tikz coordinate calculations gone wild

Question

I'm quite a bit baffled why my coordinate calculations are coming out so differently. All coordinates are where I expect them to be, except coordinate Z.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\pagestyle{empty}
\setlength{\parindent}{0pt}
\begin{document}

%% Attempt 01
\begin{tikzpicture}[x=0.25cm,y=0.25cm]
    %% why does this draw a 5 by 5 grid when I don't specify "step=1"???
    \draw[help lines,blue!20,step=1] (0,0) grid (20,20);
    %% quarter circle
    \draw[blue]  (  0:20)  arc(0:90:20);    

    %% Coordinates "B", "Bn", "Bl", "Br" all come out where I expect
    %% Coordinate  "Z" is not where I expected it.
    \path  [draw,line width=2pt,red] 
           (0,0)                 coordinate [label=135:P] (Q) 
                  -- +( 90:20)   coordinate (B)  -- +( $ (B) !-1em! (Q) $ )  coordinate (Bn) node {$Bn$}
              (B) -- +(0:20)     coordinate (Bl) -- +( $ (B) !0.5! (Bl) $ )  coordinate [label=90:Z] (Z)
                  -- +(180:20)   coordinate (Br)
              ;

    %% Coordinate "X" goes where I thought coordinate "Z" would have gone too
    \draw ( $ (B) !0.5!(Bl) $ ) coordinate [label=90:X] (X) circle (2pt);

\end{tikzpicture}

\vspace{1cm}

Same as example above except that $B$ is defined by \verb=(30:20)=
instead of \verb=(90:20)= relative to \verb=(0,0)=.  Also, I omitted
\verb!step=1! from the \verb=help lines=.

%% Attempt 02---same as attempt 01 except for (1) the placement definition
%% of coordinate "B" and (2) not "step=1" for the "help lines".
\begin{tikzpicture}[x=0.25cm,y=0.25cm]
    %% why does this draw a 5 by 5 grid when I don't specify "step=1"???
    \draw[help lines,blue!20] (0,0) grid (20,20);
    %% quarter circle
    \draw[blue]  (  0:20)  arc(0:90:20);    

    %% Coordinates "B", "Bn", "Bl", "Br" all come out where I expect
    %% Coordinate  "Z" is not where I expected it.
    \path  [draw,line width=2pt,red,opacity=0.50] 
           (0,0)                 coordinate [label=135:P] (Q) 
                  -- +( 30:20)   coordinate (B)  -- +( $ (B) !-1em! (Q) $ )  coordinate (Bn) node {$Bn$}
              (B) -- +(0:20)     coordinate (Bl) -- +( $ (B) !0.5! (Bl) $ )  coordinate [label=90:Z] (Z)
                  -- +(180:20)   coordinate (Br)
              ;

    %% Coordinate "X" goes where I thought coordinate "Z" would have gone too
    \draw ( $ (B) !0.5!(Bl) $ ) coordinate [label=90:X] (X) circle (2pt);

\end{tikzpicture}

\end{document}

Why are the coordinate calculations---for what seems to me to be the same thing---turning out so differently?

Minor question

Why must I specify "step=1" to get my help lines to look correctly? Without step=1, I only get a five-by-five grid.

Answer 1

The difference between the line which places the X node and the one which places the Z node is that in the first one you are using absolute coordinates, while in the second one you are using relative coordinates because you preceded the coordinates with a + sign.

The + is not required since you are not giving the new coordinate as a displacement with respect other, but instead you are computing the new coordinates using a calc expression. You can remove the + in all your paths (at least in this case):

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\pagestyle{empty}
\setlength{\parindent}{0pt}
\begin{document}

%% Attempt 01
\begin{tikzpicture}[x=0.25cm,y=0.25cm]
    %% why does this draw a 5 by 5 grid when I don't specify "step=1"???
    \draw[help lines,blue!20,step=1] (0,0) grid (20,20);
    %% quarter circle
    \path  (  0:20)   coordinate (A);
    \draw[blue]  (A) arc(0:90:20);

    %% Coordinates "B", "Bn", "Bl", "Br" all come out where I expect
    %% Coordinate  "Z" is not where I expected it.
    \path  [draw,line width=2pt,red]
           (0,0)                 coordinate [label=135:P] (Q)
                  -- +( 90:20)   coordinate (B)  -- ( $ (B) !-1em! (Q) $ )  coordinate (Bn) node {$B$}
              (B) -- +(0:20)     coordinate (Bl) -- ( $ (B) !0.5! (Bl) $ )  coordinate [label=90:Z] (Z)
                  -- +(180:20)   coordinate (Br)
              ;

    %% Coordinate "X" goes where I thought coordinate "Z" would have gone too
    \draw ( $ (B) !0.5!(Bl) $ ) coordinate [label=90:X] (X) circle (2pt);

\end{tikzpicture}

\vspace{1cm}

Same as example above except that $B$ is defined by \verb=(30:20)=
instead of \verb=(90:20)= relative to \verb=(0,0)=.  Also, I omitted
\verb!step=1! from the \verb=help lines=.

%% Attempt 02---same as attempt 01 except for (1) the placement definition
%% of coordinate "B" and (2) not "step=1" for the "help lines".
\begin{tikzpicture}[x=0.25cm,y=0.25cm]
    %% why does this draw a 5 by 5 grid when I don't specify "step=1"???
    \draw[help lines,blue!20] (0,0) grid (20,20);
    %% quarter circle
    \path  (  0:20)   coordinate (A);
    \draw[blue]  (A) arc(0:90:20);

    %% Coordinates "B", "Bn", "Bl", "Br" all come out where I expect
    %% Coordinate  "Z" is not where I expected it.
    \path  [draw,line width=2pt,red,opacity=0.50]
           (0,0)                 coordinate [label=135:P] (Q)
                  -- +( 30:20)   coordinate (B)  -- ( $ (B) !-1em! (Q) $ )  coordinate (Bn) node {$B$}
            (B)   -- +(0:20)     coordinate (Bl) -- ( $ (B) !0.5! (Bl) $ )  coordinate [label=90:Z] (Z)
                  -- +(180:20)   coordinate (Br)
              ;

    %% Coordinate "X" goes where I thought coordinate "Z" would have gone too
    \draw ( $ (B) !0.5!(Bl) $ ) coordinate [label=90:X] (X) circle (2pt);

\end{tikzpicture}
\end{document}

Update

Based on the comments and on a better understanding of the intended output, I suggest to completely eliminate relative coordinates (those beginning with +) in your path, and replacing those for explicit calculations (which can involve polar coordinates). So, for example, to get a point 20 units to the right of (B), use ($(B)+(0:20)$), instead of moving the origin to (B) and then write +(0:20).

Mixing absolute and relative coordinates make difficult to guess and control the result. Use explicit calculations to a better understanding and control. With this approach your code would be:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\pagestyle{empty}
\setlength{\parindent}{0pt}
\begin{document}

%% Attempt 01
\begin{tikzpicture}[x=0.25cm,y=0.25cm]
    %% why does this draw a 5 by 5 grid when I don't specify "step=1"???
    \draw[help lines,blue!20,step=1] (0,0) grid (20,20);
    %% quarter circle
    \path  (  0:20)   coordinate (A);
    \draw[blue]  (A) arc(0:90:20);

    %% Coordinates "B", "Bn", "Bl", "Br" all come out where I expect
    %% Coordinate  "Z" is not where I expected it.
    \path  [draw,line width=2pt,red]
           (0,0)                        coordinate [label=135:P] (Q)
                   -- ($(Q)+(90:20)$)   coordinate (B)  -- ( $ (B) !-1em! (Q) $ )  coordinate (Bn) node {$B$}
               (B) -- ($(B)+(0:20)$)    coordinate (Bl) -- ( $ (B) !0.5! (Bl) $ )  coordinate [label=90:Z] (Z)
                   -- ($(B)+(180:20)$)  coordinate (Br)
              ;

    %% Coordinate "X" goes where I thought coordinate "Z" would have gone too
    \draw ( $ (B) !0.5!(Bl) $ ) coordinate [label=90:X] (X) circle (2pt);

\end{tikzpicture}
\vspace{1cm}

Same as example above except that $B$ is defined by \verb=(30:20)=
instead of \verb=(90:20)= relative to \verb=(0,0)=.  Also, I omitted
\verb!step=1! from the \verb=help lines=.

%% Attempt 02---same as attempt 01 except for (1) the placement definition
%% of coordinate "B" and (2) not "step=1" for the "help lines".
\begin{tikzpicture}[x=0.25cm,y=0.25cm]
    %% why does this draw a 5 by 5 grid when I don't specify "step=1"???
    \draw[help lines,blue!20] (0,0) grid (20,20);
    %% quarter circle
    \path  (  0:20)   coordinate (A);
    \draw[blue]  (A) arc(0:90:20);

    %% Coordinates "B", "Bn", "Bl", "Br" all come out where I expect
    %% Coordinate  "Z" is not where I expected it.
    \path  [draw,line width=2pt,red,opacity=0.50]
           (0,0)                 coordinate [label=135:P] (Q)
                  -- ($(Q)+(30:20)$)    coordinate (B)  -- ( $ (B) !-1em! (Q) $ )   coordinate (Bn) node {$B$}
             (B)  -- ($(B)+(0:20)$)     coordinate (Bl) -- ( $ (B) !0.5! (Bl) $ )  coordinate [label=90:Z] (Z)
                  -- ($(B)+(180:20)$)   coordinate (Br)
              ;

    %% Coordinate "X" goes where I thought coordinate "Z" would have gone too
    \draw ( $ (B) !0.5!(Bl) $ ) coordinate [label=90:X] (X) circle (2pt);

\end{tikzpicture}
\end{document}

And the result (I hope I got it right this time):