the surface of the sphere. For this, I want to find…
ButterDog
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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…
Robert Zhang
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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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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
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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
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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…
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Are there any finite examples of metachirality?

I learned about metachirality through an amazing Vihart series ([1] [2]), and the only examples I have encountered are infinitely long screws, and this infinite graph Wikipedia mentions. Are there finite examples of metachirality? My intuition says…
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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 the tangent makes with $y=0$ and calculate that…
thampel1
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Reflection groups of division rings

My question is: Is there a classification of finite groups representable as a $\mathbb{K}$-reflection group for some division ring $\mathbb{K}$ of characteristic zero? I would also appreciate any references dealing with this problem or with…
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