{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Mobius Transformations" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline\n", "import sys; sys.path.append('..')\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "from conformalmapping import *" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Create a basic Mobius transformation." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "z3 = [1, 1j, -1]\n", "w3 = [-1, -2j, 0]\n", "m1 = Mobius.from_vectors(z3, w3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Mobius transform is a conformal map" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "m1 is a conformal map.\n" ] } ], "source": [ "if isinstance(m1, ConformalMap):\n", " print('m1 is a conformal map.')\n", "else:\n", " print('m1 is not a conformal map.')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Plotting a conformal map gives an image in the range of a grid in the domain." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAQgAAAD7CAYAAACWhwr8AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsnXW81FX6x9+HUgwsLCxccQ3EVqy1u7t79VGx21VRjHV1\n", "FdfAeuy1xVrUn63YhYmB4iLWIq6FyhrE+f3xOQPjMHPvzNyJy73zfr2+rwtzv3Fm7pznPOfJEGOk\n", "QYMGDfLRod4DaNCgQeulISAaNGhQkIaAaNCgQUEaAqJBgwYFaQiIBg0aFKQhIBo0aFCQTvUeQIYQ\n", "QsPf2qBBnYgxhnyvtxoBAYUHWQ1CCANijANq9bxymR7GOT2MEaaPcdZjjE0tzo0tRoMGDQrSEBAN\n", "GjQoSHsWEEPrPYAiGVrvARTB0HoPoEiG1nsARTC03gPIJrSWXIwQQqylDaJBgwaiqbnXnjWIBg0a\n", "NENDQDRo0KAgDQHRoEGDgjQERIMGDQrSEBANGjQoSENANGjQoCANAdGgQYOCNAREgwYNCtIQEA0a\n", 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Let's see that again in slow motion.\n", "# (NOTE:\n", "# Technically a Mobius map is an entire function, but for visualization\n", "# convenience it uses the 3-vectors it was given on construction to define\n", "# circles for its domain and range.)\n", "\n", "# Plot the original\n", "plt.subplot(1,2,1)\n", "m1.domain.grid().plot()\n", "m1.domain.plot()\n", "plt.gca().set_xlim(-1.2,1.2)\n", "plt.gca().set_ylim(-1.2,1.2)\n", "plt.gca().set_aspect('equal')\n", "\n", "# Plot the conformally mapped domain\n", "plt.subplot(1,2,2)\n", "m1(m1.domain.grid()).plot()\n", "m1.range.plot()\n", "plt.gca().set_aspect('equal')" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Another transformation for a composition example.\n", "s3 = [0, 3, 1j]\n", "m2 = Mobius.from_vectors(w3, s3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Additional work to follow ..." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.3.5" } }, "nbformat": 4, "nbformat_minor": 0 }