119 lines
3.5 KiB
TeX
119 lines
3.5 KiB
TeX
\documentclass{standalone}
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\usepackage{pgfplots} % TikZ (TeX ist kein Zeichenprogramm)
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\usetikzlibrary{calc, perspective, pgfplots.colormaps} % PGF-Plot / TikZ config
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\pgfplotsset{
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compat = newest,
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colormap = {grayscale}{color=(lightgray) color=(white) color=(lightgray)},
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colormap = {blackscale}{color=(black!70) color=(black!50) color=(black!70)},
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}
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% Define the (component) embedding into the torus
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\tikzset{declare function = { % Note: NO spaces in function argument list!
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Z(\u,\v) = 0.4 * \u * \u * cos(\v * 120);
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bx(\t) = -0.5 + 0.3 * cos(\t) + 0.05 * sin(3 * \t);
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by(\t) = 0.2 + 0.3 * sin(\t);
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}}
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% Further packages and macros
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\usepackage{amssymb, bm}
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\renewcommand{\t}[1]{{#1}^{T}}
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\newcommand{\mat}[1]{\boldsymbol{#1}}
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\newcommand{\manifold}[1]{\mathfrak{#1}}
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\begin{document}
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\begin{tikzpicture}[>=latex]
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\begin{axis}[
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axis equal image,
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hide axis,
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view = {120}{30},
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scale = 1,
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clip = false
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]
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\coordinate (O) at (0, 0, 0);
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\draw[->] (-0.1, 0, 0) -- (0.6, 0, 0) node[pos = 1.1] {};
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\draw[->] (0, -0.1, 0) -- (0, 1.2, 0) node[pos = 1.1] {};
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\draw[->] (0, 0, -0.1) -- (0, 0, 1.0) node[pos = 1.1] {};
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\addplot3[
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surf,
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shader = faceted interp,
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samples = 16,
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samples y = 16,
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domain = -1:0.4,
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domain y = -0.5:1,
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z buffer = sort,
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colormap name = grayscale,
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thin
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]
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({\x}, {\y}, {Z(\x, \y)});
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\addplot3[
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samples = 64,
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samples y = 0,
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domain = 0:360,
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color = black!40!gray,
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fill = black,
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fill opacity = 0.1,
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colormap name = blackscale,
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thick
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]
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({bx(\x)}, {by(\x)}, {Z(bx(\x), by(\x))});
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\coordinate (coordU) at ({bx(150)}, {by(150)}, {Z(bx(150), by(150))});
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\node[anchor = south west] (U) at (coordU) {$U$};
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\node[
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circle, fill=black, inner sep=0.75pt, label={$\mat{\theta}_0$}
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] (theta0) at (-0.5, 0.2, {Z(-0.5, 0.2)}) {};
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\node at (0, 0.5, 1.2) {$\Theta\subseteq\mathbb{R}^p$};
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\node (UU) at ({bx(0)}, {by(0)}, {Z(bx(0), by(0))}) {};
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\end{axis}
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\begin{scope}[shift = {(11cm, 2cm)}, scale = 2.5]
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\coordinate (O) at (-1.1, -0.3);
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\draw[step=0.1, lightgray!80, thin] (O) grid +(1.2, 1.2);
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\draw[->] ($(O) - (0.05, 0)$) -- +(1.4, 0) node[pos=1.1] {};
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\draw[->] ($(O) - (0, 0.05)$) -- +(0, 1.4) node[pos=1.1] {};
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\draw[domain=0:360, smooth, variable=\x, fill=black, fill opacity = 0.1, thick] plot ({bx(\x)}, {by(\x)});
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\node[
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circle, fill=black, inner sep=0.75pt, label={$\mat{s}_0$}
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] (s0) at (-0.5, 0.2) {};
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\coordinate (coordPhiU) at ({bx(90)}, {by(90)});
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\node[anchor = south east, outer sep = 0pt] (phiU) at (coordPhiU) {$\varphi(U)$};
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\node at (-0.5, 1.28) {$\varphi(U)\subseteq\mathbb{R}^d$};
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\node (phiUU) at ({bx(270)}, {by(270)}) {};
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\end{scope}
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\draw[->, out = 20, in = 160] (U.north east) to node[above, pos = 0.5] {$\varphi$} (phiU.north west);
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\draw[->, out = 200, in = 340] (phiU.south west) to node[above, pos = 0.5] {$\varphi^{-1}$} (U.south east);
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\node (R) at (6.1, 0) {$\mathbb{R}$};
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\draw[->, out = 270, in = 180] (UU) to node[below left, pos = 0.6] {$M$} (R);
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\draw[->, out = 270, in = 0] (phiUU) to node[below right, pos = 0.4] {$M_{\varphi}$} (R);
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\end{tikzpicture}
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\end{document}
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