Calculate Arc Radius From Distance Between Two Points In TikZ

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When working with TikZ, a powerful tool for creating graphics in LaTeX, you might encounter scenarios where you need to draw arcs with a radius determined by the distance between two defined points. While this seems straightforward, directly using the vector representing the distance between two points as the radius can lead to errors because TikZ expects a numerical value for the radius, not a coordinate vector. This article will guide you through the process of accurately calculating and utilizing the distance between two points as the radius of an arc in TikZ.

Understanding the Challenge

In TikZ, points are defined as coordinates, and the distance between them is conceptually a vector. However, the radius parameter in TikZ expects a numerical value representing the length of the radius. Simply passing the vector representing the distance won't work. The key is to extract the numerical distance from the coordinate difference.

For example, consider the scenario where you have two points, P and C, and you want to draw an arc with its center at C and a radius equal to the distance between P and C. A naive approach might involve trying to directly use the vector difference as the radius, which will result in a TikZ error. We need a method to compute the numerical distance from the coordinate information.

\documentclass{article}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
  \coordinate (P) at (0,0);
  \coordinate (C) at (3,4);
  
  % This will cause an error
  % \draw (C) circle (P); 

  \end{tikzpicture}
\end{document}

The above example highlights the problem: TikZ expects a numerical radius, not a point. The subsequent sections will delve into how to correctly calculate the distance and use it as a radius.

Calculating the Distance Between Two Points

To calculate the distance between two points in TikZ, we leverage the let operation combined with the veclen function. The let operation allows us to perform calculations on coordinates and store the results in macros. The veclen function calculates the length of a vector, which, in this context, is the distance between the two points.

The syntax for this calculation is as follows:

\path let \p1 = (P), \p2 = (C), \n1 = {veclen(\x2-\x1,\y2-\y1)} in ...;

Here’s a breakdown:

  • \p1 = (P): Assigns the coordinates of point P to the coordinate variable \p1.
  • \p2 = (C): Assigns the coordinates of point C to the coordinate variable \p2.
  • \n1 = {veclen(\x2-\x1,\y2-\y1)}: This is the crucial part. It calculates the distance. \x1 and \y1 are the x and y coordinates of \p1, and \x2 and \y2 are the x and y coordinates of \p2. The veclen function takes the difference in x-coordinates and y-coordinates and computes the Euclidean distance: (\x2−\x1)2+(\y2−\y1)2\sqrt{(\x2 - \x1)^2 + (\y2 - \y1)^2}. The result is stored in the numerical macro \n1.

Now, \n1 holds the numerical distance between points P and C, which can be used as the radius for an arc or circle.

Drawing an Arc with the Calculated Radius

Once you have calculated the distance and stored it in a macro (e.g., \n1), you can use it as the radius for drawing an arc or a circle. The syntax for drawing a circle with a specified radius is:

\draw (center) circle (radius);

For an arc, the syntax is:

\draw (center) arc (start angle:end angle:radius);

Integrating this with the distance calculation, the code might look like this:

\documentclass{article}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
  \coordinate (P) at (0,0);
  \coordinate (C) at (3,4);
  
  \path let \p1 = (P), \p2 = (C), \n1 = {veclen(\x2-\x1,\y2-\y1)} in
    \draw (C) circle (\n1);

   %visualize the points and the distance  
   \draw[red, thick] (P) -- (C);
   \node[circle, fill, inner sep=2pt, label=below:P] at (P) {};
   \node[circle, fill, inner sep=2pt, label=above right:C] at (C) {};

\end{tikzpicture}
\end{document}

This code first defines two points, P and C. Then, it calculates the distance between them using the let operation and veclen function, storing the result in \n1. Finally, it draws a circle centered at C with a radius equal to the calculated distance \n1. The addition of the red line and filled circles helps to visualize the points and the radius.

Drawing an Arc Segment

To draw an arc segment instead of a full circle, you can specify the start and end angles in the arc command. For instance, to draw an arc from 0 degrees to 180 degrees with the calculated radius:

\documentclass{article}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
  \coordinate (P) at (0,0);
  \coordinate (C) at (3,4);
  
  \path let \p1 = (P), \p2 = (C), \n1 = {veclen(\x2-\x1,\y2-\y1)} in
    \draw (C) arc (0:180:\n1);

   %visualize the points and the distance  
   \draw[red, thick] (P) -- (C);
   \node[circle, fill, inner sep=2pt, label=below:P] at (P) {};
   \node[circle, fill, inner sep=2pt, label=above right:C] at (C) {};

\end{tikzpicture}
\end{document}

This code will draw a semi-circle centered at C with a radius equal to the distance between P and C. The arc command takes three arguments: the start angle, the end angle, and the radius. The angles are measured in degrees, with 0 degrees being the positive x-axis.

Advanced Techniques and Considerations

Using the calc Library

TikZ's calc library offers an alternative way to calculate distances, which can sometimes be more readable. To use the calc library, you need to include it in your document:

\usepackage{tikz}
\usetikzlibrary{calc}

Then, you can calculate the distance using the $(...) syntax:

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

\begin{document}
\begin{tikzpicture}
  \coordinate (P) at (0,0);
  \coordinate (C) at (3,4);
  
  \pgfmathsetmacro{\distance}{veclen($(C)-(P)$)}
  \draw (C) circle (\distance);
  
  %visualize the points and the distance  
  \draw[red, thick] (P) -- (C);
  \node[circle, fill, inner sep=2pt, label=below:P] at (P) {};
  \node[circle, fill, inner sep=2pt, label=above right:C] at (C) {};
\end{tikzpicture}
\end{document}

In this example, $(C)-(P)$ calculates the vector difference between points C and P. The veclen function then calculates the length of this vector. We use \pgfmathsetmacro to store the distance in the macro \distance for later use. This method can be more intuitive for those familiar with vector notation.

Handling Units

TikZ uses a default unit of centimeters (cm). If your coordinates are in different units, you need to ensure consistency. You can explicitly specify units when defining coordinates or when using the calculated distance. For example, if your coordinates are in points (pt), you might need to convert the distance to centimeters or use points directly in the radius:

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

\begin{document}
\begin{tikzpicture}
  \coordinate (P) at (0pt,0pt);
  \coordinate (C) at (30pt,40pt);
  
  \pgfmathsetmacro{\distance}{veclen($(C)-(P)$)}
  \draw (C) circle (\distance pt);

   %visualize the points and the distance  
   \draw[red, thick] (P) -- (C);
   \node[circle, fill, inner sep=2pt, label=below:P] at (P) {};
   \node[circle, fill, inner sep=2pt, label=above right:C] at (C) {};
\end{tikzpicture}
\end{document}

Here, we've defined the coordinates in points (pt) and then used \distance pt when specifying the radius of the circle, ensuring that the units are consistent.

Naming Distances for Reuse

In complex diagrams, you might need to reuse the calculated distance multiple times. Storing the distance in a macro, as demonstrated earlier, is a good practice. You can also define styles that incorporate these calculations for even greater reusability:

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

\begin{document}
\begin{tikzpicture}
  \coordinate (P) at (0,0);
  \coordinate (C) at (3,4);
  
  \tikzset{
    distancearc/.style args={#1 and #2 at #3}{
      insert path={
        let \p1 = (#1), \p2 = (#2), \n1 = {veclen(\x2-\x1,\y2-\y1)} in
        (#3) arc (0:360:\n1)
      }
    }
  }

  \draw[distancearc={P} and {C} at {C}];

   %visualize the points and the distance  
   \draw[red, thick] (P) -- (C);
   \node[circle, fill, inner sep=2pt, label=below:P] at (P) {};
   \node[circle, fill, inner sep=2pt, label=above right:C] at (C) {};
\end{tikzpicture}
\end{document}

This code defines a TikZ style called distancearc that takes two point names and a center point as arguments. It calculates the distance between the first two points and then draws an arc (in this case, a full circle) centered at the specified center point with that distance as the radius. Styles like this can greatly simplify your TikZ code and make it more maintainable.

Conclusion

Calculating the distance between two points and using it as the radius of an arc in TikZ requires a specific approach due to how TikZ handles coordinates and numerical values. By using the let operation with the veclen function or leveraging the calc library, you can accurately determine the distance and use it for drawing circles, arcs, and other geometric figures. Understanding these techniques empowers you to create more complex and precise diagrams in LaTeX using TikZ. Mastering these techniques, along with considerations for units and reusability, will significantly enhance your ability to create sophisticated graphics in your LaTeX documents. This article has provided a comprehensive guide to these methods, equipping you with the knowledge and tools to confidently tackle such challenges in your TikZ projects.