How to reproduce 3d image of Householder Reflection similar to given in Tikz

Question

I am just getting started using Tikz and I am having some trouble. There is an image

that I think is perfect that I am trying to recreate in Tikz. I tried to define a plane and also tried looking at Drawing a Plane in 3D space but everything in Tikz seems too specific and I can't seem to find a general guide that might help me figure out how to draw something like this. As a new user it is very daunting. Everything that I have tried so far has come out looking completely wrong.

Any help would be appreciated!!

Answer 1

Welcome to TeX.SX! I don't think that you really need any libraries for that, as plain TikZ already comes with some basic 3D drawing tools. Maybe the following can help you (explanations as comments in the code):

\documentclass[border=10mm]{standalone}
\usepackage{tikz}

\begin{document}

% you can use the options x, y and z to declare your coordinate system,
% for example, x={(20:10mm)} means "1 on the x axis be at coordinate (20:10mm) on the canvas"
% (of course, you can also use cartesian coordinates to define x, y and z)
\begin{tikzpicture}[x={(20:10mm)}, y={(100:15mm)}, z={(5:10mm)}, >=stealth]

% first, we draw the plane keeping the y coordinate at zero:
\fill[black!10] (-2,0,-1) -- (-2,0,1) --  (2,0,1) -- (2,0,-1) -- cycle;

% then, we define some coordinates:
\coordinate (o) at (0,0,0);
\coordinate (a) at (0,1,0);
\coordinate (b) at (1,0,0);
\coordinate (c) at (1,-1,0);
\coordinate (x) at (1,1,0);

% finally, we draw the arrwos by connecting the coordinates and attach labels
\draw[very thick, ->] (o) -- (0,.33,0) node[midway, left] {$u$};
\draw[thick, ->, gray] (o) -- (a) node[black, left] {$(u^Hx)u$};
\draw[thick, ->, green] (o) -- (b) node[black, right] {$x-(u^Hx)u$};

\draw[thick, ->, red] (o) -- (x) node[black, above right] {$x$};
\draw[gray] (a) -- (x);

\draw[thick, ->, gray, dashed] (x) -- (b) node[black, midway, right, font=\scriptsize] {$-(u^Hx)u$};
\draw[thick, ->, gray, dashed] (b) -- (c) node[black, midway, right, font=\scriptsize] {$-(u^Hx)u$};
    
\draw[thick, ->, blue] (o) -- (c) node[black, below] {$(I-2uu^H)x$};

\end{tikzpicture}

\end{document}

Answer 2

I think TikZ (internally 2D) is enough to draw this simple task, see Jasper Habicht's answer above. The below is an alternative with 3D Asymptote. Do you see a Householder reflection by figure and by values ?

Note that in the OP's figure there is a mistake: (I-2u^H x)u instead of (I-2uu^H)x. Here the symbol H stands for Hermitian transpose, or conjugate transpose. In Asymptote, u^Hx is nothing but the scalar product dot(u,x).

// http://asymptote.ualberta.ca/
import three;
size(8cm);
currentprojection=orthographic(2,3,-1,center=true,zoom=.9);
triple u=(0,0,1), x=(0,2,2.5);
triple uh=u;
real uhx=dot(uh,x); 
triple A=x-uhx*u;
triple B=uhx*u;
triple C=x-2*uhx*u;
draw(B--x,gray);
draw(Label("$-(u^Hx)u$",Relative(.5),align=E),x--A,gray+dashed,Arrow3);
draw(Label("$-(u^Hx)u$",Relative(.5),align=E),A--C,gray+dashed,Arrow3);
draw(Label("$x-(u^Hx)u$",EndPoint,align=E),O--A,orange+1pt,Arrow3);
draw(Label("$(u^Hx)u$",EndPoint,align=W),O--B,gray+1pt,Arrow3);
draw(Label("$u$",EndPoint,align=W),O--u,gray+1pt,Arrow3);
draw(Label("$\left(I-2u^Hx\right)u$",EndPoint,align=S),O--C,blue+1pt,Arrow3);
draw(Label("$x$",EndPoint,align=NE),O--x,red+1pt,Arrow3);
draw(shift(-1.5A-1.5X)*surface(plane(4A,3X)),blue+opacity(.1));

// to see numeric values
write("u = ",u);
write("x = ",x);
write("<u,x> = ",uhx);
write("x - <u,x> u = ",A);
write("<u,x> u = ",B);
write("x - 2 <u,x> u = ",C);