Articles of reflection

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 […]

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. Under the inversion shapes get distorted in funhouse ways. But […]

Need help with linear transformations (with projection and reflection)?

Let $L$ be the line given by the equation $4x − 3y = 0$. Let $S : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be reflection through that line, and let $P : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be projection onto that line. Determine, geometrically without doing any computations, whether there exist non-zero vectors $x$ such that: $$ a) \qquad S(x) […]

Reflect point across line with matrix

What is the transformation matrix that I multiply a point by if I want to reflect that point across a line that goes through the origin in terms of the angle between the line and the x-axis? In other words, $$y = mx$$ $\theta$ is the angle between the $x$-axis and the line. The position […]