Merge pull request #58 from NaoPross/master

kugel: Eigenvalue problem and Legendre functions

7 files changed, 664 insertions, 0 deletions

diff --git a/buch/papers/kugel/figures/tikz/Makefile b/buch/papers/kugel/figures/tikz/Makefile
new file mode 100644
index 0000000..4ec4e5a
--- /dev/null
+++ b/buch/papers/kugel/figures/tikz/Makefile
@@ -0,0 +1,12 @@
+FIGURES := spherical-coordinates.pdf curvature-1d.pdf
+
+all: $(FIGURES)
+
+%.pdf: %.tex
+	pdflatex $<
+
+curvature-1d.pdf: curvature-1d.tex curvature-1d.dat
+	pdflatex curvature-1d.tex
+
+curvature-1d.dat: curvature-1d.py
+	python3 $<
diff --git a/buch/papers/kugel/figures/tikz/curvature-1d.dat b/buch/papers/kugel/figures/tikz/curvature-1d.dat
new file mode 100644
index 0000000..6622398
--- /dev/null
+++ b/buch/papers/kugel/figures/tikz/curvature-1d.dat
@@ -0,0 +1,500 @@
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diff --git a/buch/papers/kugel/figures/tikz/curvature-1d.pdf b/buch/papers/kugel/figures/tikz/curvature-1d.pdf
new file mode 100644
index 0000000..6425af6
--- /dev/null
+++ b/buch/papers/kugel/figures/tikz/curvature-1d.pdf
diff --git a/buch/papers/kugel/figures/tikz/curvature-1d.py b/buch/papers/kugel/figures/tikz/curvature-1d.py
new file mode 100644
index 0000000..4710fc8
--- /dev/null
+++ b/buch/papers/kugel/figures/tikz/curvature-1d.py
@@ -0,0 +1,32 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+
+@np.vectorize
+def fn(x):
+    return (x ** 2) * 2 / 100 + (1 + x / 4) + np.sin(x)
+
+@np.vectorize
+def ddfn(x):
+    return 2 * 5 / 100 - np.sin(x)
+
+x = np.linspace(0, 10, 500)
+y = fn(x)
+ddy = ddfn(x)
+
+cmap = ddy - np.min(ddy)
+cmap = cmap * 1000 / np.max(cmap)
+
+plt.plot(x, y)
+plt.plot(x, ddy)
+# plt.plot(x, cmap)
+
+plt.show()
+
+fname = "curvature-1d.dat"
+np.savetxt(fname, np.array([x, y, cmap]).T, delimiter="  ")
+
+# with open(fname, "w") as f:
+#     # f.write("x   y   cmap\n")
+#     for xv, yv, cv in zip(x, y, cmap):
+#         f.write(f"{xv}   {yv}   {cv}\n")
diff --git a/buch/papers/kugel/figures/tikz/curvature-1d.tex b/buch/papers/kugel/figures/tikz/curvature-1d.tex
new file mode 100644
index 0000000..6983fb0
--- /dev/null
+++ b/buch/papers/kugel/figures/tikz/curvature-1d.tex
@@ -0,0 +1,21 @@
+% vim:ts=2 sw=2 et:
+\documentclass[tikz, border=5mm]{standalone}
+\usepackage{pgfplots}
+
+\begin{document}
+\begin{tikzpicture}
+  \begin{axis}[
+      clip = false,
+      width = 8cm, height = 6cm,
+      xtick = \empty, ytick = \empty,
+      colormap name = viridis,
+      axis lines = middle,
+      axis line style = {ultra thick, -latex}
+    ]
+    \addplot+[
+      smooth, mark=none, line width = 3pt, mesh,
+      point meta=explicit,
+    ] file {curvature-1d.dat};
+  \end{axis}
+\end{tikzpicture}
+\end{document}
diff --git a/buch/papers/kugel/figures/tikz/spherical-coordinates.pdf b/buch/papers/kugel/figures/tikz/spherical-coordinates.pdf
new file mode 100644
index 0000000..1bff016
--- /dev/null
+++ b/buch/papers/kugel/figures/tikz/spherical-coordinates.pdf
diff --git a/buch/papers/kugel/figures/tikz/spherical-coordinates.tex b/buch/papers/kugel/figures/tikz/spherical-coordinates.tex
new file mode 100644
index 0000000..3a45385
--- /dev/null
+++ b/buch/papers/kugel/figures/tikz/spherical-coordinates.tex
@@ -0,0 +1,99 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{bm}
+\usepackage{lmodern}
+\usepackage{tikz-3dplot}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+
+\tdplotsetmaincoords{60}{130}
+\pgfmathsetmacro{\l}{2}
+
+\begin{tikzpicture}[
+    >=latex,
+    tdplot_main_coords,
+    dot/.style = {
+      black, fill = black, circle,
+      outer sep = 0, inner sep = 0,
+      minimum size = .8mm 
+    },
+    round/.style = {
+      draw = orange, thick, circle,
+      minimum size = 1mm,
+      inner sep = 0pt, outer sep = 0pt,
+    },
+    cross/.style = {
+      cross out, draw = magenta, thick,
+      minimum size = 1mm, 
+      inner sep = 0pt, outer sep = 0pt
+    },
+  ]
+
+  % origin
+  \coordinate (O) at (0,0,0);
+
+  % poles
+  \coordinate (NP) at (0,0,\l);
+  \coordinate (SP) at (0,0,-\l);
+  
+  % \draw (SP) node[dot, gray] {};
+  % \draw (NP) node[dot, gray] {};
+
+  % gray unit circle
+  \tdplotdrawarc[gray]{(O)}{\l}{0}{360}{}{};
+  \draw[gray, dashed] (-\l, 0, 0) to (\l, 0, 0);
+  \draw[gray, dashed] (0, -\l, 0) to (0, \l, 0);
+
+  % axis
+  \draw[->] (O) -- ++(1.25*\l,0,0) node[left] {\(\mathbf{\hat{x}}\)};
+  \draw[->] (O) -- ++(0,1.25*\l,0) node[right] {\(\mathbf{\hat{y}}\)};
+  \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})
+  
+  % point A
+  \coordinate (A) at (\px,\py,\pz);
+  \coordinate (Ap) at (\px,\py, 0);
+  
+  % lines
+  \draw[red!80!black, ->] (O) -- (A);
+	\draw[red!80!black, densely dashed] (O) -- (Ap) -- (A)
+		node[above right] {\(\mathbf{\hat{r}}\)};
+  
+  % arcs
+  \tdplotdrawarc[blue!80!black, ->]{(O)}{.8\l}{0}{\phi}{}{};
+  \node[below right, blue!80!black] at (.8\l,0,0) {\(\bm{\hat{\varphi}}\)};
+  
+  \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) {\(\bm{\hat{\vartheta}}\)};
+
+
+  % dots
+  \draw (O) node[dot] {};
+  \draw (A) node[dot, fill = red!80!black] {};
+
+\end{tikzpicture}
+\end{document}
+% vim:ts=2 sw=2 et: