２次関数の最大値(その1/2)「数学チャレンジ問題長方形DBFEの面積の最大値は？」をとsympyとFreeCADとChatGPTとWolframAlphaでやってみたい。

Last updated at 2024-06-29Posted at 2024-06-29

・問題文は2次元ですが、３次元FreeCADのマクロで、XY平面上に作図しました。
・(その2/2)に続きます。
https://qiita.com/mrrclb48z/items/5dc3518f948cc3d53f82

オリジナル

・YUUU0123 様 (0:00〜3:36)

sympyで(オリジナル様の方法を参考に)

Ver.0.1 平方完成で

「平方完成(へいほうかんせい、英: completing the square) 」を調べた。ChatGPT以後を追加
https://qiita.com/mrrclb48z/items/38e849f41f46bf0c8f9c

# Ver.0.1 平方完成で
from sympy import *
var('x,h,k',real=True,nonnegative=True)
def complete_square(expr):
    a, b, c = Poly(expr, x).all_coeffs()
    
    h = -b / (2 * a)
    k = expr.subs(x, h)
    
    return h, k
y=(2-x)*2*x
h,k=complete_square(y.expand())
print("#",h,k)
plot(y,(x,0,2))
# １２

Ver.0.2 sympyのmaximumで

sympy.calculus.util.maximum(f, symbol, domain=Reals)
https://docs.sympy.org/latest/modules/calculus/index.html#sympy.calculus.util.maximum

# Ver.0.2 sympyのmaximumで
from sympy import *
var('x',real=True,nonnegative=True)
y=(2-x)*2*x
y_max=maximum(y,x,Interval(0,2))
x_max=solve(Eq(y,y_max),x)[0]
print("#",x_max,y_max)
# plot(y,(x,0,2))
# # 1 2

Ver.0.3 微分で

＞強いて、微分で
To take derivatives, use the diff function.
https://docs.sympy.org/latest/tutorials/intro-tutorial/calculus.html#derivatives

# Ver.0.3 微分で
from sympy import *
var('x',real=True,nonnegative=True)
y=(2-x)*2*x
x_max=solve(Eq(diff(y),0),x)[0]
y_max=y.subs({x:x_max})
print("#",x_max,y_max)
# # 1 2

Ver.0.4 ベクトルで

・点Eの座標より。

# Ver.0.4 ベクトルで
from sympy import *
var('t',real=True,nonnegative=True)
A=Point(0,2)
B=Point(0,0)
C=Point(4,0)
E=C+t*(A-C)
area=E.x*E.y
area_max=maximum(area,t,Interval(0,1))
area_t  =solve(Eq(area,area_max),t)[0]
E=E.subs({t:area_t})
print("#",E,area_max)
plot(area,(t,0,1))
# Point2D(2, 1) 2

FreeCADのマクロで作図

・計算部分は、Ver.0.4のコピー貼り付けです。

import FreeCAD
import Part
import DraftTools
import Draft
import Mesh
############################################################################
# Ver.0.4 ベクトルで
from sympy import *
var('t',real=True,nonnegative=True)
A=Point(0,2)
B=Point(0,0)
C=Point(4,0)
E=C+t*(A-C)
area=E.x*E.y
area_max=maximum(area,t,Interval(0,1))
area_t  =solve(Eq(area,area_max),t)[0]
E=E.subs({t:area_t})
print("#",E,area_max)
# plot(area,(t,0,1))
# Point2D(2, 1) 2
############################################################################
# 作図用
D=Point(0,E.y)
F=Point(E.x,0)
############################################################################
# 3D作図 z=0 XY平面に作図しました。
############################################################################
def myXYZ2Txt_2D(A):
    return '(' + str(A.x) + ',' + str(A.y) +  ')'
    # return ""
def myTxtXYZ_2D(A,myWedgei):
    P5x=float(A.x)
    P5y=float(A.y)
    P5z=0.0
    p5 = FreeCAD.Vector(P5x, P5y, P5z)
    myText = Draft.makeText(myWedgei, p5)
    myText.Label = myWedgei
    # FreeCADGui.ActiveDocument.ActiveObject.FontSize = '0.1 mm'
    FreeCADGui.ActiveDocument.ActiveObject.FontSize = '0.3 mm'
    return
def myTxtXYZ_S_2D(*xy_tx):
    for i in range(1,int(len(xy_tx)/2)+1):
        myTxtXYZ_2D(xy_tx[2*i-2],xy_tx[2*i-1]+myXYZ2Txt_2D(xy_tx[2*i-2]) )
    return
def myLine_2D(A,B):
    Ax,Ay,Az=float(A.x),float(A.y),0.0
    Bx,By,Bz=float(B.x),float(B.y),0.0
    pl = FreeCAD.Placement()
    pl.Rotation.Q = (0.4247081540122249, 0.17592004639554645, 0.33985110062924484, 0.8204732460821097)
    pl.Base = FreeCAD.Vector(-3.9166066876399563, -2.1670824762243774, 1.7495260956243028)
    points = [FreeCAD.Vector(Ax,Ay,Az), FreeCAD.Vector(Bx,By,Bz)]
    line = Draft.make_wire(points, placement=pl, closed=False, face=True, support=None)
    Draft.autogroup(line)
    return
def myLine_S_2D(*args):
    for i in range(1,len(args)):
        myLine_2D(args[i-1],args[i])
    return
def myLine_C_2D(*args):
    for i in range(1,len(args)):
        myLine_2D(args[i-1],args[i])
    myLine_2D(args[i],args[0])
    return
#######################################################################################
myLine_C_2D  (A,D,B,F,C,E)
myLine_S_2D  (D,E,F)
myTxtXYZ_S_2D(A,"A",D,"D",B,"B", F,"F",C,"C",E,"E", )
#######################################################################################
doc = App.activeDocument()
App.ActiveDocument.addObject("App::Origin", "Origin")
# App.ActiveDocument.getObject('Origin').Visibility = True
App.ActiveDocument.recompute()
Gui.activeDocument().activeView().viewAxonometric()
Gui.SendMsgToActiveView("ViewFit")