python - 如何四面体每个面的体积偏移
问题描述
四面体平面的体积如何偏移？
计算三角锥每个面的平行面，计算每个面在四个点的交点。体积是根据交点计算的坐标计算的。
①我该怎么办？
from sympy import *
def myIntersection_of_3_planes(a,b,c):
ab = a.intersection(b)
abc = ab[0].intersection(c)
return abc[0]
def myUnitVector(C):
D=Matrix([[0], [0], [0]])
myL=sqrt(C[0]**2+C[1]**2+C[2]**2)
D[0]=C[0]/myL
D[1]=C[1]/myL
D[2]=C[2]/myL
return D
def my3Point_Offset(myT0,myT1,myT2,my03):
mmy03=myCrossProduct(Matrix(myT1)-Matrix(myT0),Matrix(myT2)-Matrix(myT0))
mmy03=myUnitVector(mmy03)
mmy03[0]=float(mmy03[0])*my03
mmy03[1]=float(mmy03[1])*my03
mmy03[2]=float(mmy03[2])*my03
myT0S=Matrix([[0], [0], [0]])
myT1S=Matrix([[0], [0], [0]])
myT2S=Matrix([[0], [0], [0]])
myT0S[0]=myT0[0]-mmy03[0]
myT0S[1]=myT0[1]-mmy03[1]
myT0S[2]=myT0[2]-mmy03[2]
myT1S[0]=myT1[0]-mmy03[0]
myT1S[1]=myT1[1]-mmy03[1]
myT1S[2]=myT1[2]-mmy03[2]
myT2S[0]=myT2[0]-mmy03[0]
myT2S[1]=myT2[1]-mmy03[1]
myT2S[2]=myT2[2]-mmy03[2]
return myT0S,myT1S,myT2S
def my4Point_Intersection(myT0,myT1,myT2,myT3,my01,my02,my03):
aa=my3Point_Offset(myT0, myT1, myT2, my03[0])
bb=my3Point_Offset(myT0, myT2, myT3, my01[0])
cc=my3Point_Offset(myT0, myT3, myT1, my02[0])
p11, p12, p13 = map(Point3D, [aa[0],aa[1],aa[2]])
p21, p22, p23 = map(Point3D, [bb[0],bb[1],bb[2]])
p31, p32, p33 = map(Point3D, [cc[0],cc[1],cc[2]])
a = Plane(p11, p12, p13)
b = Plane(p21, p22, p23)
c = Plane(p31, p32, p33)
ans=myIntersection_of_3_planes(a, b, c)
return ans
def myCrossProduct(A, B):
return expand(Matrix([
[A[1,0]*B[2,0]-A[2,0]*B[1,0]],
[A[2,0]*B[0,0]-A[0,0]*B[2,0]],
[A[0,0]*B[1,0]-A[1,0]*B[0,0]]
]))
def myIntersection_of_3_planes(a,b,c):
ab = a.intersection(b)
abc = ab[0].intersection(c)
return abc[0]
def myVolumeTetrahedron(myTetrahedron):
my_PQRS=Matrix(myTetrahedron)
my_S =Matrix([myTetrahedron[3],myTetrahedron[3],myTetrahedron[3]])
return float(det(my_PQRS[:3, :3].T - my_S.T)/6)
myTetrahedron=[[0,0,0],[10,0,0],[0,10,0],[0,0,10]]
myOffset =[[0] ,[0] ,[0] ,[0] ]
myT0=my4Point_Intersection(myTetrahedron[0],myTetrahedron[1],myTetrahedron[2],myTetrahedron[3], myOffset[1],myOffset[2],myOffset[3])
myT1=my4Point_Intersection(myTetrahedron[1],myTetrahedron[2],myTetrahedron[3],myTetrahedron[0], myOffset[2],myOffset[3],myOffset[0])
myT2=my4Point_Intersection(myTetrahedron[2],myTetrahedron[3],myTetrahedron[0],myTetrahedron[1], myOffset[3],myOffset[0],myOffset[1])
myT3=my4Point_Intersection(myTetrahedron[3],myTetrahedron[0],myTetrahedron[1],myTetrahedron[2], myOffset[0],myOffset[1],myOffset[2])
myTetrahedron=[myT0,myT1,myT2,myT3]
print("#",'{:.3f}'.format(myVolumeTetrahedron(myTetrahedron)))
#
myTetrahedron=[[0,0,0],[10,0,0],[0,10,0],[0,0,10]]
myOffset =[[1] ,[1] ,[1] ,[1] ]
myT0=my4Point_Intersection(myTetrahedron[0],myTetrahedron[1],myTetrahedron[2],myTetrahedron[3], myOffset[1],myOffset[2],myOffset[3])
myT1=my4Point_Intersection(myTetrahedron[1],myTetrahedron[2],myTetrahedron[3],myTetrahedron[0], myOffset[2],myOffset[3],myOffset[0])
myT2=my4Point_Intersection(myTetrahedron[2],myTetrahedron[3],myTetrahedron[0],myTetrahedron[1], myOffset[3],myOffset[0],myOffset[1])
myT3=my4Point_Intersection(myTetrahedron[3],myTetrahedron[0],myTetrahedron[1],myTetrahedron[2], myOffset[0],myOffset[1],myOffset[2])
myTetrahedron=[myT0,myT1,myT2,myT3]
print("#",'{:.3f}'.format(myVolumeTetrahedron(myTetrahedron)))
# (10*10)/2*10/3=166.667
# -166.667
# -119.877
②我在找一个有官方公式的网页
波纹管
我尝试使用 numpy。
from sympy import *
import numpy as np
def myIntersection_of_3_planes(a,b,c):
ab = a.intersection(b)
abc = ab[0].intersection(c)
return abc[0]
def myUnitVector(C):
return np.array(C)/sqrt(C[0]**2+C[1]**2+C[2]**2)
def my3Point_Offset(myT0,myT1,myT2,my03):
mmy03=my03*np.array(myUnitVector(myCrossProduct(Matrix(myT1)-Matrix(myT0),Matrix(myT2)-Matrix(myT0))))
return Matrix((np.array(myT0) - np.array(mmy03))[0]), \
Matrix((np.array(myT1) - np.array(mmy03))[0]), \
Matrix((np.array(myT2) - np.array(mmy03))[0])
def my4Point_Intersection(myT0,myT1,myT2,myT3,my01,my02,my03):
aa=my3Point_Offset(myT0, myT1, myT2, my03[0])
bb=my3Point_Offset(myT0, myT2, myT3, my01[0])
cc=my3Point_Offset(myT0, myT3, myT1, my02[0])
p11, p12, p13 = map(Point3D, [aa[0],aa[1],aa[2]])
p21, p22, p23 = map(Point3D, [bb[0],bb[1],bb[2]])
p31, p32, p33 = map(Point3D, [cc[0],cc[1],cc[2]])
a = Plane(p11, p12, p13)
b = Plane(p21, p22, p23)
c = Plane(p31, p32, p33)
ans=myIntersection_of_3_planes(a, b, c)
return ans
def myCrossProduct(A, B):
return expand(Matrix([
[A[1,0]*B[2,0]-A[2,0]*B[1,0]],
[A[2,0]*B[0,0]-A[0,0]*B[2,0]],
[A[0,0]*B[1,0]-A[1,0]*B[0,0]]
]))
def myIntersection_of_3_planes(a,b,c):
ab = a.intersection(b)
abc = ab[0].intersection(c)
return abc[0]
def myVolumeTetrahedron(myTetrahedron):
my_PQRS=Matrix(myTetrahedron)
my_S =Matrix([myTetrahedron[3],myTetrahedron[3],myTetrahedron[3]])
return float(det(my_PQRS[:3, :3].T - my_S.T)/6)
myTetrahedron=[[0,0,0],[10,0,0],[0,10,0],[0,0,10]]
myOffset =[[0] ,[0] ,[0] ,[0] ]
myT0=my4Point_Intersection(myTetrahedron[0],myTetrahedron[1],myTetrahedron[2],myTetrahedron[3], myOffset[1],myOffset[2],myOffset[3])
myT1=my4Point_Intersection(myTetrahedron[1],myTetrahedron[2],myTetrahedron[3],myTetrahedron[0], myOffset[2],myOffset[3],myOffset[0])
myT2=my4Point_Intersection(myTetrahedron[2],myTetrahedron[3],myTetrahedron[0],myTetrahedron[1], myOffset[3],myOffset[0],myOffset[1])
myT3=my4Point_Intersection(myTetrahedron[3],myTetrahedron[0],myTetrahedron[1],myTetrahedron[2], myOffset[0],myOffset[1],myOffset[2])
myTetrahedron=[myT0,myT1,myT2,myT3]
print("#",'{:.3f}'.format(myVolumeTetrahedron(myTetrahedron)))
#
myTetrahedron=[[0,0,0],[10,0,0],[0,10,0],[0,0,10]]
myOffset =[[1] ,[1] ,[1] ,[1] ]
myT0=my4Point_Intersection(myTetrahedron[0],myTetrahedron[1],myTetrahedron[2],myTetrahedron[3], myOffset[1],myOffset[2],myOffset[3])
myT1=my4Point_Intersection(myTetrahedron[1],myTetrahedron[2],myTetrahedron[3],myTetrahedron[0], myOffset[2],myOffset[3],myOffset[0])
myT2=my4Point_Intersection(myTetrahedron[2],myTetrahedron[3],myTetrahedron[0],myTetrahedron[1], myOffset[3],myOffset[0],myOffset[1])
myT3=my4Point_Intersection(myTetrahedron[3],myTetrahedron[0],myTetrahedron[1],myTetrahedron[2], myOffset[0],myOffset[1],myOffset[2])
myTetrahedron=[myT0,myT1,myT2,myT3]
print("#",'{:.3f}'.format(myVolumeTetrahedron(myTetrahedron)))
# (10*10)/2*10/3=166.667
# -166.667
# -165.517
解决方案
你会如何用微积分做这样的事情？在考虑 3 维空间之前，请先考虑 2d：抛物线下被一条线切割的区域。
y1=(x-1)^2
y2=x
您可以很容易地看到面积是线积分和抛物线积分之间的差。
integral(y1) = 1/3(x-1)^3
integral(y2) = 1/2(x^2) ...
integral(y1)|(bounds) – integral(y2)|(bounds) = 1/3(x-1)^3 |(bounds) – 1/2(x^2) |(bounds)
但是，我们会注意到此示例包括 x 轴下方的负区域。这可以用绝对值来解决。
同样的原理也适用于 n 维空间。在您的情况下，您的四面体是被 2d 形状切割的 3d 形状。使用绝对值将使您的形状保持在 x 轴上方（因此为正）。
推荐阅读
- npm - 为什么 npm run serve 在使用 vue-cli 的 WSL 上会出错？
- javascript - Fullcalendar 3.1.0 js/c# 找到事件后不显示更新对话框
- docker - 詹金斯。在远程 docker 上构建容器镜像
- php - Laravel 在某些图像中上传图像失败，但在其他图像中没有
- python - 使用 GroupBy 应用时保留 NaN 组
- firebase - FIrebase Auth - 使用不同的域登录
- ios - iOS中的Adyen hpp页面集成总是给商家签名无效错误
- javascript - 克隆类型并使用它的键
- bootloader - STM32F4：从自定义引导加载程序跳转到用户应用程序触发 HardFault 异常
- javascript - 如何使用java脚本动态更改对象的键并将多个类似对象的数组合并为单个数组？