| 1 | #!/usr/bin/python |
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| 2 | # -*- coding: utf-8 -*- |
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| 3 | # |
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| 4 | # Pyromaths |
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| 5 | # Un programme en Python qui permet de créer des fiches d'exercices types de |
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| 6 | # mathématiques niveau collège ainsi que leur corrigé en LaTeX. |
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| 7 | # Copyright (C) 2006 -- Jérôme Ortais (jerome.ortais@pyromaths.org) |
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| 8 | # |
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| 9 | # This program is free software; you can redistribute it and/or modify |
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| 10 | # it under the terms of the GNU General Public License as published by |
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| 11 | # the Free Software Foundation; either version 2 of the License, or |
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| 12 | # (at your option) any later version. |
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| 13 | # |
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| 14 | # This program is distributed in the hope that it will be useful, |
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| 15 | # but WITHOUT ANY WARRANTY; without even the implied warranty of |
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| 16 | # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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| 17 | # GNU General Public License for more details. |
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| 18 | # |
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| 19 | # You should have received a copy of the GNU General Public License |
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| 20 | # along with this program; if not, write to the Free Software |
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| 21 | # Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA |
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| 22 | # |
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| 23 | from random import randrange |
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| 24 | |
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| 25 | def genere_points(xmin, xmax, ymin, ymax, simple=False): |
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| 26 | |
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| 27 | if simple: # génère 3 intervalles de variations /\/ ou \/\ |
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| 28 | if randrange(1): |
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| 29 | variation = [0, 1, 0] |
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| 30 | else: |
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| 31 | variation = [1, 0, 1] |
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| 32 | longueur = xmax - xmin |
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| 33 | hauteur = ymax - ymin |
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| 34 | x_variation = [xmin] |
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| 35 | x_variation.append(x_variation[0] + randrange(int(0.25 * longueur), int(0.4 * longueur))) |
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| 36 | x_variation.append(x_variation[1] + randrange(int(0.25 * longueur), int(0.4 * longueur))) |
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| 37 | x_variation.append(xmax) |
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| 38 | liste_points = [(x_variation[0], randrange(ymin + (1 - variation[0]) * int(0.4 * hauteur), ymax - variation[0] * int(0.4 * hauteur)))] |
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| 39 | for i in [1, 2, 3]: |
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| 40 | if variation[i - 1]: |
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| 41 | liste_points.append((x_variation[i], randrange(liste_points[i - 1][1], ymax))) |
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| 42 | else: |
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| 43 | liste_points.append((x_variation[i], randrange(ymin, liste_points[i - 1][1]))) |
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| 44 | else: |
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| 45 | liste_points = [(xmin, 0)] |
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| 46 | for i in range(10): |
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| 47 | liste_points.append((liste_points[i][0] + randrange(2, 5), randrange(ymin, ymax))) |
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| 48 | return liste_points |
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| 49 | |
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| 50 | def bezier(simple=False, xmin=-15, xmax=15, ymin=-12, ymax=12): |
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| 51 | liste_points = genere_points(simple=simple, xmin=xmin, xmax=xmax, ymin=ymin, ymax=ymax) |
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| 52 | variation = [] |
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| 53 | largeur = liste_points[-1][0] - liste_points[0][0] |
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| 54 | points = [liste_points[0]] |
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| 55 | for i in range(1, len(liste_points)): |
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| 56 | if liste_points[i - 1][1] < liste_points[i][1]: |
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| 57 | points.append(liste_points[i]) |
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| 58 | elif liste_points[i - 1][1] > liste_points[i][1]: |
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| 59 | points.append(liste_points[i]) |
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| 60 | deltaxx = 0.3 * (points[1][0] - points[0][0]) |
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| 61 | nderiv = float(points[1][1] - points[0][1]) / (points[1][0] - points[0][0]) |
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| 62 | apres = [(points[0][0] + 0.5, points[0][1] + deltaxx * nderiv)] |
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| 63 | avant = [] |
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| 64 | if points[0][1] < points[1][1]: |
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| 65 | variation += '+' |
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| 66 | else: |
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| 67 | variation += '-' |
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| 68 | x_variation = [points[0][0]] |
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| 69 | y_variation = [points[0][1]] |
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| 70 | |
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| 71 | for i in range(1, len(points) - 1): |
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| 72 | if (points[i][1] <= points[i - 1][1] and points[i][1] <= points[i + 1][1]): |
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| 73 | nderiv = 0 |
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| 74 | x_variation.append(points[i][0]) |
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| 75 | y_variation.append(points[i][1]) |
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| 76 | variation += '+' |
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| 77 | elif (points[i][1] >= points[i - 1][1] and points[i][1] >= points[i + 1][1]): |
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| 78 | nderiv = 0 |
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| 79 | x_variation.append(points[i][0]) |
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| 80 | y_variation.append(points[i][1]) |
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| 81 | variation += '-' |
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| 82 | else: |
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| 83 | nderiv = float(points[i - 1][1] - points[i + 1][1]) / (points[i - 1][0] - points[i + 1][0]) |
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| 84 | deltax = 0.3 * (points[i - 1][0] - points[i][0]) |
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| 85 | deltaxx = 0.3 * (points[i + 1][0] - points[i][0]) |
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| 86 | avant.append((points[i][0] + deltax, points[i][1] + deltax * nderiv)) |
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| 87 | apres.append((points[i][0] + deltaxx, points[i][1] + deltaxx * nderiv)) |
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| 88 | avant.append((points[len(points) - 1][0] - 0.5, points[len(points) - 1][1] - 0)) |
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| 89 | string = str(points[0]) |
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| 90 | for i in range(len(points) - 1): |
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| 91 | string += str(apres[i]) + str(avant[i]) + str(points[i + 1]) |
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| 92 | textbezier = "\\input{tabvar}\n\ |
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| 93 | \psset{unit=" + str(14.0 / largeur) + "cm}\n\ |
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| 94 | \\begin{pspicture}" |
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| 95 | textbezier += "(" + str(xmin) + "," + str(ymin) + ")(" + str(liste_points[-1][0]) + "," + str(ymax) + ")\n" |
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| 96 | textbezier += "\psgrid[subgriddiv=0,griddots=10,gridlabels=0pt]\n\ |
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| 97 | \psaxes[linewidth=1pt,%\n\ |
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| 98 | ticks=none,%\n\ |
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| 99 | %labels=none\n\ |
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| 100 | Dx=2\n\ |
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| 101 | ]{->}(0,0)" |
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| 102 | textbezier += "(" + str(xmin) + "," + str(ymin) + ")(" + str(liste_points[-1][0]) + "," + str(ymax) + ")\n" |
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| 103 | impression = textbezier + "\psbezier" + string + "\n" + "\\end{pspicture}\n\\vspace{2cm}\n\n" |
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| 104 | impression += "\\begin{tikzpicture}\n\\tkzTabInit[espcl=1]\n{$x$ / .5,$f'\\,(x)$ / .5}\n" |
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| 105 | impression += "{$" + "$,$".join([str(i) for i in x_variation]) + "$}\n" |
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| 106 | impression += "\\tkzTabLine{," + ",".join(y_variation) + "}\n\end{tikzpicture}\n" |
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| 107 | #=============================================================================== |
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| 108 | # {$x$ / 1,$g(x)$ / 1}% |
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| 109 | # {$0$,$1$,$2$}% |
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| 110 | # \tkzTabLine{d,+,0,-,d} |
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| 111 | # \end{tikzpicture}" |
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| 112 | #=============================================================================== |
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| 113 | impression += "$$\n\ |
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| 114 | \\tabvar{%\n\ |
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| 115 | \\tx{x}&\\tx{" + str(x_variation[0]) + "}" |
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| 116 | for i in range(1, len(x_variation)): |
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| 117 | impression += "&& \\tx{" + str(x_variation[i]) + "}" |
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| 118 | impression += "&& \\tx{" + str(points[-1][0]) + "}\\cr\n" |
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| 119 | impression += "\\tx{f(x)}" |
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| 120 | for i in range(len(variation)): |
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| 121 | if variation[i] == '+': |
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| 122 | impression += "&\\txb{" + str(y_variation[i]) + "}&\\fm" |
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| 123 | else: |
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| 124 | impression += "&\\txh{" + str(y_variation[i]) + "}&\\fd" |
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| 125 | if variation[-1] == '+': |
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| 126 | impression += "&\\txh{" + str(points[-1][1]) + "}" |
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| 127 | else: |
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| 128 | impression += "&\\txb{" + str(points[-1][1]) + "}" |
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| 129 | return impression + "\\cr\n}$$\n\n" |
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| 130 | |
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| 131 | |
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| 132 | def tab_var(fonc, Intervalle=["-\\infty", "+\\infty"]): |
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| 133 | from pyromaths.classes.Polynome import Polynome |
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| 134 | if isinstance(fonc, Polynome): |
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| 135 | return tab_var_poly(fonc, Intervalle[0], Intervalle[1]) |
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| 136 | else: |
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| 137 | return _(u"Je ne sais pas étudier les variations de cette fonction.") |
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| 138 | |
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| 139 | def tab_var_poly(P, borneinf="-\\infty" , bornesup=u"+\\infty"): |
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| 140 | Intervalle = [borneinf, bornesup] |
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| 141 | # print"borneinf=",borneinf |
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| 142 | tab_var = "\\input{tabvar}\n" |
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| 143 | tab_var += "$$\\tabvar{%\n" |
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| 144 | if P.deg == 0: |
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| 145 | tab_var += tabvar_x(Intervalle, P.var) |
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| 146 | Cste = str(P(0)) |
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| 147 | tab_var += "\\tx{f(" + P.var + ")}&\\tx{" + Cste + "}&\\fhm&\\tx{" + Cste + "}\\cr\n" |
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| 148 | return tab_var + "}$$\n" |
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| 149 | elif P.deg == 1: |
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| 150 | tab_var += tabvar_x(Intervalle, P.var) |
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| 151 | if P.dictio[1] > 0: |
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| 152 | tab_var += "\\tx{f(" + P.var + ")}& &\\fm& \\cr\n" |
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| 153 | else: |
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| 154 | tab_var += "\\tx{f(" + P.var + ")}& &\\fd&\\cr\n" |
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| 155 | return tab_var + "}$$\n" |
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| 156 | elif P.deg == 2: |
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| 157 | alpha = -P.dictio[1] / (2 * P.dictio[2]) |
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| 158 | extremum = P(alpha) |
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| 159 | if (borneinf == "-\\infty" or Intervalle[0] < alpha) and\ |
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| 160 | (Intervalle[1] == "+\\infty" or Intervalle[1] > alpha): |
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| 161 | listex = [Intervalle[0], alpha, Intervalle[1]] |
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| 162 | if P.dictio[0] > 0: |
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| 163 | tab_var_y = "\\tx{f(" + P.var + ")}& &\\fd&\\txb{" + nombre_TeX(extremum) + "}&\\fm&\\cr\n" |
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| 164 | else: |
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| 165 | tab_var_y = "\\tx{f(" + P.var + ")}& &\\fm&\\txh{" + nombre_TeX(extremum) + "}&\\fd&\\cr\n" |
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| 166 | else: |
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| 167 | listex = Intervalle |
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| 168 | if P.dictio > 0: |
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| 169 | if not(borneinf == "-\\infty") and Intervalle[0] > alpha: |
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| 170 | tab_var_y = "\\tx{f(" + P.var + ")}& &\\fm& \\cr\n" |
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| 171 | else: |
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| 172 | tab_var_y = "\\tx{f(" + P.var + ")}& &\\fd&\\cr\n" |
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| 173 | else: |
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| 174 | if not(borneinf == "-\\infty") and Intervalle[0] > alpha: |
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| 175 | tab_var_y = "\\tx{f(" + P.var + ")}& &\\fd&\\cr\n" |
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| 176 | else: |
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| 177 | tab_var_y = "\\tx{f(" + P.var + ")}& &\\fm& \\cr\n" |
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| 178 | tab_var += tabvar_x(listex, P.var) |
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| 179 | tab_var += tab_var_y |
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| 180 | return tab_var + "}$$\n" |
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| 181 | elif P.deg == 3: |
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| 182 | return _(u"attends un peu pour le degré 3") |
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| 183 | # #Les fonctions développées pour les équations du second degré doivent aider |
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| 184 | else: |
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| 185 | return _(u"Degré trop élevé") |
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| 186 | # #Cas particulier où P' a une racine évidente, ou polynome bicarré |
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| 187 | |
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| 188 | |
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| 189 | def tabvar_x(listex, var): |
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| 190 | '''imprime la premiere ligne du tabvar''' |
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| 191 | var_x = "\\tx{" + var + "}&\\tx{" + nombre_TeX(listex[0]) + "}" |
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| 192 | for i in range(1, len(listex)): |
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| 193 | var_x += "&&\\tx{" + nombre_TeX(listex[i]) + "}" |
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| 194 | return var_x + "\\cr\n" |
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| 195 | |
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| 196 | def nombre_TeX(nombre): # A remplacer par sepmillier |
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| 197 | from pyromaths.classes.Fractions import Fraction |
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| 198 | if type(nombre) in [type(int), type(float)]: |
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| 199 | return str(nombre) |
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| 200 | elif isinstance(nombre, Fraction): |
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| 201 | if nombre.denominateur == 1: |
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| 202 | return nombre_TeX(nombre.numerateur) |
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| 203 | else: |
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| 204 | return "\\dfrac{" + str(nombre.numerateur) + "}{" + str(nombre.denominateur) + "}" |
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| 205 | else: |
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| 206 | return str(nombre) |
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