3 files changed, 70 insertions, 1 deletions
diff --git a/buch/papers/kugel/figures/tikz/Makefile b/buch/papers/kugel/figures/tikz/Makefile
index 4ec4e5a..3b2df59 100644
--- a/buch/papers/kugel/figures/tikz/Makefile
+++ b/buch/papers/kugel/figures/tikz/Makefile
@@ -1,4 +1,4 @@
-FIGURES := spherical-coordinates.pdf curvature-1d.pdf
+FIGURES := spherical-coordinates.pdf curvature-1d.pdf legendre-substitution.pdf
 
 all: $(FIGURES)
 
diff --git a/buch/papers/kugel/figures/tikz/legendre-substitution.pdf b/buch/papers/kugel/figures/tikz/legendre-substitution.pdf
new file mode 100644
index 0000000..f77b6cb
--- /dev/null
+++ b/buch/papers/kugel/figures/tikz/legendre-substitution.pdf
diff --git a/buch/papers/kugel/figures/tikz/legendre-substitution.tex b/buch/papers/kugel/figures/tikz/legendre-substitution.tex
new file mode 100644
index 0000000..3a699b8
--- /dev/null
+++ b/buch/papers/kugel/figures/tikz/legendre-substitution.tex
@@ -0,0 +1,69 @@
+
+% vim: ts=2 sw=2 et :
+\documentclass[tikz, border=2mm]{standalone}
+
+\usepackage{amsmath}
+\usepackage{bm}
+
+\usepackage{times}
+\usepackage{txfonts}
+
+\usepackage{tikz-3dplot}
+
+\tdplotsetmaincoords{60}{130}
+\pgfmathsetmacro{\l}{2}
+
+\begin{document}
+  \begin{tikzpicture}[
+      >=latex,
+      tdplot_main_coords,
+    ]
+
+    % origin and poles
+    \coordinate (O) at (0,0,0);
+    \coordinate (NP) at (0,0,\l);
+    \coordinate (SP) at (0,0,-\l);
+
+    % gray unit circle
+    \tdplotdrawarc[lightgray, dashed]{(O)}{\l}{0}{360}{}{};
+    \draw[lightgray, dashed] (-\l, 0, 0) to (\l, 0, 0);
+    \draw[lightgray, dashed] (0, -\l, 0) to (0, \l, 0);
+
+    % axis
+    \draw[->] (O) -- ++(0,0,1.25*\l) node[above] {\(\mathbf{\hat{z}}\)};
+
+    % meridians
+    \foreach \phi in {0, 30, 60, ..., 150}{
+      \tdplotsetrotatedcoords{\phi}{90}{0};
+      \tdplotdrawarc[lightgray, densely dotted, tdplot_rotated_coords]{(O)}{\l}{0}{360}{}{};
+    }
+
+    % dot above and its projection
+    \pgfmathsetmacro{\phi}{120}
+    \pgfmathsetmacro{\theta}{40}
+
+    \pgfmathsetmacro{\px}{cos(\phi)*sin(\theta)*\l}
+    \pgfmathsetmacro{\py}{sin(\phi)*sin(\theta)*\l}
+    \pgfmathsetmacro{\pz}{cos(\theta)*\l})
+
+    % Special meridian
+    \tdplotsetrotatedcoords{\phi-90}{90}{0};
+    \tdplotdrawarc[gray, tdplot_rotated_coords]{(O)}{\l}{0}{360}{}{};
+    
+    % point A
+    \coordinate (A) at (\px,\py,\pz);
+    \coordinate (Ap) at (\px,\py, 0);
+    
+    % lines
+    \draw[red!80!black, thick, ->] (O) -- (A);
+    \draw[red!80!black]
+      (O) -- node[midway, below, font=\small, sloped, fill=white] {$\sqrt{1 - z^2}$}
+      (Ap) -- node[midway, right, font=\small, sloped, fill=white, anchor=north] {$z = \cos \vartheta$}
+      (A) node[above right, fill=white] {$r = 1$};
+    
+    % theta arc
+    \tdplotsetrotatedcoords{\phi-90}{-90}{0};
+    \tdplotdrawarc[blue!80!black, ->, tdplot_rotated_coords]{(O)}{.95\l}{0}{\theta}{}{};
+    \node[above right = 1mm, blue!80!black] at (0,0,.8\l) {\(\bm{\hat{\vartheta}}\)};
+  \end{tikzpicture}
+\end{document}