12 votes
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How many non-infinite plane curves with infinite reflectional symmetry?

Claim 1. Let $\ell_1,\ell_2,\ell_3$ be three line in the plane that do not intersect in a single point. Then the only bounded subset of the plane that is symmetric to all three lines is the empty set. ...
Hagen von Eitzen's user avatar
12 votes
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What is a Coxeter Group?

I'll try to sketch the connections between your bullet points with an easy example of a Coxeter group. You probably want to study some basic algebraic knowledge to follow this. We examine the ...
Babelfish's user avatar
  • 1,842
10 votes
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Normal of a point on the surface of an ellipsoid

The $\nabla$ operator is known as the gradient operator. It's a vector of all of the partial derivatives of the function with respect to all of its variables, i.e., $$\nabla f=\bigg<\frac{\partial ...
Rushabh Mehta's user avatar
9 votes

Interpretation Reflection principle

For $a>0$ denote by $$\tau_a :=\tau_a^B:= \inf\{t>0; B_t = a\}$$ the first time the Brownian motion hits the line $y=a$.The reflection principle, which you stated in your question, is then ...
saz's user avatar
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7 votes
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Relation between reflection group and coxeter group

We have $(s_i)^2 = e$ because if we repeat the same reflection twice in a row we end up back where we started. Since $s_i s_j$ is an element of a group, it has an order (possibly infinity), which we ...
Daniel McLaury's user avatar
7 votes
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Point within the interior of a given angle

Denote the mid-point of $OM$ be $N$. Then $N$ is the center of the circle passing through the points $O,P,M,K$. We have: $$\angle PNK = 2\alpha,\quad \angle PMK = \pi-\alpha$$ By Cosine Theorem, $$PK^...
player3236's user avatar
  • 16.4k
6 votes

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

Given a line $\overline r+t\cdot\overline{v}$ and a point to be reflected $\overline p$, we must first find the closest point on the line to $\overline{p}$. Let $t_0$ correspond to that point. Its ...
Felix's user avatar
  • 296
6 votes
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Functional equation $P(X)=P(1-X)$ for polynomials

First, if $P(X) = P(1-X)$ holds on $\Bbb R$ then the same relation holds on $\Bbb C$, due to the identity principle for holomorphic functions. With $Q(x) := P(x + \frac 12)$, the condition $P(X) = P(...
Martin R's user avatar
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6 votes
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What does it mean for a Coxeter system to be of "spherical" type?

Let $\Gamma$ be a Coxeter graph and let $(W,S)$ be the Coxeter system of $\Gamma$. We say $\Gamma$ is $\textit{of spherical type}$ if $W_{\Gamma}$ is finite. Note that if $\Gamma_1,\ldots, \Gamma_{\...
Mee Seong Im's user avatar
  • 3,190
6 votes

Reflection in Geometric Algebra

Here's a less convoluted version of your calculation that should make your error more immediately obvious: $$\mathrm{Ref}_{v-w}(v) = −(v−w)v(v−w)^{−1} = -\frac{(v−w)v}{v−w} = -v$$ In short, you've ...
celtschk's user avatar
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6 votes
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Find the $\angle ACB$ of $\triangle ABC$.

Sometimes, geometry consists of dropping the right line or introducing the accurate point... Denote by $D$ the point on $AP$ such that $\angle PDC=90°$ and let $BP=x$. Note now that the triangle $\...
Dr. Mathva's user avatar
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6 votes
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Reflection reverses a root string

It follows from the definition of $W_\alpha$ that for $k \in \mathbb Z$, $W_\alpha(\beta+k\alpha)$ is of the form $\beta + \ell(k) \alpha$ for some $\ell(k) \in \mathbb Z$. Consequently, that root ...
Torsten Schoeneberg's user avatar
6 votes

Find the shortest path from a point inside the square touching its 3 sides

We can reflect the original square in some directions. For example, if you think the solution should bump into the sides top right, top left and bottom left, then we can make three reflections and ...
AnilCh's user avatar
  • 1,077
5 votes

What is the difference between and projection and a reflection, in vector transformation?

The (orthogonal) projection onto a line "compresses" every point in the plane onto the line. If you drop the perpendicular from the point to the line, the image of the point after projection is the ...
rschwieb's user avatar
  • 153k
5 votes
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What is the reflection across a parabola?

Locally near the original curve, the Schwarz reflection is the only anti-holomorphic function that fixes the curve. When you look at it globally then you're looking at its analytic continuation (or ...
mercio's user avatar
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5 votes
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Periodic light rays in ellipse

I'm going out on a limb and suggesting that this is true and easy (which means it doesn't start addressing the really hard questions about billiards on elliptical tables). I argue from continuity. ...
Ethan Bolker's user avatar
  • 93.6k
5 votes

A geometry problem with the reflection of the incenter

@Alex Zhao's proof is incorrect, but I think I've managed to make the idea work. As in the original, this is just a big angle chase. Let $K$ be the intersection of $(I)$ and $BC$ closest to $C$. The ...
brainjam's user avatar
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5 votes
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Distance of ray reflection inside a unit cube

Suppose to watch the cube "from above", so that emitting vertex (red in diagram below) and receiving vertex are opposite vertices of a unit square. We can reflect this square about its sides, so that ...
Intelligenti pauca's user avatar
5 votes

What's the equation of the graph $y=x^3-x^2+x-2$ after it is reflected in the x axis, and then the y axis as well?

Reflecting wrt $x$ axis, inverts the sign of $y$ coordinate. So After reflecting wrt $x$ axis, the equation is: $$ -y = x^3-x^2+x-2 $$ Now reflecting wrt $y$ axis, inverts the sign of $x$. This ...
artha's user avatar
  • 484
5 votes
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How can I prove mathematically the reflection matrix has only the eigenvalues 1 or -1?

$M=I-2P$. Then $M^2=(I-2P)(I-2P)=I^2-4P+4P^2=I-4P+4P=I$ and $M^2x=λ^2x=Ix=x$. Then $λ^2=1$. Hence $λ=±1$
Kubrick's user avatar
  • 334
5 votes
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An artist needs help from mathematicians! Angles of reflections: should I paint these distant trees in the water's reflections?

In a simple mathematical model, the surface of the pond is a plane $P$; the viewer's eye is a point $E$ "above" $P$. A point $S$ in the scene is visible to $E$ as a reflection in the pond if ...
Andrew D. Hwang's user avatar
5 votes
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Finite reflection group: Reflections with the same reflecting hyperplane are equal?

Here is a more direct argument, which avoids using Maschke's Theorem (it also works in infinite-dimensional case). Let $s, t\in GL(V)$ be involutions fixing the same hyperplane $H\subset V$. Then $s, ...
Moishe Kohan's user avatar
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5 votes
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Reflections of Graphs?

For any function $g(x) = \text{ expression}$, you evaluate $g(1)$ by replacing $x$ with $1$ in the $\text{ expression}$. You can do this for any number instead of $1$. Suppose I say $\text{ ...
D S's user avatar
  • 3,576
5 votes

Function with same symmetry as planar hexagonal lattice

A correct answer needs an extra term. Define the function $\,f:\mathbb{C}\to\mathbb{R}\,$ by $$ f(z) := g(z) + g(\omega\,z) + g(\omega^2\,z) $$ where $\,z = x + i\,y\, $ is a complex number, $\,\omega ...
Somos's user avatar
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5 votes
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Unitary operators and a product of reflections

This is akin to a typographical error. As you note the stated result is not true in complex vector spaces, at least if "Householder transformation"/"reflection" is defined as a ...
leslie townes's user avatar
4 votes

Determining whether an orthogonal matrix represents a rotation or reflection

Here is a general guideline for $2 \times 2$ orthogonal matrices. They have one of the two forms $$\text{Either} \ \ R = \begin{bmatrix} a &-b\\[0.3em] b & \ \ \ a\\[0.3em] ...
Jean Marie's user avatar
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4 votes
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What algebra is generated by $\mathrm{O}(2)$?

Well, as you said, any "complex number" $\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$ is a linear combinations of rotations. This implies that any matrix of the form $\begin{pmatrix} 1 & ...
PseudoNeo's user avatar
  • 9,621
4 votes

How many non-infinite plane curves with infinite reflectional symmetry?

Here is a theorem along the lines you are asking for. To avoid worrying about what the definition of "finite curve" should be, I will simply consider arbitary compact subsets of $\mathbb{R}^2$. ...
Eric Wofsey's user avatar
4 votes

Symmetry of point about a line in 3d

Let $\frac{x+1}{4}=\frac{y+1}{-3}=\frac{z-15}{16}=t$ and $(x,y,z)$ be the needed point. Hence, $$\frac{x+t_x}{2}=-1+4t,$$ $$\frac{y+t_y}{2}=-1-3t$$ and $$\frac{z+t_z}{2}=15+16t,$$ which gives, $$x=-2+...
Michael Rozenberg's user avatar
4 votes

A geometry problem with the reflection of the incenter

Denote circumcircle of $\triangle ABC$ as $\Omega$ and circle with center $M$ and radius $MI$ as $\Gamma$. It's well-known that $\Gamma$ is the circumcircle of $\triangle BIC$. Let $\Phi$ be the ...
richrow's user avatar
  • 4,092

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