{"id":9798,"date":"2024-05-18T15:36:13","date_gmt":"2024-05-18T13:36:13","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=9798"},"modified":"2024-05-19T09:38:42","modified_gmt":"2024-05-19T07:38:42","slug":"trouver-lequation-dune-parabole-passant-par-3-points","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/","title":{"rendered":"Trouver l&rsquo;\u00e9quation d&rsquo;une parabole passant par 3 points"},"content":{"rendered":"\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Dans ma vie professionnelle, et pas plus tard qu&rsquo;hier, j&rsquo;ai souvent eu \u00e0 trouver l&rsquo;\u00e9quation d&rsquo;une parabole passant par 3 points. Pourquoi ? Parce que dans les exercices que l&rsquo;on propose aux \u00e9l\u00e8ves, c&rsquo;est pratique!<\/p>\n\n\n\n<!--more-->\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Le probl\u00e8me est que le logiciel <a href=\"https:\/\/www.geogebra.org\/\" target=\"_blank\" rel=\"noreferrer noopener\">geobegra<\/a> ne permet pas de le faire \u00e0 ma connaissance. <\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph has-black-color has-text-color has-background has-link-color wp-elements-47d906cb4111ef180a0419f709595a82 wp-block-paragraph\" style=\"background-color:#abb7c266\"><strong><em>Edit:<\/em><\/strong> <em>un lecteur m&rsquo;informe qu&rsquo;il est facile de trouver une \u00e9quation connaissant 3 points sous geogebra \u00e0 l&rsquo;aide de la fonction Polynome(A,B,C). En effet, cela fonctionne mais les coefficients sont des flottants. Ainsi, si l&rsquo;on veut proposer une \u00e9quation \u00e0 des \u00e9l\u00e8ves, cela ne convient pas car les coefficients sont souvent des valeurs approch\u00e9es.<\/em><\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">J&rsquo;ai donc d\u00fb m&rsquo;y coller en Python&#8230;<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/#Trouver_lequation_dune_parabole_passant_par_3_points_aspect_mathematique\" >Trouver l&rsquo;\u00e9quation d&rsquo;une parabole passant par 3 points: aspect math\u00e9matique<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/#Formallisation\" >Formallisation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/#Resolution_algebrique\" >R\u00e9solution alg\u00e9brique<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/#Trouver_lequation_dune_parabole_passant_par_3_points_implementation_en_Python\" >Trouver l&rsquo;\u00e9quation d&rsquo;une parabole passant par 3 points: impl\u00e9mentation en Python<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/#Des_coefficients_rationnels\" >Des coefficients rationnels<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/#Trace_de_la_parabole_en_Python\" >Trac\u00e9 de la parabole en Python<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/#Trace_de_la_courbe_en_LaTeX\" >Trac\u00e9 de la courbe en \\(LaTeX\\)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/#Le_script_complet\" >Le script complet<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Trouver_lequation_dune_parabole_passant_par_3_points_aspect_mathematique\"><\/span>Trouver l&rsquo;\u00e9quation d&rsquo;une parabole passant par 3 points: aspect math\u00e9matique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Voyons avant tout l&rsquo;aspect math\u00e9matique du probl\u00e8me. C&rsquo;est assez simple: on connait les coordonn\u00e9es de trois points, et on cherche les coefficients <em>a<\/em>, <em>b<\/em> et <em>c<\/em> de la fonction <em>f<\/em>(<em>x<\/em>) = <em>ax<\/em>\u00b2 + <em>bx<\/em> + <em>c<\/em> de sorte que sa courbe repr\u00e9sentative passe par les trois points.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Formallisation\"><\/span>Formallisation<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On pose \\(A(x_A;y_A)\\), \\(B(x_B;y_B)\\) et \\(C(x_C;y_C)\\) nos trois points. On a alors le syst\u00e8me:$$\\left\\{\\begin{array}{l}ax_A^2+bx_A+c=y_A\\\\ax_B^2+bx_B+c=y_B\\\\ax_C^2+bx_C+c=y_C \\end{array}\\right.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Resolution_algebrique\"><\/span>R\u00e9solution alg\u00e9brique<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Ce syst\u00e8me est un syst\u00e8me lin\u00e9aire de trois \u00e9quations \u00e0 trois inconnues, donc rien de bien m\u00e9chant \u00e0 r\u00e9soudre&#8230;C&rsquo;est juste un peu long si on le fait \u00e0 la main. Mais nous n&rsquo;allons pas le faire, bien s\u00fbr&#8230; car nous sommes des sacripants&#8230; \ud83d\ude42<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Nous allons \u00e9crire ce syst\u00e8me de fa\u00e7on matricielle:$$\\begin{pmatrix}x_A^2 &amp; x_A &amp; 1\\\\x_B^2 &amp; x_B &amp; 1\\\\x_C^2 &amp; x_C &amp; 1\\\\ \\end{pmatrix}\\begin{pmatrix}a\\\\b\\\\c\\end{pmatrix}=\\begin{pmatrix}y_A\\\\y_B\\\\y_C\\end{pmatrix}.$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On en d\u00e9duit alors:$$\\begin{pmatrix}a\\\\b\\\\c\\end{pmatrix}=\\begin{pmatrix}y_A\\\\y_B\\\\y_C\\end{pmatrix}\\begin{pmatrix}x_A^2 &amp; x_A &amp; 1\\\\x_B^2 &amp; x_B &amp; 1\\\\x_C^2 &amp; x_C &amp; 1\\\\ \\end{pmatrix}^{-1}.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Trouver_lequation_dune_parabole_passant_par_3_points_implementation_en_Python\"><\/span>Trouver l&rsquo;\u00e9quation d&rsquo;une parabole passant par 3 points: impl\u00e9mentation en Python<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Il existe une m\u00e9thode pour r\u00e9soudre de tels syst\u00e8mes lin\u00e9aires en Python. Avec le module <em>numpy<\/em> et <em>linalg.solve<\/em>, l&rsquo;affaire est pli\u00e9e&#8230; Sauf que la plupart du temps, le syst\u00e8me est r\u00e9solu sur les flottants et donne donc comme r\u00e9sultats des nombres en valeurs approch\u00e9es! Scrogneugneu! C&rsquo;est carr\u00e9ment naze! Il nous faut donc mettre ce qui nous sert de cerveau en mode <em>cogitum<\/em>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Des_coefficients_rationnels\"><\/span>Des coefficients rationnels<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">L&rsquo;id\u00e9e est de manipuler des rationnels, et non des flottants. On ne va donc pas utiliser <em>numpy<\/em> mais <em>sympy<\/em> qui, d&rsquo;ailleurs, est quand-m\u00eame appropri\u00e9 pour faire de l&rsquo;alg\u00e8bre.<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"false\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from sympy import Matrix, Rational\n\ndef parabola(A,B,C):\n    L = [A,B,C]\n    matrice = Matrix([[Rational(a**2),Rational(a),Rational(1)] for a,b in L])\n    other = Matrix([[Rational(b)] for a,b in L])\n\n    return matrice.solve(other)\n\nif __name__ == '__main__':\n    A, B, C = (-5,4), (0,-3), (7,5)\nP = parabola(A,B,C)<\/pre>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Voil\u00e0! Avec la fonction <em>parabola<\/em>, j&rsquo;ai la matrice des coefficients <em>a<\/em>, <em>b<\/em>, <em>c<\/em>. Il nous faut maintenant peaufiner tout \u00e7a.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Trace_de_la_parabole_en_Python\"><\/span>Trac\u00e9 de la parabole en Python<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Une fois l&rsquo;expression de la parabole obtenue, je vais la tracer. Je vais d\u00e9finir alors les deux fonctions suivantes.<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"false\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from sympy import Matrix, Rational, latex\nimport matplotlib.pyplot as plt\nimport numpy as np\n\ndef f(P,x):\n    return P[0]*x*x + P[1]*x + P[2]\n    \ndef draw_parabola(A, B, C):\n    P = parabola(A,B,C)\n    x = np.linspace(-10, 10, 400)\n    y = f(P,x)\n\n    # Convertir les coefficients en cha\u00eenes de caract\u00e8res LaTeX\n    P_latex = [latex(P[i]) for i in range(3)]\n    chaine = rf'$f(x) = {P_latex[0]}x^2'\n    if P[1] &lt; 0:\n        chaine += rf' {P_latex[1]}x'\n    else:\n        chaine += rf' + {P_latex[1]}x'\n    if P[2] &lt; 0:\n        chaine += rf' {P_latex[2]}$'\n    else:\n        chaine += rf' + {P_latex[2]}$'\n        \n    plt.plot(x, y, label=chaine)\n\n    # Ajouter les points\n\n    for a,b in [A,B,C]:\n        plt.scatter(a, b, color='red', zorder=5)\n        plt.annotate(f'({a},{b})', (a, b), textcoords=\"offset points\", xytext=(-15,-10), ha='center', color='red')\n    plt.title('Graphique de la fonction f(x)')\n    plt.xlabel('x')\n    plt.ylabel('f(x)')\n    plt.legend()\n\n    # Afficher la grille\n    plt.grid(True)\n\n    # Afficher le graphique\n    plt.show()<\/pre>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">L\u00e0, une magnifique courbe se dessine alors, faisant appara\u00eetre les trois points impos\u00e9s.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-13.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"218\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-13-300x218.png\" alt=\"trouver parabole passant par 3 points python\" class=\"wp-image-9801\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-13-300x218.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-13.png 382w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Trace_de_la_courbe_en_LaTeX\"><\/span>Trac\u00e9 de la courbe en \\(LaTeX\\)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Si on souhaite cr\u00e9er de beaux devoirs pour nos chers ch\u00e9rubins, le graphique \u00e0 la <em>matplotlib<\/em> ne suffira pas n\u00e9cessairement. Le top du top, c&rsquo;est tout de m\u00eame d&rsquo;avoir un graphique en \\(\\LaTeX\\). Alors soit! <\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On va cr\u00e9er une fonction qui cr\u00e9e  la courbe en \\(\\LaTeX\\) \u00e0 l&rsquo;aide de TiKZ.<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"false\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">import os\n\ndef export_latex(A, B, C, x_range=(-10, 10), num_points=100):\n    P = parabola(A,B,C)\n    # Convertir les coefficients en cha\u00eenes de caract\u00e8res LaTeX\n    P_latex = [ '({}\/{})'.format(P[i].numerator,P[i].denominator) for i in range(3)]\n    \n    y_min, y_max = 0,0\n    x = x_range[0]\n    while x &lt;= x_range[1]:\n        if P[0]*x*x+P[1]*x+P[2] &lt; y_min:\n            y_min = P[0]*x*x+P[1]*x+P[2]\n        if P[0]*x*x+P[1]*x+P[2] > y_max:\n            y_max = P[0]*x*x+P[1]*x+P[2]\n        x += (x_range[1]-x_range[0])\/num_points\n    \n    y_min, y_max = y_min-1, y_max+1\n    points_latex = ''\n    \n    for x,y in [A,B,C]:\n        points_latex += f\"\"\"\n            \\\\fill[red] ({x},{y}) circle (4pt);\\n\n        \"\"\"\n    \n    tikz_code = f\"\"\"\n\\\\documentclass{{standalone}}\n\\\\usepackage{{tikz}}\n\\\\begin{{document}}\n\\\\begin{{tikzpicture}}\n    \\\\draw[gray,dashed] ({x_range[0]},{y_min}) grid ({x_range[1]},{y_max});\n    \\\\draw[thick,->,>=latex] ({x_range[0]},0) -- ({x_range[1]},0);\n    \\\\draw[thick,->,>=latex] (0,{y_min}) -- (0,{y_max});\n    \\\\draw[thick] (1,0.1) -- (1,-0.1) node[below] {{1}};\n    \\\\draw[thick] (0.1,1) -- (-0.1,1) node[left] {{1}};\n    \\\\draw[thick, smooth, domain={x_range[0]}:{x_range[1]}, samples={num_points}] plot (\\\\x, {{ {P_latex[0]}*\\\\x*\\\\x + {P_latex[1]}*\\\\x + {P_latex[2]} }});\n    {points_latex}\n\\\\end{{tikzpicture}}\n\\\\end{{document}}\n\"\"\"\n    with open('parabola_plot.tex', 'w') as f:\n        f.write(tikz_code)\n        \n    cmd = \"pdflatex  --shell-escape -synctex=1 -interaction=nonstopmode parabola_plot.tex\"\n    os.system(cmd)\n    \n    readpdf = \"START parabola_plot.pdf\"\n    os.system(readpdf)<\/pre>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Voili-voilou!<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-14.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"224\" height=\"300\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-14-224x300.png\" alt=\"trouver parabole passant par 3 points LaTeX TiKZ\" class=\"wp-image-9803\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-14-224x300.png 224w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-14-600x804.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-14.png 677w\" sizes=\"auto, (max-width: 224px) 100vw, 224px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Cette fonction exporte le fichier \\(\\LaTeX\\), puis le compile via PdfLaTeX. Ainsi, on peut le retoucher si besoin est.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Il y a fort \u00e0 parier que ce code est \u00e0 conserver dans vos outils de profs, n&rsquo;est-ce pas ? \u00e7a vous laissera plus de temps pour regarder Netflix \ud83d\ude42 (bon, l\u00e0, je parle pour moi&#8230;).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_script_complet\"><\/span>Le script complet<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"true\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\"># -*- coding: utf-8 -*-\n\"\"\"\nCreated on Fri May 17 18:13:53 2024\n\n@author: St\u00e9phane Pasquet\n@url: https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/\n\"\"\"\n\nfrom sympy import Matrix, Rational, latex\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport os\n\n# retourne a, b, x de y = ax\u00b2 + bx + c, o\u00f9 A, B, C sont sur la parabole\n\ndef parabola(A,B,C):\n    L = [A,B,C]\n    matrice = Matrix([[Rational(a**2),Rational(a),Rational(1)] for a,b in L])\n    other = Matrix([[Rational(b)] for a,b in L])\n\n    return matrice.solve(other)\n    \ndef f(P,x):\n    return P[0]*x*x + P[1]*x + P[2]\n    \n\ndef draw_parabola(A, B, C):\n    P = parabola(A,B,C)\n    x = np.linspace(-10, 10, 400)\n    y = f(P,x)\n\n    # Convertir les coefficients en cha\u00eenes de caract\u00e8res LaTeX\n    P_latex = [latex(P[i]) for i in range(3)]\n    chaine = rf'$f(x) = {P_latex[0]}x^2'\n    if P[1] &lt; 0:\n        chaine += rf' {P_latex[1]}x'\n    else:\n        chaine += rf' + {P_latex[1]}x'\n    if P[2] &lt; 0:\n        chaine += rf' {P_latex[2]}$'\n    else:\n        chaine += rf' + {P_latex[2]}$'\n        \n    plt.plot(x, y, label=chaine)\n\n    # Ajouter les points\n\n    for a,b in [A,B,C]:\n        plt.scatter(a, b, color='red', zorder=5)\n        plt.annotate(f'({a},{b})', (a, b), textcoords=\"offset points\", xytext=(-15,-10), ha='center', color='red')\n    plt.title('Graphique de la fonction f(x)')\n    plt.xlabel('x')\n    plt.ylabel('f(x)')\n    plt.legend()\n\n    # Afficher la grille\n    plt.grid(True)\n\n    # Afficher le graphique\n    plt.show()\n\ndef export_latex(A, B, C, x_range=(-10, 10), num_points=100):\n    P = parabola(A,B,C)\n    # Convertir les coefficients en cha\u00eenes de caract\u00e8res LaTeX\n    P_latex = [ '({}\/{})'.format(P[i].numerator,P[i].denominator) for i in range(3)]\n    \n    y_min, y_max = 0,0\n    x = x_range[0]\n    while x &lt;= x_range[1]:\n        if P[0]*x*x+P[1]*x+P[2] &lt; y_min:\n            y_min = P[0]*x*x+P[1]*x+P[2]\n        if P[0]*x*x+P[1]*x+P[2] > y_max:\n            y_max = P[0]*x*x+P[1]*x+P[2]\n        x += (x_range[1]-x_range[0])\/num_points\n    \n    y_min, y_max = y_min-1, y_max+1\n    points_latex = ''\n    \n    for x,y in [A,B,C]:\n        points_latex += f\"\"\"\n            \\\\fill[red] ({x},{y}) circle (4pt);\\n\n        \"\"\"\n    \n    tikz_code = f\"\"\"\n\\\\documentclass{{standalone}}\n\\\\usepackage{{tikz}}\n\\\\begin{{document}}\n\\\\begin{{tikzpicture}}\n    \\\\draw[gray,dashed] ({x_range[0]},{y_min}) grid ({x_range[1]},{y_max});\n    \\\\draw[thick,->,>=latex] ({x_range[0]},0) -- ({x_range[1]},0);\n    \\\\draw[thick,->,>=latex] (0,{y_min}) -- (0,{y_max});\n    \\\\draw[thick] (1,0.1) -- (1,-0.1) node[below] {{1}};\n    \\\\draw[thick] (0.1,1) -- (-0.1,1) node[left] {{1}};\n    \\\\draw[thick, smooth, domain={x_range[0]}:{x_range[1]}, samples={num_points}] plot (\\\\x, {{ {P_latex[0]}*\\\\x*\\\\x + {P_latex[1]}*\\\\x + {P_latex[2]} }});\n    {points_latex}\n\\\\end{{tikzpicture}}\n\\\\end{{document}}\n\"\"\"\n    with open('parabola_plot.tex', 'w') as f:\n        f.write(tikz_code)\n        \n    cmd = \"pdflatex  --shell-escape -synctex=1 -interaction=nonstopmode parabola_plot.tex\"\n    os.system(cmd)\n    \n    readpdf = \"START parabola_plot.pdf\"\n    os.system(readpdf)\n\n\nif __name__ == '__main__':\n    A, B, C = (-5,4), (0,-3), (7,5)\n    draw_parabola(A, B, C)\n    export_latex(A, B, C)<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Dans ma vie professionnelle, et pas plus tard qu&rsquo;hier, j&rsquo;ai souvent eu \u00e0 trouver l&rsquo;\u00e9quation d&rsquo;une parabole passant par 3 points. Pourquoi ? Parce que dans les exercices que l&rsquo;on propose aux \u00e9l\u00e8ves, c&rsquo;est pratique!<\/p>\n","protected":false},"author":1,"featured_media":9804,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,6,5],"tags":[135,421,101,422],"class_list":["post-9798","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-latex","category-mathematiques","category-python","tag-matplotlib","tag-parabole","tag-python","tag-sympy"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.8 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Trouver l&#039;\u00e9quation d&#039;une parabole passant par 3 points - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Dans ma vie professionnelle, et pas plus tard qu&#039;hier, j&#039;ai souvent eu \u00e0 trouver l&#039;\u00e9quation d&#039;une parabole passant par 3 points...\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Trouver l&#039;\u00e9quation d&#039;une parabole passant par 3 points - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Dans ma vie professionnelle, et pas plus tard qu&#039;hier, j&#039;ai souvent eu \u00e0 trouver l&#039;\u00e9quation d&#039;une parabole passant par 3 points...\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/18\/trouver-lequation-dune-parabole-passant-par-3-points\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathweb.fr\" \/>\n<meta property=\"article:published_time\" content=\"2024-05-18T13:36:13+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-05-19T07:38:42+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/head-articles-python-parabole-3-points.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"740\" \/>\n\t<meta property=\"og:image:height\" content=\"198\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"St\u00e9phane Pasquet\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"St\u00e9phane Pasquet\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2024\\\/05\\\/18\\\/trouver-lequation-dune-parabole-passant-par-3-points\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2024\\\/05\\\/18\\\/trouver-lequation-dune-parabole-passant-par-3-points\\\/\"},\"author\":{\"name\":\"St\u00e9phane Pasquet\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"headline\":\"Trouver l&rsquo;\u00e9quation d&rsquo;une parabole passant par 3 points\",\"datePublished\":\"2024-05-18T13:36:13+00:00\",\"dateModified\":\"2024-05-19T07:38:42+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2024\\\/05\\\/18\\\/trouver-lequation-dune-parabole-passant-par-3-points\\\/\"},\"wordCount\":708,\"commentCount\":2,\"publisher\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2024\\\/05\\\/18\\\/trouver-lequation-dune-parabole-passant-par-3-points\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2024\\\/05\\\/head-articles-python-parabole-3-points.jpg\",\"keywords\":[\"matplotlib\",\"parabole\",\"python\",\"sympy\"],\"articleSection\":[\"LaTeX\",\"Math\u00e9matiques\",\"Python\"],\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2024\\\/05\\\/18\\\/trouver-lequation-dune-parabole-passant-par-3-points\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2024\\\/05\\\/18\\\/trouver-lequation-dune-parabole-passant-par-3-points\\\/\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2024\\\/05\\\/18\\\/trouver-lequation-dune-parabole-passant-par-3-points\\\/\",\"name\":\"Trouver l'\u00e9quation d'une parabole passant par 3 points - 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