Questions tagged [reflection]

Question 1

If given a point $(x,y)$, and told to reflect it across a given line $y=mx+b$, there are several ways that this can be achieved. For one, the most simple way is to just count the diagonals, but this ...

Question 2

I'm trying to understand to Wythoffian constructions. In particular, how to show that when additional mirror is activated, all the previous faces remain (translated and dilated) and are separated by ...

Question 3

From a previous post, the following fact has been brought to my attention: For any two continuous functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$, the following function is an isotopy of $\mathbb{...

Question 4

A straight line with a negative slope passes through the point $P(8,1)$ and meets x-axis at $A$ and the y-axis at $B$. Find the minimum possible value of the perimeter of $\triangle AOB$. My Attempt ...

Question 5

Let $A$ be a subset of $\mathbb{R}$ and let $f: A \rightarrow \mathbb{R}$ be a continuous function. If $f(x) = mx$ for some positive number $m > 0$, then the function $H: \mathbb{R}^{2} \times [0, ...

Question 6

Suppose $\mathbf{s}=(s_1,\dots,s_k)$ is a word in $S$. Define $w_i\in W$ by $w_0=1$ and $w_i=s_1\cdots s_i$, and $r_i\in R$ by $r_i=w_{i-1}s_iw_{i-1}^{-1}$. Set $\Phi(\mathbf{s}):=(r_1,\dots,r_k)$. ...

Question 7

$\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}\DeclareMathOperator\ASL{ASL}\...

Question 8

I was able to find the following formula when reflecting a point $(x,y)$ to point $(x^\prime,y^\prime)$ over the line $y = mx$: $$(x^\prime,y^\prime)=\left(\frac{1-m^2}{1+m^2}x+\frac{2m}{1+m^2}y,\frac{...

Question 9

Let two mirror surfaces be defined by the graphs of functions $y=\frac{1}{x}$ and $y=-\frac{1}{x}$ for all $x>0$. A beam of light originates at the point $(0, a)$ $(a>0)$ and travels ...

Question 10

Is there a standard mapping notation for rotations and reflections? Some sources I've looked at use R for rotations and r for reflections, others don't really make a distinction and may use the same ...

Question 11

Suppose I have a finite set $S$ in $\mathbb{R}^n$ and reflections $\sigma_j$ defines as follow:$$\sigma_ j x=\sigma_j(x_1,\ldots,x_{n})=(x_1,\ldots,-x_j,\ldots,x_{n})$$ Formerly, I tried to decompose $...

Question 12

Translations are easy in Cartesian coordinates since each point P can be moved to its corresponding point P$^\prime$ with either a 2-component vector on the plane or a 3-component vector in space. ...

Question 13

As shown in the diagram, a laser light originates vertically from point $A$ and must reach point $T$. There are two rotating mirrors that adjust their angles based on a given relationship with $\theta$...

Question 14

To the best of my knowledge, if $R$ is a (reduced) irreducible root system in some real vector space $V$, then its Weyl group $W(R)$ is the finite Coxeter group generated by the reflections associated ...

Question 15

I am reading "Reflecting Barriers" related chapters in Cox and Miller's book "The theory of stochastic processes". On page 224 it says that: $"$ Consider a Wiener process $X(t)$ ...

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Questions tagged [reflection]

Reflecting a point $(x,y)$ about a line $y=mx+b$

Wythoffian constructions and polyhedra expansion

Isotopy of $\mathbb{R}^{2}$ that swaps two functions (or a function and its reflection)

Minimum Perimeter of triangle with co-ordinate axes as two sides and the third side passing through the point $(8,1)$

Ambient Isotopy of $\mathbb{R}^{2}$ Taking a Function to Its Reflection About the $x$-Axis

Deriving the inequality $|R(1,w)|\geq k$ via wall-crossing parity in Davis's Coxeter Groups book

Exceptional complex reflection groups acting irreducibly on the adjoint representation

algebra error occurs when I generalize a reflection formula

Number of reflections between two hyperbolic mirrors

standard mapping notation for rotations and reflections

Decompose a set into reflectional symmetric sets

looking for general rotation and reflection formulas for cartesian coordinate systems

Reflection of light to certain point using mirrors

A problem on conjugations in Weyl groups.

Question about Reflecting Barriers

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