What does "$ (b)!1cm!270:(a) $"mean?

Question

I have latex code:

\documentclass{minimal}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\pgfdeclarelayer{background}
\pgfdeclarelayer{foreground}
\pgfsetlayers{background,main,foreground}
\xdefinecolor{darkgreen}{RGB}{175, 193, 36}
\newcounter{cntShader}
\newcounter{cntRoot}
\setcounter{cntShader}{20}
\def\couleur{darkgreen}
\begin{tikzpicture}
    \foreach \y in {86,38,15}{
        \setcounter{cntShader}{1}
        \coordinate (a) at (0,0);
        \coordinate (b) at (0:1);
        \foreach \x in {1,...,\y}{%
            \coordinate (c) at ($ (b)!1cm!270:(a) $);
            \begin{pgfonlayer}{background}
                \draw[fill=\couleur!\thecntShader] (a)--(b)--(c)--cycle;
            \end{pgfonlayer}
            \setcounter{cntRoot}{\x}
            \addtocounter{cntRoot}{1}
            \node[fill=white,draw,circle,inner sep=1pt] at (c)
                {$\sqrt{\thecntRoot}$};
            \coordinate (b) at (c);
            \pgfmathsetcounter{cntShader}{\thecntShader+4}
            \setcounter{cntShader}{\thecntShader}
       }
    }
    \node[fill=white,draw,circle,inner sep=1pt] at (0:1) {$\sqrt{1}$};
\end{tikzpicture}
\end{document}

it turns out the following picture:

I don't know the line \coordinate (c) at ($ (b)!1cm!270:(a) $); mean. And I am confused about why to use \pgfdeclarelayer{foreground} and \pgfsetlayers{background,main,foreground}. I just see the code use background layer in \begin{pgfonlayer}{background}.

Do not use \documentclass{minimal} - it is a special class -not for minimal examples. - use standalone or article. — hpekristiansen, Sep 22 '21 at 03:39
Just for reference: https://tex.stackexchange.com/questions/42114/why-should-the-minimal-class-be-avoided?noredirect=1&lq=1 and https://tex.meta.stackexchange.com/questions/228/ive-just-been-asked-to-write-a-minimal-working-example-mwe-what-is-that — hpekristiansen, Sep 22 '21 at 03:51

hpekristiansen · Answer 1 · 2021-09-22T03:56:38.227

P.151 in the manual:

⟨coordinate⟩!⟨dimension⟩!⟨angle⟩:⟨second coordinate⟩ When you write ⟨a⟩!⟨dimension⟩!⟨b⟩, this means the following: Use the point that is distanced ⟨dimension⟩ from ⟨a⟩ on the straight line from ⟨a⟩ to ⟨b⟩

As before, if you use a ⟨angle⟩, the ⟨second coordinate⟩ is rotated by this much around the ⟨coordinate⟩ before it is used.

Examples can be seen on page 151 in the manual.

\coordinate (c) at ($ (b)!1cm!270:(a) $);

means stand at (b) look in the direction of (a), turn 270 degrees and walk 1 cm. At this position place coordinate (c).

Yes the code would work without the foreground layer, but why not keep it as a standard way of having useful layers.

What does "$ (b)!1cm!270:(a) $"mean?

1 Answers1