hw6_106207409_0512.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. 定義二維平面上的區域  𝑆⊆ℝ2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "a,b = 7,5\n",
    "c,d = -5,2\n",
    "\n",
    "n,m = 10,20\n",
    "x = np.linspace(a,b,n)\n",
    "y = np.linspace(c,d,n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x25cb471b5c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "S_points = []\n",
    "for yj in y:\n",
    "    for xi in x:\n",
    "        plt.scatter(xi,yj,s=20,c=0)\n",
    "        S_points.append([xi,yj])\n",
    "\n",
    "plt.xlim(0,10)\n",
    "plt.ylim(-10,10)\n",
    "S_points = np.array(S_points).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 100)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S_points.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "xx, yy = np.meshgrid(x, y)\n",
    "xx, yy = xx.flatten(), yy.flatten()\n",
    "S_meshgrid = np.vstack([xx, yy])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-10, 10)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x25cb4933cf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(xx,yy)\n",
    "plt.title('$S$')\n",
    "plt.xlim(0,10)\n",
    "plt.ylim(-10,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 100)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S_meshgrid.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "發現S_meshgrid跟上面的S_points大小一樣"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(S_meshgrid, S_points)#用來比對，發現真的一樣"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. 將線性轉換 Φ:R2→R2 定義為"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Φ(x,y) = ((1/2)x-(1/3)y,(1/3)x-(1/4)y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Φ的矩陣表示法:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[0.5,  2],\n",
       "[0.5, -4]])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sp\n",
    "sp.Matrix([[1/2,2],[1/2,-4]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.array([[1/2,2],[1/2,4]])\n",
    "\n",
    "def phi(x):\n",
    "    assert A.shape[1]==x.shape[0],'incompatible dimension'\n",
    "    return A@x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. 畫出Φ(S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'$\\\\Phi(S)$')"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x25cb69f59e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "T = phi(S_meshgrid)\n",
    "u, v = phi(S_meshgrid)\n",
    "\n",
    "plt.scatter(u, v, c='r')\n",
    "plt.title('$\\Phi(S)$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 將 S 與 phi(S) 畫在一起做比較，並將 S 的透明度降低"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-10, 10)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x25cb6a4e860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(xx, yy, alpha=0.5)\n",
    "plt.scatter(u, v, c='r')\n",
    "plt.title('$S$ and $\\Phi(S)$')\n",
    "\n",
    "plt.xlim(0,10)\n",
    "plt.ylim(-10,10)"
   ]
  }
 ],
 "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": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}