How to compute xslant and yslant

Question

There must be an easy way to compute the required values of xslant and yslant to obtain the text in the "correct plane". Below I have guessed at then, but that is a very tedious process. The desired results should be that the following lines should not have any magic numbers:

\tikzset{yz slant style/.style={red,    yslant=-0.5,  xslant=-0.1}}
\tikzset{xy slant style/.style={blue,   yslant=-0.25, xslant=1   }}
\tikzset{xz slant style/.style={orange, yslant=1,     xslant=0.05}}

and instead should be based on the values of \xc,\xs,\yc, and \ys (or \elevation and \anglerot). My guessed value produce something which I think are close:

Notes:

• I realize that this is more of a math question, but without fully understanding how tikz/pgf works I don't know how to compute them.
• The preamble code is from 3D Arrow Tips for TikZ/pgfplots.

Code:

\documentclass{scrartcl}
\usepackage{xcolor}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
%% https://tex.stackexchange.com/a/51388/4301
\newcommand\elevation{14}
\newcommand\anglerot{-50}
\pgfmathsetmacro\xc{cos(\anglerot)}

\pgfmathsetmacro\xs{sin(\anglerot)}

\pgfmathsetmacro\yc{cos(\elevation)} 
\pgfmathsetmacro\ys{sin(\elevation)} 
\newcommand\axexthreed{\xs1cm,-\xc1cm} 
\newcommand\axeythreed{\yc1cm,-\ys1cm}
\newcommand*\axezthreed{0cm,1cm}
\newcommand{\CoordX}{3}%
\newcommand{\CoordY}{4}%
\newcommand*{\CoordZ}{5}%
%% How do I compute these:
\tikzset{yz slant style/.style={red, yslant=-0.5, xslant=-0.1}}
\tikzset{xy slant style/.style={blue, yslant=-0.25, xslant=1}}
\tikzset{xz slant style/.style={orange, yslant=1, xslant=0.05}}
\begin{tikzpicture}[x = {(\axexthreed)},
                    y = {(\axeythreed)},
                    z = {(\axezthreed)},
                    ]
\draw[black, ultra thick,-latex] (-1,0,0) -- (\CoordX,0,0) node[black,left=6pt]  {$x$}; 
\draw[black, ultra thick,-latex] (0,-1,0) -- (0,\CoordY,0) node[black,right=6pt] {$y$}; 
\draw[black, ultra thick,-latex] (0,0,-1) -- (0,0,\CoordZ) node[black,above=6pt] {$z$};       

\foreach \x in {1,...,\CoordX} {
    \draw [thin, draw=gray] (\x,\CoordY,0) -- (\x,0,0)
        node [left, yz slant style] at ($(\x,0,0)+(-0.2,0,0)$) {\x};
}
\foreach \x in {1,...,\CoordY} {
    \draw [thin, draw=gray] (\CoordX,\x,0) -- (0,\x,0)
        node [right, xy slant style] at ($(0,\x,0)+(-0.4,-0.3,0)$) {\x};
}
\foreach \x in {1,...,\CoordZ} {
    \draw [thin, draw=gray] (\CoordX,0,\x) -- (0,0,\x)
        node [right, xz slant style] at ($(0,0,\x)+(0,-0.0,-0.2)$) {\x};
}


\end{tikzpicture}
\end{document}

Answer 1

Okay, yslant to spin vertical letters, rotate and xslant to lay letters flat, xscale and yscale for forehortening.

\tikzset{yz slant style/.style={red, yslant=-tan(\elevation), xscale=cos(\elevation)}}
\tikzset{xy slant style/.style={blue, xslant=-tan(\elevation+\anglerot), rotate=-\elevation, yscale=cos(\anglerot)}}
\tikzset{xz slant style/.style={orange, yslant=-cot(\anglerot), xscale=sin(-\anglerot)}}