# Connect spy with tangent lines

I am using TikZ spy library to magnify part of my plot, and I would like to connect the spy point and the magnifying glass with two tangents to both circles (see second example). I found and implemented an algorithm to compute such lines (or, better, the four interesting points in the two circles), but I cannot understand how to pass to it the coordinates and radii of spy's circles. It seems to me that there is a mismatch between numbers, point and centimeters.

Is there a way to get what I described?

\documentclass[crop,tikz,margin=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{intersections}
\usetikzlibrary{spy}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}[
spy using outlines = {circle,size=3cm,magnification=5,connect spies},
]
\begin{axis}
\coordinate (spy point) at (axis cs: 0, 0);
\coordinate (magnifying glass) at (rel axis cs: -0.4, 0.2);
\end{axis}

\spy on (spy point) in node at (magnifying glass);
\end{tikzpicture}
\begin{tikzpicture}

\def\xa{5}
\def\ya{3}

\def\xb{0}
\def\yb{0}

\coordinate (magnifying glass) at (\xa, \ya);
\coordinate (spy point) at (\xb, \yb);

\pgfmathsetmacro\distancea{sqrt((\xp - \xa) * (\xp - \xa) + (\yp - \ya) * (\yp - \ya) - \radiusa * \radiusa))}
\pgfmathsetmacro\distanceb{sqrt((\xp - \xb) * (\xp - \xb) + (\yp - \yb) * (\yp - \yb) - \radiusb * \radiusb))}
\pgfmathsetmacro\denoma{(\xp - \xa)*(\xp - \xa) + (\yp - \ya)*(\yp - \ya)}
\pgfmathsetmacro\denomb{(\xp - \xb)*(\xp - \xb) + (\yp - \yb)*(\yp - \yb)}

\pgfmathsetmacro\xc{(\radiusa * \radiusa * (\xp - \xa) + \radiusa * (\yp - \ya) * \distancea) / \denoma + \xa}
\pgfmathsetmacro\yc{(\radiusa * \radiusa * (\yp - \ya) - \radiusa * (\xp - \xa) * \distancea) / \denoma + \ya}

\pgfmathsetmacro\xe{(\radiusa * \radiusa * (\xp - \xa) - \radiusa * (\yp - \ya) * \distancea) / \denoma + \xa}
\pgfmathsetmacro\ye{(\radiusa * \radiusa * (\yp - \ya) + \radiusa * (\xp - \xa) * \distancea) / \denoma + \ya}

\pgfmathsetmacro\xd{(\radiusb * \radiusb * (\xp - \xb) + \radiusb * (\yp - \yb) * \distanceb) / \denomb + \xb}
\pgfmathsetmacro\yd{(\radiusb * \radiusb * (\yp - \yb) - \radiusb * (\xp - \xb) * \distanceb) / \denomb + \yb}

\pgfmathsetmacro\xf{(\radiusb * \radiusb * (\xp - \xb) - \radiusb * (\yp - \yb) * \distanceb) / \denomb + \xb}
\pgfmathsetmacro\yf{(\radiusb * \radiusb * (\yp - \yb) + \radiusb * (\xp - \xb) * \distanceb) / \denomb + \yb}

% \draw (\xa, \ya) node[scale=3, green] {.};
% \draw (\xb, \yb) node[scale=3, green] {.};
% \draw (\xp, \yp) node[scale=3, blue] {.};
% \draw (\xc, \yc) node[scale=3, red] {.};
% \draw (\xd, \yd) node[scale=3, red] {.};
% \draw (\xe, \ye) node[scale=3, red] {.};
% \draw (\xf, \yf) node[scale=3, red] {.};

% \draw (\xa, \ya) -- (\xp, \yp);
% \draw (\xb, \yb) -- (\xp, \yp);

\draw (\xc, \yc) -- (\xd, \yd);
\draw (\xe, \ye) -- (\xf, \yf);
\end{tikzpicture}
\end{document}

Here is a proposal without external programs. I am using your methods to compute the tangents but would like to remark that this has also been done in this answer. The coordinates of the relevant nodes are extracted with calc, and the conversion to cm is as simple as a multiplication by 1pt/1cm.

\documentclass[crop,tikz,margin=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{intersections,calc}
\usetikzlibrary{spy}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}[
spy using outlines = {circle,size=3cm,magnification=5,connect spies},
get coords/.code={\xdef\xa{\n1}\xdef\ya{\n2}
\xdef\xb{\n3}\xdef\yb{\n4}}
]
\begin{axis}
\coordinate (spy point) at (axis cs: 0, 0);
\coordinate (magnifying glass) at (rel axis cs: -0.4, 0.2);
\end{axis}

\spy[spy connection path={
\path let \p1=(tikzspyonnode),\p2=(tikzspyinnode),
\n1={\x1*1pt/1cm},\n2={\y1*1pt/1cm},\n3={\x2*1pt/1cm},\n4={\y2*1pt/1cm} in [get coords];
\pgfmathsetmacro\distancea{sqrt((\xp - \xa) * (\xp - \xa) + (\yp - \ya) * (\yp - \ya) - \radiusa * \radiusa))}
\pgfmathsetmacro\distanceb{sqrt((\xp - \xb) * (\xp - \xb) + (\yp - \yb) * (\yp - \yb) - \radiusb * \radiusb))}
\pgfmathsetmacro\denoma{(\xp - \xa)*(\xp - \xa) + (\yp - \ya)*(\yp - \ya)}
\pgfmathsetmacro\denomb{(\xp - \xb)*(\xp - \xb) + (\yp - \yb)*(\yp - \yb)}
\pgfmathsetmacro\xc{(\radiusa * \radiusa * (\xp - \xa) + \radiusa * (\yp - \ya) * \distancea) / \denoma + \xa}
\pgfmathsetmacro\yc{(\radiusa * \radiusa * (\yp - \ya) - \radiusa * (\xp - \xa) * \distancea) / \denoma + \ya}
\pgfmathsetmacro\xe{(\radiusa * \radiusa * (\xp - \xa) - \radiusa * (\yp - \ya) * \distancea) / \denoma + \xa}
\pgfmathsetmacro\ye{(\radiusa * \radiusa * (\yp - \ya) + \radiusa * (\xp - \xa) * \distancea) / \denoma + \ya}
\pgfmathsetmacro\xd{(\radiusb * \radiusb * (\xp - \xb) + \radiusb * (\yp - \yb) * \distanceb) / \denomb + \xb}
\pgfmathsetmacro\yd{(\radiusb * \radiusb * (\yp - \yb) - \radiusb * (\xp - \xb) * \distanceb) / \denomb + \yb}
\pgfmathsetmacro\xf{(\radiusb * \radiusb * (\xp - \xb) - \radiusb * (\yp - \yb) * \distanceb) / \denomb + \xb}
\pgfmathsetmacro\yf{(\radiusb * \radiusb * (\yp - \yb) + \radiusb * (\xp - \xb) * \distanceb) / \denomb + \yb}
\draw (\xc, \yc) -- (\xd, \yd);
\draw (\xe, \ye) -- (\xf, \yf);}]
on (spy point) in node at (magnifying glass);
\end{tikzpicture}
\end{document}

ADDENDUM: A way to save this as a style. Clearly, one can improve on this e.g. by not hard-coding the radii.

\documentclass[crop,tikz,margin=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{intersections,calc}
\usetikzlibrary{spy}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}[
spy using outlines = {circle,size=3cm,magnification=5,connect spies},
get coords/.code={\xdef\xa{\n1}\xdef\ya{\n2}
\xdef\xb{\n3}\xdef\yb{\n4}},
my connector/.store in=\myconnector,
my connector={
\path let \p1=(tikzspyonnode),\p2=(tikzspyinnode),
\n1={\x1*1pt/1cm},\n2={\y1*1pt/1cm},\n3={\x2*1pt/1cm},\n4={\y2*1pt/1cm} in [get coords];
\pgfmathsetmacro\distancea{sqrt((\xp - \xa) * (\xp - \xa) + (\yp - \ya) * (\yp - \ya) - \radiusa * \radiusa))}
\pgfmathsetmacro\distanceb{sqrt((\xp - \xb) * (\xp - \xb) + (\yp - \yb) * (\yp - \yb) - \radiusb * \radiusb))}
\pgfmathsetmacro\denoma{(\xp - \xa)*(\xp - \xa) + (\yp - \ya)*(\yp - \ya)}
\pgfmathsetmacro\denomb{(\xp - \xb)*(\xp - \xb) + (\yp - \yb)*(\yp - \yb)}
\pgfmathsetmacro\xc{(\radiusa * \radiusa * (\xp - \xa) + \radiusa * (\yp - \ya) * \distancea) / \denoma + \xa}
\pgfmathsetmacro\yc{(\radiusa * \radiusa * (\yp - \ya) - \radiusa * (\xp - \xa) * \distancea) / \denoma + \ya}
\pgfmathsetmacro\xe{(\radiusa * \radiusa * (\xp - \xa) - \radiusa * (\yp - \ya) * \distancea) / \denoma + \xa}
\pgfmathsetmacro\ye{(\radiusa * \radiusa * (\yp - \ya) + \radiusa * (\xp - \xa) * \distancea) / \denoma + \ya}
\pgfmathsetmacro\xd{(\radiusb * \radiusb * (\xp - \xb) + \radiusb * (\yp - \yb) * \distanceb) / \denomb + \xb}
\pgfmathsetmacro\yd{(\radiusb * \radiusb * (\yp - \yb) - \radiusb * (\xp - \xb) * \distanceb) / \denomb + \yb}
\pgfmathsetmacro\xf{(\radiusb * \radiusb * (\xp - \xb) - \radiusb * (\yp - \yb) * \distanceb) / \denomb + \xb}
\pgfmathsetmacro\yf{(\radiusb * \radiusb * (\yp - \yb) + \radiusb * (\xp - \xb) * \distanceb) / \denomb + \yb}
\draw (\xc, \yc) -- (\xd, \yd);
\draw (\xe, \ye) -- (\xf, \yf);}
]
\begin{axis}
\coordinate (spy point) at (axis cs: 0, 0);
\coordinate (magnifying glass) at (rel axis cs: -0.4, 0.2);
\end{axis}

\spy[spy connection path=\myconnector]
on (spy point) in node at (magnifying glass);
\end{tikzpicture}
\end{document}
• That is almost perfect! And it is not necessary to wrap the plot with a scope, as it was with my method. Is there a way to refactor that big spy connection path={...} as a reusable style with \tikzset{...}? I tried it but it reverted back to the default single line connection. – Claudio Oct 16 '18 at 13:27
• @Claudio I added a way to make this a style. (I have no time for polishing this right now. However, it works.) – user121799 Oct 16 '18 at 15:05
• Couldn’t this also be added to the 'spy'-library to be easily available as an option? I can see many people wanting this. – gr4nt3d Oct 16 '18 at 15:46
• @gr4nt3d Could this be added? Well, this would require 3 steps: 1. clean this up (I plan to do that later this week); 2. test this in many constellations (could be done if many users use a polished version in their code and report bugs/problems). 3. Send a feature request to the authors (I guess that the chances that this will be implemented are slim). – user121799 Oct 16 '18 at 16:20

I ended up implementing the algorithm in a short Python script and manually extracting the points' coordinated from TikZ source.

Note that I used an inner scope otherwise the tangent lines itself were visible inside the magnifying glass. There is also a visible difference in line width between the tangents and the spy circle, so the figure needs to be tuned a bit.

It is still less convenient than having everything implemented in TikZ.

\documentclass[crop,tikz,margin=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{intersections}
\usetikzlibrary{spy}
\usepackage{pgfplots}
\makeatletter
\newcommand\xcoord[2][center]{{%
\pgfpointanchor{#2}{#1}%
\pgfmathparse{\pgf@x/\pgf@xx}%
\pgfmathprintnumber{\pgfmathresult}%
}}
\newcommand\ycoord[2][center]{{%
\pgfpointanchor{#2}{#1}%
\pgfmathparse{\pgf@y/\pgf@yy}%
\pgfmathprintnumber{\pgfmathresult}%
}}
\makeatother
\begin{document}
\begin{tikzpicture}
\begin{scope}[
spy using outlines = {circle,size=3cm,magnification=5},
]
\begin{axis}
\coordinate (spy point) at (axis cs: 0, 0);
\coordinate (magnifying glass) at (rel axis cs: -0.4, 0.2);
\coordinate (a) at (rel axis cs: 0.5, 1.1);
\end{axis}
\node at (a) {spy point: \xcoord{spy point}, \ycoord{spy point}, glass: \xcoord{magnifying glass}, \ycoord{magnifying glass}};

\spy on (spy point) in node at (magnifying glass);
\end{scope}
\coordinate (c) at (-1.6589690159337693, 0.10010961563788023);
\coordinate (d) at (0.7862061968132461, 2.6420219231275763);
\coordinate (e) at (-2.962541913303562, 2.6233998438799935);
\coordinate (f) at (0.5254916173392875, 3.1466799687759988);
\draw (c) -- (d);
\draw (e) -- (f);
\end{tikzpicture}
\end{document}

from math import sqrt
import matplotlib.pyplot as plt

def main():
xa = -2.74
ya = 1.14
xb = 0.57
yb = 2.85

figure, ax = plt.subplots()
circlea = plt.Circle((xa, ya), radiusa, color='C0')
circleb = plt.Circle((xb, yb), radiusb, color='C0')

(xc, yc), (xd, yd), (xe, ye), (xf, yf) = compute(xa, ya, radiusa, xb, yb, radiusb)

ax.plot([xc, xd], [yc, yd], color='C0')
ax.plot([xe, xf], [ye, yf], color='C0')

ax.set_xlim(
)
ax.set_ylim(
)

print("\\coordinate (c) at ({}, {});".format(xc, yc))
print("\\coordinate (d) at ({}, {});".format(xd, yd))
print("\\coordinate (e) at ({}, {});".format(xe, ye))
print("\\coordinate (f) at ({}, {});".format(xf, yf))

plt.show()

distancea = sqrt((xp - xa) * (xp - xa) + (yp - ya) * (yp - ya) - radiusa * radiusa)
distanceb = sqrt((xp - xb) * (xp - xb) + (yp - yb) * (yp - yb) - radiusb * radiusb)
denoma = (xp - xa)*(xp - xa) + (yp - ya)*(yp - ya)
denomb = (xp - xb)*(xp - xb) + (yp - yb)*(yp - yb)

xc = (radiusa * radiusa * (xp - xa) + radiusa * (yp - ya) * distancea) / denoma + xa
yc = (radiusa * radiusa * (yp - ya) - radiusa * (xp - xa) * distancea) / denoma + ya

xe = (radiusa * radiusa * (xp - xa) - radiusa * (yp - ya) * distancea) / denoma + xa
ye = (radiusa * radiusa * (yp - ya) + radiusa * (xp - xa) * distancea) / denoma + ya

xd = (radiusb * radiusb * (xp - xb) + radiusb * (yp - yb) * distanceb) / denomb + xb
yd = (radiusb * radiusb * (yp - yb) - radiusb * (xp - xb) * distanceb) / denomb + yb

xf = (radiusb * radiusb * (xp - xb) - radiusb * (yp - yb) * distanceb) / denomb + xb
yf = (radiusb * radiusb * (yp - yb) + radiusb * (xp - xb) * distanceb) / denomb + yb

return (xc, yc), (xd, yd), (xe, ye), (xf, yf)

if __name__ == '__main__':
main()