# tikz arithmetic with numbers and lengths

Consider this example:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
\def\starty{3}
\def\length{1};
\coordinate(a1) at (1, \starty);
\coordinate(b1) at ($(a1) + (0, -\length)$);
\coordinate(a2) at (2, \starty - \length);
\coordinate(b2) at ($(a2) + (0, \length)$);
\draw[red, ->](a1) -- (b1);
\draw[red, ->](b2) -- (a2);
\draw (0, 0) -- (3, 0);
\draw (0, 0) -- (0, 3);
\end{tikzpicture}
\end{document}

The result is 2 horizontally displaced arrows as it should be from a simple arithmetical inspection:

However, when I replace

\def\length{1};

by

\def\length{1cm};

the result is unexpected:

What is causing the discrepancy and how should I fix this example?

• If you add dimensionless+something with units, dimensionless is interpreted in pt.
– user194703
Nov 30, 2019 at 21:48

The issue is that you add/combine expressions with and without units. TikZ distinguishes between expressions with and without units. I recommend reading this answer. If you have

\path (x,y) coordinate (p);

with x and y dimensionless, then the point p will be at x*(x unit vector)+y*(y unit vector). The initial values of these unit vectors are (1cm,0) and (0,1cm), respectively, but you can change them, e.g. with x=(1cm,0.2cm). (These changes are tricky if you do not supply units because if one uses x={({cos(20)},{(sin(20)})},y={({cos(20+90)},{(sin(20+90)})}, then one does not get just a rotated coordinate system. Rather, when y=... is parsed, it already uses the redefined x unit vector. This is why packages like tikz-3dplot attach units to define the rotated coordinate systems.)

If you have

\path (x,y) coordinate (p);

where x and y carry units, then the point p will be at x to the right and y up (modulo transformations like rotations, of course). For the initial values of the unit vectors

\path (1,2) coordinate (p);

and

\path (1cm,2cm) coordinate (p);

yield the same results, but in general they don't. You can also have one coordinate with units and the other one without, e.g.

\path (1cm,2) coordinate (p);

will lead to a point 1cm to the right and shifted by twice the y unit vector.

Now, coming to your question, if you present TikZ a mix

\path (a+b,y) coordinate (p);

where a carries units and b does not, then TikZ will attach units pt to b. So, for instance, in

\path (1cm+1,2) coordinate (p);

p will have an x coordinate of 1cm+1pt, while in

\path (1+1,2) coordinate (p);

it will have an x coordinate of 2 times the x unit vector.

To illustrate this, I compare the coordinates of your MWE with those in which I appended pt to the dimensionless expressions, and show that they match.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}

\subsection*{No units}

\begin{tikzpicture}
\def\starty{3}
\def\length{1};
\coordinate(a1) at (1, \starty);
\coordinate(b1) at ($(a1) + (0, -\length)$);
\coordinate(a2) at (2, \starty - \length);
\coordinate(b2) at ($(a2) + (0, \length)$);
\draw[red, ->](a1) -- (b1);
\draw[red, ->](b2) -- (a2);
\draw (0, 0) -- (3, 0);
\draw (0, 0) -- (0, 3);
\end{tikzpicture}

\subsection*{Mix of expressions with and without units}

\begin{tikzpicture}
\def\starty{3}
\def\length{1cm};
\coordinate(a1) at (1, \starty);
\coordinate(b1) at ($(a1) + (0, -\length)$);
\coordinate(a2) at (2, \starty - \length);
\coordinate(b2) at ($(a2) + (0, \length)$);
\draw[red, ->](a1) -- (b1);
\draw[red, ->](b2) -- (a2);
\draw (0, 0) -- (3, 0);
\draw (0, 0) -- (0, 3);
\draw[<->,blue] (2,3pt-1cm) -- ++ (1,0) -- (2,3pt);
\end{tikzpicture}

\end{document}