Articles of reflection

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 have an object that is colliding with a circle and I need it to […]

Set in $\mathbb R^²$ with an axis of symmetry in every direction

Let $A\subset \mathbb R^2$ be a set that has an axis of symmetry in every direction, that is, for any $n\in S^1$, there exists a line $D$ orthogonal to $n$, such that $A$ is invariant under the (affine) reflection of axis $D$. It is easy to show that if all of these axes intersect at […]

Range of a standard brownian motion, using reflection principle

With a standard brownian motion $B_t$, I’m trying to find the distribution of the “range”: $$R_{t} = \sup_{0 \leq s \leq t} B_s – \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The reflection principle gives, for $a > 0$, $P(\overline{M_t} \geq a) = 2 P(B_t \geq a)$ (as stated by @A.S. as comment on […]

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… $U=I-\frac{uu^T}{\beta}$ $\beta=\frac{\left \| u \right \|_2^2}{2}$ $Ux=x-u$ $\beta=\frac{16}{2}=8$ $u=\begin{bmatrix} 0\\ 2\\ 3 \end{bmatrix}$ $U=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & […]

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 geometrically/mathematically prove that this will always happen?

Functional equation $P(X)=P(1-X)$ for polynomials

I have encountered the following problem : Find all polynomials $P$ such as $P(X)=P(1-X)$ on $\mathbb{C}$ and then $\mathbb{R}$. I have found that on $\mathbb{C}$ such polynomials have an even degree. Because for each $a$ root, $1-a$ must be a root too. I struggle to find whether we can do something with polynomials of degree […]

Reflection of a curve around a slant line

a fifth-degree function: y = 80*x^5-225*x^4+350*x^3-300*x^2+150*x-20 (the green curve in the image) needs to be reflected/mirrored around the line y=55x-20 (the blue line) and I am only interested in the segment [0,1]. While there is plenty of content on the internet on how to reflect around the axes or vertical/horizontal lines, I have not found […]

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 want to do this in […]

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 wrong. The right answer is $f(x) = 3^{-x+6}$ but I […]

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 don’t understand the significance of doubling […]