Add jypiter notebook to serie1
This commit is contained in:
Hussein Kabbout
parent
69c9d4d465
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d302bcbf87
294
serie1/flaechen.ipynb
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294
serie1/flaechen.ipynb
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Zweidimensionale Flächen darstellen\n",
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"\n",
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"In diesem Tutorial wollen wir uns damit beschäftigen, wie wir Flächen darstellen können. Eine zweidimensionale Fläche ist zum Beispiel durch den Graphen einer Funktion $f\\colon \\mathbb{R}^2 \\to \\mathbb{R}$ gegeben. Wir definieren uns also zuerst eine Funktion $f(x,y) = x^2 + y^2$.\n",
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"\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"def f(x,y):\n",
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" return x**2 + y**2"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Nun berechnen wir die Koordinaten für eine Menge von Stützpunkten. Dazu legen wir mit dem Befehl `numpy.meshgrid()` ein Gitter in die x-y-Ebene und berechnen anschliessend für jeden Punkt den Funktionswert $z=f(x,y)$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"\n",
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"[x,y] = np.meshgrid(np.linspace(-5,5), np.linspace(-5,5));\n",
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"z = f(x,y)\n",
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"\n",
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"print(np.shape(x))\n",
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"print(np.shape(y))\n",
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"print(np.shape(z))\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Wir sehen, dass x, y und z hier jeweils 50x50 Matrizen sind. Für jedes $x\\in[-5,5]$ und $y\\in[-5,5]$ steht der entsprechende Funktionswert $z=f(x,y)$ in dieser Matrix z, die zugehörige $x$-Koordinate in der Matrix x resp. die zugehörige $y$-Koordinate in der Matrix y. Nun stehen uns in Python verschiedene Möglichkeiten zu Verfügung, um diese Fläche $z$ (resp. die Matrix z) zu visualisieren. Wir schauen uns hier mal eine kleine Auswahl an."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Höhenlinien\n",
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"\n",
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"Die uns bekannte Bibliothek `matplotlib` stellt die Funktion `contour` zur Verfügung, welche die Höhenlinien der Fläche einzuzeichnet. "
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"import matplotlib.pyplot as plt\n",
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"\n",
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"plt.contour(x, y, z)\n",
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"\n",
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"plt.title('Höhenlinien')\n",
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"plt.xlabel('x')\n",
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"plt.ylabel('y')\n",
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"\n",
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"plt.show()\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Wir können auch eine andere Einfärbung wählen und dazu eine Legende generieren."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"from matplotlib import cm\n",
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"\n",
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"fig = plt.figure(0)\n",
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"cont = plt.contour(x, y, z, cmap=cm.coolwarm)\n",
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"fig.colorbar(cont, shrink=0.5, aspect=5)\n",
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"\n",
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"plt.title('Höhenlinien')\n",
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"plt.xlabel('x')\n",
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"plt.ylabel('y')\n",
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"\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Auch die Niveaus können selber ausgewähl werden."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"fig = plt.figure(1)\n",
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"cont = plt.contour(x, y, z, [1,4,9,16,25,36,49,64,81,100,121,144,169],cmap=cm.coolwarm)\n",
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"fig.colorbar(cont, shrink=0.5, aspect=5)\n",
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"\n",
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"plt.title('Höhenlinien')\n",
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"plt.xlabel('x')\n",
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"plt.ylabel('y')\n",
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"\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Oberfläche im 3D\n",
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"\n",
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"Nun wollen wir die Fläche im dreidimensionalen Raum darstellen. Dazu verwenden wir das Paket `mpl_toolkits.mplot3d`, welches uns eine interaktive Ansicht bietet. Dazu müssen wir hier in jupyter noch mit dem Befehlt `%matplotlib notebook` die Einstellungen anpassen."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"%matplotlib notebook\n",
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"from mpl_toolkits.mplot3d import Axes3D\n",
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"\n",
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"fig = plt.figure(2)\n",
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"ax = fig.add_subplot(111, projection='3d')\n",
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"ax.plot_surface(x,y,z)\n",
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"\n",
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"plt.title('Fläche')\n",
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"ax.set_xlabel('x')\n",
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"ax.set_ylabel('y')\n",
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"ax.set_zlabel('z')\n",
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"\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Mit der Option `projection` wählen wir den dreidimensionalen Plot aus. Auch hier können wir wieder mit der Einfärbung spielen."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"fig = plt.figure(3)\n",
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"ax = fig.add_subplot(111, projection='3d')\n",
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"surf = ax.plot_surface(x,y,z, cmap=cm.coolwarm, linewidth=0, antialiased=False)\n",
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"\n",
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"fig.colorbar(surf, shrink=0.5, aspect=5)\n",
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"\n",
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"plt.title('Fläche')\n",
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"ax.set_xlabel('x')\n",
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"ax.set_ylabel('y')\n",
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"ax.set_zlabel('z')\n",
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"\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Um den Durchblick zu behalten, können wir alternativ auch nur ein Gitter plotten."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"fig = plt.figure(4)\n",
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"ax = fig.add_subplot(111, projection='3d')\n",
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"ax.plot_wireframe(x,y,z)\n",
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"\n",
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"plt.title('Gitter')\n",
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"ax.set_xlabel('x')\n",
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"ax.set_ylabel('y')\n",
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"ax.set_zlabel('z')\n",
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"\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Dabei lässt sich natürlich der Abstand zwischen den Linien einstellen."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"fig = plt.figure(5)\n",
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"ax = fig.add_subplot(111, projection='3d')\n",
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"ax.plot_wireframe(x,y,z, rstride=5, cstride=5)\n",
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"\n",
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"plt.title('Gitter')\n",
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"ax.set_xlabel('x')\n",
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"ax.set_ylabel('y')\n",
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"ax.set_zlabel('z')\n",
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"\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Zum Schluss zeichnen wir auch noch die Höhenlinien im dreidimensionalen Raum ein."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"fig = plt.figure(6)\n",
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"ax = fig.add_subplot(111, projection='3d')\n",
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"surf = ax.contour(x,y,z, cmap=cm.coolwarm)\n",
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"\n",
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"fig.colorbar(surf, shrink=0.5, aspect=5)\n",
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"\n",
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"plt.title('Fläche')\n",
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"ax.set_xlabel('x')\n",
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"ax.set_ylabel('y')\n",
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"ax.set_zlabel('z')\n",
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"\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.5"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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