用拉格朗日乘數法求條件極值

目標函式：L^2=(x-x0)^2+(y-y0)^2+(z-z0)^2

約束條件：Ax+By+Cz+D=0

F=(x-x0)^2+(y-y0)^2+(z-z0)^2+λ(Ax+By+Cz+D)

2(x-x0)+λA=0 ①

2(y-y0)+λB=0 ②

Ax+By+Cz+D=0 ④

①②移項相除：(x-x0)/(y-y0)=A/B x=A(y-y0)/B+x0

代入④：A^2(y-y0)/B+Ax0+By+C^2(y-y0)/B+Cz0+D=0

(A^2+C^2)(y-y0)/B+By-By0+Ax0+By0+Cz0+D=0

(A^2+B^2+C^2)(y-y0)/B+Ax0+By0+Cz0+D=0

(y-y0)=-(Ax0+By0+Cz0+D)*B/(A^2+B^2+C^2)

x=x0-(Ax0+By0+Cz0+D)*A/(A^2+B^2+C^2)

y=y0-(Ax0+By0+Cz0+D)*B/(A^2+B^2+C^2)

z=z0-(Ax0+By0+Cz0+D)*C/(A^2+B^2+C^2)

L^2=(Ax0+By0+Cz0+D)^2/(A^2+B^2+C^2)^2(A^2+B^2+C^2)

∴L=|Ax0+By0+Cz0+D|/√(A^2+B^2+C^2)