TikZ-3dplot and rotation of coordinates

Question

In the tikz-3dplot documentation (page 15) \tdplotsetmaincoords{a}{b} is suggested to rotate the system of coordinates. But it apparently realizes rotations only about the x and z axes.

How to realize a system of coordinates where x points up, z points right and y points almost "out of the page"? (A rotation about the y axis would be needed).

Despite trying several values of a and b, I didn't obtain this. My code so far (partially derived from this very useful answer):

\documentclass[border=2mm,tikz]{standalone}
\usepackage{tikz}
\usepackage{tikz-3dplot}

\begin{document}

\tdplotsetmaincoords{70}{70}
\begin{tikzpicture}[tdplot_main_coords,font=\sffamily]

\draw[-latex] (-7.5,0,0) -- (7.5,0,0)  node[above right]  {$x$};
\draw[-latex] (0,-7.5,0) -- (0,7.5,0)  node[below] {$y$};
\draw[-latex] (0,0,-7.5) -- (0,0,7.5)  node[above left]  {$z$};

\draw[fill=gray,opacity=0.4] (0,-7.5,7.5) -- (0,7.5,7.5) -- (0,7.5,-7.5) -- (0,-7.5,-7.5) -- cycle;
\draw[fill=gray,opacity=0.2] (-7.5,0,-7.5) -- (-7.5,0,7.5) -- (7.5,0,7.5) -- (7.5,0,-7.5) -- cycle;

\end{tikzpicture}

\end{document}

The z axis should replace the current y axis, the x axis should replace the current z axis, and the y axis should replace the current x axis.

As an additional attempt, I tried \tdplotsetrotatedcoords{0}{40}{0} before the last two draws, but it had no effect at all.

Answer 1

The easiest way to go would be to just relabel the axes in such a way that x points up and so on. But this is probably not what you want. So the way to go is to use rotated coordinates.

\documentclass[border=2mm,tikz]{standalone}
\usepackage{tikz-3dplot}

\begin{document}

\tdplotsetmaincoords{70}{70}
\begin{tikzpicture}[tdplot_main_coords,font=\sffamily]
\tdplotsetrotatedcoords{-90}{-90}{0}
\begin{scope}[tdplot_rotated_coords]
\draw[-latex] (-7.5,0,0) -- (7.5,0,0)  node[above right]  {$x$};
\draw[-latex] (0,-7.5,0) -- (0,7.5,0)  node[below] {$y$};
\draw[-latex] (0,0,-7.5) -- (0,0,7.5)  node[above left]  {$z$};
\end{scope}
\draw[fill=gray,opacity=0.4] (0,-7.5,7.5) -- (0,7.5,7.5) -- (0,7.5,-7.5) -- (0,-7.5,-7.5) -- cycle;
\draw[fill=gray,opacity=0.2] (-7.5,0,-7.5) -- (-7.5,0,7.5) -- (7.5,0,7.5) -- (7.5,0,-7.5) -- cycle;
\end{tikzpicture}
\end{document}

First of all, yes, at first sight it is odd that \tdplotsetmaincoords takes only two parameters. If you think a bit more, you realize that, in order to describe a rotation in 3D, you need only two angles. But I agree with you that adjusting the angles in \tdplotsetmaincoords won't allow you to access all possible rotations. That's why I propose using \tdplotsetrotatedcoords. Notice that there are ambiguities, and from the manual it is not clear in which order these rotations are being applied. (More specifically, rotating first about the x-axis and then about the y-axis yields a different result than rotating first about the y-axis and then about the x-axis. That is, the problem is that SO(3) is non-Abelian.) Using these tricks I was so far able to get any desired rotation.

Answer 2

I believe a combination of \tdplotsetmaincoords{100}{10} with the option rotate=269.7 will do what you ask!

\documentclass[border=2mm,tikz]{standalone}
\usepackage{tikz}
\usepackage{tikz-3dplot}

\begin{document}

\tdplotsetmaincoords{80}{175}
\begin{tikzpicture}[tdplot_main_coords,font=\sffamily,rotate=269.2]

\draw[-latex] (-7.5,0,0) -- (7.5,0,0)  node[above right]  {$x$};
\draw[-latex] (0,-7.5,0) -- (0,7.5,0)  node[below] {$y$};
\draw[-latex] (0,0,-7.5) -- (0,0,7.5)  node[above left]  {$z$};

\draw[fill=gray,opacity=0.4] (0,-7.5,7.5) -- (0,7.5,7.5) -- (0,7.5,-7.5) -- (0,-7.5,-7.5) -- cycle;
\draw[fill=gray,opacity=0.2] (-7.5,0,-7.5) -- (-7.5,0,7.5) -- (7.5,0,7.5) -- (7.5,0,-7.5) -- cycle;

\end{tikzpicture}

\end{document}