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{
"cells": [
{
"cell_type": "markdown",
"id": "e26c0a4c",
"metadata": {},
"source": [
"# Machine Learning\n",
"## Gradient and Tangent Planes\n",
"Given a hypersurface by $f(\\boldsymbol{x})=constant$, its gradient $\\nabla f$ is orthogonal to the hypersurface. Thus, the tangent hyperplane at point $\\boldsymbol{x}_0$ is obtained by: <br>\n",
"$\\nabla f(\\boldsymbol{x}_0)\\cdot (\\boldsymbol{x}-\\boldsymbol{x}_0)=0$\n",
"<hr>\n",
"The Python code at: https://github.com/ostad-ai/Machine-Learning\n",
"<br> Explanation: https://www.pinterest.com/HamedShahHosseini/machine-learning/background-knowledge/"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "624127b5",
"metadata": {},
"outputs": [],
"source": [
"# importing required modules\n",
"import numpy as np\n",
"from matplotlib import pyplot as plt"
]
},
{
"cell_type": "markdown",
"id": "efeb84ed",
"metadata": {},
"source": [
"The level curve for $f(x,y)=x^2+y^2$ defined by $f(x,y)=r^2$ is given.\n",
"We want to find the tangent line to the point of interest, $(x_0,y_0)$, on the curve.\n",
"<br> For this purpose, we take the gradient of $f$:\n",
"<br>$\\nabla f(x_0,y_0)=[2x_0,2y_0]^T$\n",
"<br> Now, we put the gradient into the equation for hyperplanes:\n",
"<br> $[2x_0,2y_0]^T\\cdot[x-x_0,y-y_0]^T$=0\n",
"<br> simplifiyng above equation leads to:\n",
"<br> $x_0(x-x_0)+y_0(y-y_0)=0 \\rightarrow xx_0+yy_0=x^2_0+y^2_0=r^2$\n",
"<br> Finally, we get the tangent line:\n",
"<br>$xx_0+yy_0=r^2$"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "6d80e3a8",
"metadata": {},
"outputs": [
{
"data": {
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",
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# tangent line to the level curve x^2+y^2=r^2\n",
"teta=np.linspace(0,2*np.pi,100)\n",
"# radius of a circle and angle of the point\n",
"for r,alpha in [(9,3.9),(15,.6),(25,1.9)]:\n",
" # xs and ys are the points on the circle\n",
" xs=r*np.cos(teta)\n",
" ys=r*np.sin(teta)\n",
" # the point of interest with angle and radius\n",
" # x_0=r*cos(alpha), y_0=r*sin(alpha)\n",
" #--tangent line\n",
" # x*cos(alpha)+y*sin(alpha)=r\n",
" # xt and yt are the points on the tangent line\n",
" xt=np.linspace(-1.3*r,1.3*r,100)\n",
" yt=(r-xt*np.cos(alpha))/np.sin(alpha)\n",
" plt.plot(xs,ys,'-'); plt.plot(xt,yt,'-')\n",
" plt.plot(r*np.cos(alpha),r*np.sin(alpha),'*')\n",
"plt.axis('equal')\n",
"plt.title('Tangent lines to the level curves: $x^2+y^2=r^2$')\n",
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2a4c1855",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"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": "3.11.1"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
|