{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 行列と幾何変換 I\n", "\n", "本日の授業では、行列を使って平面上の点のリストの扱いを勉強します。特に、行列を道具として点の幾何変換を行います。\n", "\n", "点の幾何変換とは、点の実数倍、平行移動、回転などの変換です。\n", "\n", "\n", "## 内容の概要\n", "\n", "- 1回目：平面上の点とグラフの表示、線形変換\n", "- 2回目：アフィン変換、同次座標系" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 1. 平面上の点とグラフの表示\n", "\n", "Pythonでは、matplotlib.pyplotというモジュールを導入することで点の表示ができます。\n", "\n", "使い方：\n", "\n", "- matplotlib.pyplot.plot(点のｘ座標のリスト, 点のｙ座標のリスト, プロットのフォーマット)\n", "\n", "
\n", "matplotlib.pyplot.plot(...)
\n", "と書くのは長いので、matplotlib.pyplotを導入する行において
\n", "import matplotlib.pyplot as plt
\n", "とすると、asに続く文字列で省略表記を定義することができます。\n", "
\n", "\n", "### 例1：点のリストを描く\n", "\n", "２つの点(1,0)と(3,1)を赤丸で表示するために、以下のコードを使用します。\n", "\n", "
\n",
    "import matplotlib.pyplot as plt\n",
    "plt.plot([1,3], [0,1], 'ro')\n",
    "
\n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "#２つの点の表示、markersizeによって点のサイズを調整できます。\n", "plt.plot([1,3], [0,1], 'ro', markersize=20)\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### ｘ軸とｙ軸のアスペクト比\n", "\n", "以下の命令によって、ｘ軸とｙ軸のアスペクト比を１にすることができます。\n", "\n", "``\n", "plt.axes().set_aspect('equal')\n", "``\n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "\n", "plt.axes().set_aspect('equal')\n", "\n", "plt.plot([1,3], [0,1], 'ro', markersize=20)\n", "plt.grid()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "行列を使って、点列を描画します。\n", "\n", "### 点のリストとNumpyの行列\n", "\n", "これからの計算では、$n$個の点からなる点列を$2\\times n$の形の行列で表現します。そうすると、行列の各列は点列の点に対応します。\n", "行列として処理する場合、Pythonライブラリにあるnumpyを使うと便利です。\n", "\n", "\n", "numpyでは、「:」は「すべて」の意味をもっています。即ちここでは、「すべての行」または「すべての列」です。\n", "\n", "例: p_list = np.array([[0, 1, 1], [0, 0, 1]])について、\n", "\n", "列を取るために「すべての行」を使います。\n", "- p_list[:,0] はp_listのすべての行、第0列を示しています。即ち、点列の0番目(0から数える)の点です。\n", "- p_list[:,1] はp_listのすべての行、第1列を示しています。即ち、点列の1番目(0から数える)の点です。\n", "- p_list[:,2] はp_listのすべての行、第2列を示しています。即ち、点列の2番目(0から数える)の点です。\n", "\n", "行を取るために「すべての列」を使います。\n", "- p_list[0,:] はp_listのすべての列、第0行を示しています。即ち、すべての点のｘ座標のリストです。\n", "- p_list[1,:] はp_listのすべての列、第1行を示しています。即ち、すべての点のｙ座標のリストです。\n", "\n", "### 例2：三角形を描く\n", "\n", "３つの点(0,0), (1,0), (1,1)が作る三角形を描いてみます。ｘ軸とｙ軸のアスペクト比を１とします。\n", "\n", "閉じている多角形を描くためには、点のリストの最後にはじめの点を入れる必要があります。即ち、点(0,0), (1,0), (1,1), (0,0)というリストが必要です。" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p_listの0行目(x座標) [0 1 1 0]\n", "p_listの1行目(y座標) [0 0 1 0]\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "p_list = np.array([[0, 1, 1, 0],\n", " [0, 0, 1, 0]])\n", "print(\"p_listの0行目(x座標)\",p_list[0,:])\n", "print(\"p_listの1行目(y座標)\",p_list[1,:])\n", "\n", "plt.axes().set_aspect('equal')\n", "plt.plot(p_list[0,:], p_list[1,:], 'r-')\n", "plt.grid()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 「家」の輪郭を描く\n", "\n", "以下の三角の点列と四角の点列を使って、シンプルな「家」を描くことができます。\n", "閉じている多角形を描くために、点のリストの最後にはじめの点を入れる必要があることに注意してください。\n", "\n", "- roof_nodes：(-3,2),(3,2),(0,3) \n", "- wall_nodes：(-2,2),(-2,0),(2,0),(2,2)\n", "\n", "### 点の並び方と行列の転置\n", "\n", "以下のコードでは、点の入力を楽にするために行列の転置を使っています。\n", "\n", "roof_nodesの点のリストを用意するとき、一点ずつ書き並べる方が簡単なので、縦の行列を先に作成します。その後に、次のコードで行列の転置を行います。\n", "``\n", "roof_nodes = roof_nodes.T\n", "``\n", "\n", "### 例3：「家」を描く\n" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "roof_nodes = np.array([[-3,2],[3,2],[0,3],[-3,2]])\n", "roof_nodes = roof_nodes.T\n", "wall_nodes = np.array([[-2,2],[-2,0],[2,0],[2,2],[-2,2]])\n", "wall_nodes = wall_nodes.T\n", "\n", "plt.axes().set_aspect('equal')\n", "\n", "#家の描画\n", "plt.plot(roof_nodes[0,:], roof_nodes[1,:], 'ro-')\n", "plt.plot(wall_nodes[0,:], wall_nodes[1,:], 'ro-')\n", "plt.grid()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 演習1\n", "\n", "上記の「家」の例とは異なる、自分なりの「家」を描いてください。\n", "\n", "適当に窓やドアを追加しても良いです。その場合、家のそれぞれの構成部分のために点のリストを用意してください。\n", "\n", "例えば、window_nodes、door_nodes。" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "#ここにコードを書いてください。\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 2. 線形変換\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2.1 拡大・縮小変換" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "点の$x$座標または$y$座標を実数倍することで、図形の拡大と縮小ができます。\n", "\n", "$x$方向に$\\alpha$倍にする変換\n", "\n", "\n", "$$\n", "x'=\\alpha x,\\ y' = y \\iff\n", "\\left(\\begin{array}{c} x' \\\\ y' \\end{array}\\right) = A_x \n", "\\left(\\begin{array}{c} x \\\\ y \\end{array}\\right), \\quad \n", "A_x=\\left(\\begin{array}{cc} \\alpha & 0 \\\\ 0 & 1 \\end{array}\\right)\n", "$$\n", "\n", "$y$方向に$\\beta$倍にする変換\n", "\n", "$$\n", "x'=x,\\ y' = \\beta y \\iff\n", "\\left(\\begin{array}{c} x' \\\\ y' \\end{array}\\right) = A_y \n", "\\left(\\begin{array}{c} x \\\\ y \\end{array}\\right), \\quad \n", "A_y=\\left(\\begin{array}{cc} 1 & 0 \\\\ 0 & \\beta \\end{array}\\right) \n", "$$\n", "\n", "$x$方向に$\\alpha$倍、$y$方向に$\\beta$倍にする変換\n", "\n", "$$\n", "x'=\\alpha x,\\ y' = \\beta y \\iff\n", "\\left(\\begin{array}{c} x' \\\\ y' \\end{array}\\right) = A \n", "\\left(\\begin{array}{c} x \\\\ y \\end{array}\\right), \\quad \n", "A=\\left(\\begin{array}{cc} \\alpha & 0 \\\\ 0 & \\beta \\end{array}\\right) = A_x A_y\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 例：\n", "\n", "以下の例では、「家」を$x$方向に$2$倍、$y$方向に$1.5$倍に伸ばしています。変換前の家は赤、変換後の家は青で描かれています。" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "roof_nodes = np.array([[-3,2],[3,2],[0,3],[-3,2]]).T\n", "wall_nodes = np.array([[-2,2],[-2,0],[2,0],[2,2],[-2,2]]).T\n", "\n", "plt.axes().set_aspect('equal')\n", "\n", "#変換前の家\n", "plt.plot(roof_nodes[0,:], roof_nodes[1,:], 'ro-')\n", "plt.plot(wall_nodes[0,:], wall_nodes[1,:], 'ro-')\n", "plt.grid()\n", "\n", "#変換行列\n", "A = np.array([[2,0],[0,1.5]])\n", "\n", "#変換後の家\n", "new_roof_nodes = np.dot(A,roof_nodes)\n", "new_wall_nodes = np.dot(A,wall_nodes)\n", "plt.plot(new_roof_nodes[0,:], new_roof_nodes[1,:], 'bo-')\n", "plt.plot(new_wall_nodes[0,:], new_wall_nodes[1,:], 'bo-')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 演習2\n", "\n", "演習1で考えた「家」を、$x$方向と$y$方向に適当に伸ばして描いてください。" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "#ここにコードを書いてください。\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## レポート課題\n", "\n", "演習1で考えた「家」をベースにして、拡大・縮小変換によって２つのサイズの異なる「家」を描いてください。" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "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": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }