# 1.平面方程为一般式

$$Ax+By+Cz+D = 0$$

$$\cases{x=x_i-At \cr y=y_i-Bt \cr z=z_i-Ct}$$

$$t=\dfrac {Ax_i+By_i+Cz_i+D}{A^2+B^2+C^2}$$

# 2.平面由法向量和平面上一点构成

1)$$V_i{V_i}^\prime \parallel \overrightarrow n$$, 2)$$O{V_i}^\prime\perp \overrightarrow n$$

$$\dfrac{ {x}^\prime -x }a = \dfrac{ {y}^\prime -y }b = \dfrac{ {z}^\prime -z }c=t$$ $$\Rightarrow$$ $$\cases{{x}^\prime=x+at \cr {y}^\prime=y+bt \cr {z}^\prime=z+ct}$$ (1)

$$a({x}^\prime-x_0)+b(y^\prime-y_0)+c(z^\prime-z_0)=0$$ $$\Rightarrow$$ $$ax^\prime+by^\prime+cz^\prime=ax_0+by_0+cz_0$$ (2)

$$t=\dfrac{ ax_0+by_0+cz_0-(ax+by+cz) }{a^2+b^2+c^2}$$ (3)

# 3.平面由三个不共线的点构成

$$\overrightarrow{n}=\overrightarrow{OP_1}\times\overrightarrow{OP_2}=\begin{vmatrix}i & j & k\cr {x_1-x_0 } & {y_1-y_0} & {z_1-z_0}\cr {x_2-x_0 } & y_2-y_0 & z_2-z_0 \end{vmatrix}$$

$$\cases{a=(y_1-y_0)(z_2-z_0)-(y_2-y_0)(z_1-z_0) \cr b=(x_2-x_0)(z_1-z_0)-(x_1-x_0)(z_2-z_0) \cr c=(x_1-x_0)(y_2-y_0)-(x_2-x_0)(y_1-y_0)}$$

posted @ 2016-12-08 15:11  NobodyZhou  阅读(55713)  评论(0编辑  收藏  举报