已知平面内三点，求其平面的法向量

设三点坐标为A(x1,y1,z1),B(x2,y2,z2),C(x3,y3,z3)
向量AB=(x2-x1,y2-y1,z2-z1),AC=(x3-x1,y3-y1,z3-z1)
AB、AC所在平面的法向量即AB×AC=(a,b,c),其中：
a=(y2-y1)(z3-z1)-(z2-z1)(y3-y1)
b=(z2-z1)(x3-x1)-(z3-z1)(x2-x1)
c=(x2-x1)(y3-y1)-(x3-x1)(y2-y1)

设 a=(ax,ay,az)， b=(bx,by,bz)。i，j，k分别是X，Y，Z轴方向的单位向量，则：
a×b=(aybz-azby)i+(azbx-axbz)j+(axby-aybx)k
a·b=(axbx+ayby+az*bz)

Python代码

def normal_vector(p1, p2, p3):x1, y1, z1 = p1x2, y2, z2 = p2x3, y3, z3 = p3a = (y2 - y1) * (z3 - z1) - (z2 - z1) * (y3 - y1)b = (z2 - z1) * (x3 - x1) - (x2 - x1) * (z3 - z1)c = (x2 - x1) * (y3 - y1) - (y2 - y1) * (x3 - x1)return (a, b, c)if __name__ == '__main__':p1 = 1.0, 5.2, 0.0p2 = 2.8, 3.9, 1.0p3 = 7.6, 8.4, 2.0p4 = normal_vector(p1, p2, p3)print(p4)# output (-5.800000000000001, 3.0, 14.340000000000002)

C++代码

#include<iostream>using namespace std;//三维double矢量
struct Vec3d
{double x, y, z;Vec3d(){x = 0.0;y = 0.0;z = 0.0;}Vec3d(double dx, double dy, double dz){x = dx;y = dy;z = dz;}void Set(double dx, double dy, double dz){x = dx;y = dy;z = dz;}
};//计算三点成面的法向量
void Cal_Normal_3D(const Vec3d& v1, const Vec3d& v2, const Vec3d& v3, Vec3d &vn)
{//v1(n1,n2,n3);//平面方程: na * (x – n1) + nb * (y – n2) + nc * (z – n3) = 0 ;double na = (v2.y - v1.y)*(v3.z - v1.z) - (v2.z - v1.z)*(v3.y - v1.y);double nb = (v2.z - v1.z)*(v3.x - v1.x) - (v2.x - v1.x)*(v3.z - v1.z);double nc = (v2.x - v1.x)*(v3.y - v1.y) - (v2.y - v1.y)*(v3.x - v1.x);//平面法向量vn.Set(na, nb, nc);
}int main()
{	// Vec3d v1(1.0, 5.2, 3.7);// Vec3d v2(2.8, 3.9, 4.5);// Vec3d v3(7.6, 8.4, 6.2);   // 法向量为：-5.81 0.78 14.34;Vec3d v1(1.0, 5.2, 0.0);Vec3d v2(2.8, 3.9, 1.0);Vec3d v3(7.6, 8.4, 2.0);// 法向量为：-5.8 3 14.34Vec3d vn;Cal_Normal_3D(v1, v2, v3, vn);// cout <<"法向量为："<< vn.x << '\t' << vn.y << '\t' << vn.z << '\n';cout <<"法向量为："<< vn.x << " " << vn.y << " " << vn.z << '\n';return 0;
}