Ifthenelse variable equality

Question

Last week I made simple coordinate transformation in order to define \tdplotsetcoord in Cartesian coord. system. My work and support of exp. user can be viewed in this topic. The code looks as follows:

\def\Sx{0} \def\Sy{0} \def\Sz{-3}
\tdplotsetcoord{S}{sqrt((\Sx)^(2) + (\Sy)^(2) + (\Sz)^(2))}{acos(\Sz/(sqrt((\Sx)^(2) + (\Sy)^(2) + (\Sz)^(2))))}{ifthenelse(\Sy<0,-acos(\Sx/(sqrt((\Sx)^(2) + (\Sy)^(2)))),acos(\Sx/(sqrt((\Sx)^(2) + (\Sy)^(2)))))} % {radius}{polar}{asimuth}

However there is a problem with (0,0,z) Cartesian coordinates due to zero division in those math expressions. Also I'm trying to define transformation from zero C. coordinates manually using ifthenelse command:

1) Why variable = 0 doesn't work? I'm getting PGF Math Error: Unknown operator '=' or '=0'. This is the example of polar angle transformation:

\pgfmathsetmacro\polar{ifthenelse(\Sx=0 \AND \Sy=0 \AND \Sz=0,0,acos(\Sz/(sqrt(\Sx^(2) + \Sy^(2) + \Sz^(2)))))}

2) I failed also with \equal command getting bunch of errors (log file). Can you show me the correct syntax? Thanks.

\pgfmathsetmacro\polar{ifthenelse(\equal{\Sx}{0} \AND \equal{\Sy}{0} \AND \equal{\Sz}{0},0,acos(\Sz/(sqrt(\Sx^(2) + \Sy^(2) + \Sz^(2)))))}

Edit: Working example does not differ from previous topic.
MWE:

    \documentclass[a4paper,fleqn,leqno]{article}
    \usepackage{tikz,tikz-3dplot}
    \usetikzlibrary{arrows.meta,decorations.markings}
    \begin{document}
    \begin{tikzpicture}[cm={-1,-1,1,0,(0,0)},x=3.85mm,z=-1cm]
    \draw[-{Classical TikZ Rightarrow[scale=1.2]},thick] (-2,0,0) -- (5,0,0) node[anchor=north,xshift=-3pt] {$x$};
    \draw[-{Classical TikZ Rightarrow[scale=1.2]},thick] (0,-2,0) -- (0,5,0) node[anchor=west] {$y$};
    \draw[-{Classical TikZ Rightarrow[scale=1.2]},thick] (0,0,-2) -- (0,0,5) node[anchor=south] {$z$};
    \def\Sx{0} \def\Sy{0} \def\Sz{-3}

% code without dealing with zero division
    \tdplotsetcoord{S}{sqrt((\Sx)^(2) + (\Sy)^(2) + (\Sz)^(2))}{acos(\Sz/(sqrt((\Sx)^(2) + (\Sy)^(2) + (\Sz)^(2))))}{ifthenelse(\Sy<0,-acos(\Sx/(sqrt((\Sx)^(2) + (\Sy)^(2)))),acos(\Sx/(sqrt((\Sx)^(2) + (\Sy)^(2)))))}

% code implementing zero coordinations definition
    \pgfmathsetmacro\polar{ifthenelse(\Sx=0 \AND \Sy=0 \AND \Sz=0,0,acos(\Sz/(sqrt(\Sx^(2) + \Sy^(2) + \Sz^(2)))))}
% plus asimuth angle zero definition...
    \tdplotsetcoord{S}{sqrt((\Sx)^(2) + (\Sy)^(2) + (\Sz)^(2))}{\polar}{...}

    \draw[-{Stealth[scale=1.5,width=3pt]},color=red,thick] (0,0,0) -- (S) node[midway,anchor=east,font=\footnotesize] {${\vec{r}}_{1}$};
    \draw[dashed,color=red!70!white,semithick] (0,0,0) -- (Sxy);
    \draw[dashed,color=red!70!white,semithick] (Sxy) -- (S);
    \end{tikzpicture}
    \end{document}

Answer 1

The expression

\pgfmathsetmacro\polar{
   ifthenelse(\Sx=0 \AND \Sy=0 \AND \Sz=0,
     0,
     acos(\Sz/(sqrt(\Sx^(2) + \Sy^(2) + \Sz^(2)))))}

has several errors. The ifthenelse keyword implemented by pgf has a different syntax from the \ifthenelse macro from the ifthen package. You seem to be trying to mix the two, hence the issue. For the ifthenelse keyword, logical 'and' is denoted by && and numerical equality by ==. You also need to ensure that your squaring applies to the entirety of e.g. \Sz by adding parentheses:

\pgfmathsetmacro\polar{
    ifthenelse(\Sx==0 && \Sy==0 && \Sz==0,
      0,
      acos(\Sz/(sqrt((\Sx)^(2) + (\Sy)^(2) + (\Sz)^(2)))))}

You also can't do text comparisons with the ifthenelse keyword, so \EQUAL is not applicable at all.

Answer 2

Also the final code looks like this, for someone to facilitate his work:

\def\Sx{-2} \def\Sy{-3} \def\Sz{2}
\pgfmathsetmacro\Sρ{"sqrt((\Sx)^(2)+(\Sy)^(2)+(\Sz)^(2))"}
\pgfmathsetmacro\Sθ{ifthenelse(\Sx==0&&\Sy==0&&\Sz==0,0,"acos(\Sz/(sqrt((\Sx)^(2)+(\Sy)^(2)+(\Sz)^(2))))")}
\pgfmathsetmacro\SΦ{ifthenelse(\Sx==0&&\Sy==0,0,ifthenelse(\Sy<0,"-acos(\Sx/(sqrt((\Sx)^(2)+(\Sy)^(2))))","acos(\Sx/(sqrt((\Sx)^(2)+(\Sy)^(2))))"))}
\tdplotsetcoord{S}{\Sρ}{\Sθ}{\SΦ}

Last three math expressions have to be returned as "text strings" (also it has to be done in another variable to evaluate them in tdplotsetcoord), otherwise ifthenelse command evaluates those expressions (->getting zero division error) even when second option is to return.