空間図形「21年 兵庫県大 商経・情報 3」をsympyとFreeCADでやってみた。

座標空間内の8点.....平行四辺形....
オリジナル

https://kamelink.com/blog/2022/12/03/9-1-21-hyogokendai-shokei-johou-3/

上と同じです。大学入試数学問題集成＞【３】

https://mathexamtest.web.fc2.com/2021/202111613/2021116130100mj.html#top-0104

# sympyで(オリジナルのやり方)
勉強中

# sympyで(sympy的?安易なやり方)

```python
from sympy import *
var('p q t')
def myTaisekiGyouretuSiki(PTO,PTA,PTB,PTC):
    return Matrix([[PTA.x-PTO.x, PTA.y-PTO.y, PTA.z-PTO.z], \
                   [PTB.x-PTO.x, PTB.y-PTO.y, PTB.z-PTO.z], \
                   [PTC.x-PTO.x, PTC.y-PTO.y, PTC.z-PTO.z]]).det()/6
def myMensekiVector3D(P,Q):
    return Rational(1, 2) * sqrt(P.distance(Point(0, 0, 0)) ** 2 * Q.distance(Point(0, 0, 0)) ** 2 \
                                 - P.dot(Q) ** 2)
O=Point3D(0,0,0)
A=Point3D(0,0,1)
B=Point3D(1,0,1)
C=Point3D(1,1,1)
D=Point3D(0,1,1)
E=Point3D(1,0,0)
F=Point3D(1,1,0)
P=Point3D(1,0,p)
Q=Point3D(0,1,q)
Hp=O+t*(C-O)
lp=P.distance(Hp).subs({t:solve((Hp-P).dot(C-O),t)[0]}).simplify()
print("#(1)",lp.subs({p:0}))
print("#(2)",lp)
ans=solve(myTaisekiGyouretuSiki(O,P,C,Q),q)[0]
print("#(3)",ans)
S =(myMensekiVector3D((O-P),(C-P))*2).subs({q:ans})
S2=(S**2).simplify()
ans=solve(diff(S2,p))[0]
print("#(4)最小値",ans,S.subs({p:ans}))
print("#(4)最大値",0  ,S.subs({p:0}))
print("#(4)最大値",1  ,S.subs({p:1}))
plot(S,(p,0,1))
#(1) sqrt(6)/3
#(2) sqrt(6*p**2 - 6*p + 6)/3
#(3) 1 - p
#(4)最小値 1/2 sqrt(6)/2
#(4)最大値 0 sqrt(2)
#(4)最大値 1 sqrt(2)
```
以下の、on line sympyで、上記のソースコードを貼り付けて実行できました。Plotもでました。
私の環境は,pycharmです。　

https://live.sympy.org/

グラフの縦横比は、調整してありません。pとSの関係。

![Figure_1.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/109698/09840c70-f3a9-02bb-a1e6-b4ff482f575d.png)


# FreeCADのマクロで
マクロからsympyを使っています。
(4)の最小値の条件で作図しました。
ワイヤーフレームです。Part Boxは、使っていません。



![png1.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/109698/4e20a47a-b953-28d0-8acd-7a4bc657a20a.png)

```python
import FreeCAD
import Part
import Draft
import Mesh
#########################################################################################################
from sympy import *
var('p q t')
def myTaisekiGyouretuSiki(PTO,PTA,PTB,PTC):
    return Matrix([[PTA.x-PTO.x, PTA.y-PTO.y, PTA.z-PTO.z], \
                   [PTB.x-PTO.x, PTB.y-PTO.y, PTB.z-PTO.z], \
                   [PTC.x-PTO.x, PTC.y-PTO.y, PTC.z-PTO.z]]).det()/6
def myMensekiVector3D(P,Q):
    return Rational(1, 2) * sqrt(P.distance(Point(0, 0, 0)) ** 2 * Q.distance(Point(0, 0, 0)) ** 2 \
                                 - P.dot(Q) ** 2)
O=Point3D(0,0,0)
A=Point3D(0,0,1)
B=Point3D(1,0,1)
C=Point3D(1,1,1)
D=Point3D(0,1,1)
E=Point3D(1,0,0)
F=Point3D(1,1,0)
G=Point3D(0,1,0)
P=Point3D(1,0,p)
Q=Point3D(0,1,q)
Hp=O+t*(C-O)
lp=P.distance(Hp).subs({t:solve((Hp-P).dot(C-O),t)[0]}).simplify()
print("#(1)",lp.subs({p:0}))
print("#(2)",lp)
ans=solve(myTaisekiGyouretuSiki(O,P,C,Q),q)[0]
print("#(3)",ans)
S =(myMensekiVector3D((O-P),(C-P))*2).subs({q:ans})
S2=(S**2).simplify()
ans=solve(diff(S2,p))[0]
print("#(4)最小値",ans,S.subs({p:ans}))
print("#(4)最大値",0  ,S.subs({p:0}))
print("#(4)最大値",1  ,S.subs({p:1}))
#########################################################################################################
# 怪しいです。
P=P.subs({p:ans})
Q=Q.subs({q:1-ans})
#########################################################################################################
# 3D作図
def myXYZ2Txt(A):
    return '(' + str(A.x) + ',' + str(A.y) + ',' + str(A.z) + ')'
def myTxtXYZ(A,myWedgei):
    P5x=float(A.x)
    P5y=float(A.y)
    P5z=float(A.z)
    p5 = FreeCAD.Vector(P5x, P5y, P5z)
    myText = Draft.makeText(myWedgei, p5)
    myText.Label = myWedgei
    FreeCADGui.ActiveDocument.ActiveObject.FontSize = '0.1 mm'
def myTxtXYZ_S(*xy_tx):
    for i in range(1,int(len(xy_tx)/2)+1):
        myTxtXYZ(xy_tx[2*i-2],xy_tx[2*i-1]+myXYZ2Txt(xy_tx[2*i-2]) )
def myLine(A,B):
    Ax,Ay,Az=float(A.x),float(A.y),float(A.z)
    Bx,By,Bz=float(B.x),float(B.y),float(B.z)
    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)
def myLine_S(*args):
    for i in range(1,len(args)):
        myLine(args[i-1],args[i])
myTxtXYZ_S(O,"O",A,"A",B,"B",C,"C",D,"D",E,"E",F,"F",G,"G",P,"P",Q,"Q")
myLine_S(O,E,F,G,O)
myLine_S(A,B,C,D,A)
myLine(A,O)
myLine(B,E)
myLine(C,F)
myLine(D,G)
myLine_S(O,P,C,Q,O)
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")
```