版块 14 空间解析几何

A. 实用解题工具

直线方程有如下类型

{A_{1} x + B_{1} y + C_{1} x + D_{1} = 0 A_{2} x + B_{2} y + C_{2} x + D_{2} = 0 平面 π_{1} 平面 π_{2}

标准式方程: $\frac{x - x _{0}}{l} = \frac{y - y _{0}}{m} = \frac{z - z _{0}}{n}$
两点式方程: $\frac{x - x _{1}}{x _{2} - x _{1}} = \frac{y - y _{1}}{y _{2} - y _{1}} = \frac{z - z _{1}}{z _{2} - z _{1}}$
参数式方程:

⎩ ⎨ ⎧ x = x_{0} + l t y = y_{0} + m t z = z_{0} + n t

⎩ ⎨ ⎧ 定义 : {A_{1} x + B_{1} y + C_{1} x + D_{1} = 0 A_{2} x + B_{2} y + C_{2} x + D_{2} = 0 平面 π_{1} 平面 π_{2} 两平面平行 : \Leftrightarrow \frac{A _{1}}{A _{2}} = \frac{B _{1}}{B _{2}} = \frac{C _{1}}{C _{2}} 两平面垂直 : \Leftrightarrow A_{1} A_{2} + B_{1} B_{2} + C_{1} C_{2} = 0 平面间夹角 θ : cos θ = \frac{A _{1} A _{2} + B _{1} B _{2} + C _{1} C _{2}}{A _{1}^{2} + B _{1}^{2} + C _{1}^{2} A _{2}^{2} + B _{2}^{2} + C _{2}^{2}}

⎩ ⎨ ⎧ 定义 : {直线 L : \frac{x - x _{0}}{l} = \frac{y - y _{0}}{m} = \frac{z - z _{0}}{n} 平面 π_{1} : A_{1} x + B_{1} y + C_{1} z + D_{1} = 0 平行 :\Leftrightarrow A l + B m + C n = 0 垂直 :\Leftrightarrow \frac{A}{l} = \frac{B}{m} = \frac{C}{n} 线面夹角 θ : sin θ = \frac{A l + B m + C n}{A ^{2} + B ^{2} + C ^{2} l ^{2} + m ^{2} + n ^{2}}

⎩ ⎨ ⎧ 定义 : {L_{1} : \frac{x - x _{1}}{l _{1}} = \frac{y - y _{1}}{m _{1}} = \frac{z - z _{1}}{n _{1}} L_{2} : \frac{x - x _{2}}{l _{2}} = \frac{y - y _{2}}{m _{2}} = \frac{z - z _{2}}{n _{2}} 平行 :\Leftrightarrow \frac{l _{1}}{l _{2}} = \frac{m _{1}}{m _{2}} = \frac{n _{1}}{n _{2}} 垂直 :\Leftrightarrow l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0 直线夹角 θ : cos θ = \frac{∣ l _{1} l _{2} + m _{1} m _{2} + n _{1} n _{2} ∣}{l _{1}^{2} + m _{1}^{2} + n _{1}^{2} l _{2}^{2} + m _{2}^{2} + n _{2}^{2}}

{球 : x^{2} + y^{2} + z^{2} = a^{2} 椭球 : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} + \frac{z ^{2}}{c ^{2}} = 1

y y^{2} {z = x^{2} + y^{2} z = - x^{2} + y^{2}

⎩ ⎨ ⎧ 旋转抛物面 : z = x^{2} + y^{2} 椭圆抛物面 : z = \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} 双曲抛物面 : z = \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}}

{单 . 叶 : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} - \frac{z ^{2}}{c ^{2}} = 1 双叶 : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} - \frac{z ^{2}}{c ^{2}} = 1

方法: 空间曲线 $Γ$ 的方程写成两个曲面联立的形式

{F (x, y, z) = 0 G (x, y, z) = 0

。若要求在坐标面 $x O y$ 上的投影,先在方程

{F (x, y, z) = 0 G (x, y, z) = 0

中消去变量 $z$ ,得到 $f (x, y) = 0$ 。再写成

{f (x, y) = 0 z = 0

的形式。其他坐标面投影类似。

例题 1. 空间直线 $x = y = z$ 绕 $z$ 轴旋转得到的曲面方程为___

【解】设空间曲线上任意点 $(x_{0}, y_{0}, z_{0})$ 满足 $x_{0} = y_{0} = z_{0}$ ,绕 $z$ 轴旋转,所得出的点的坐标始终满足 $z = z_{0}$ ,

且 $x^{2} + y^{2} = x_{0}^{2} + y_{0}^{2}$ 。整理两式,消除 $x_{0}, y_{0}, z_{0}$ 可得到 $x^{2} + y^{2} = x_{0}^{2} + y_{0}^{2} = z_{0}^{2} + z_{0}^{2} = 2 z$

即, $x^{2} + y^{2} = 2 z$ 为所求。

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