高等数学公式之平面的方程和二次曲面方程

高等数学公式 — 平面的方程和二次曲面

一、平面的方程

$A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$，法向量 $\vec{n} = (A, B, C)$

$Ax + By + Cz + D = 0$文章源自公式库网-https://www.gongshiku.com/html/201808/gaodengshuxuegongshizhipingmiandefangchengheerciqumianfangcheng.html

$\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1$（$a,b,c$ 分别为 $x,y,z$ 轴上的截距）文章源自公式库网-https://www.gongshiku.com/html/201808/gaodengshuxuegongshizhipingmiandefangchengheerciqumianfangcheng.html

$d = \dfrac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$文章源自公式库网-https://www.gongshiku.com/html/201808/gaodengshuxuegongshizhipingmiandefangchengheerciqumianfangcheng.html

$\cos\theta = \dfrac{|A_1A_2 + B_1B_2 + C_1C_2|}{\sqrt{A_1^2+B_1^2+C_1^2}\cdot\sqrt{A_2^2+B_2^2+C_2^2}}$文章源自公式库网-https://www.gongshiku.com/html/201808/gaodengshuxuegongshizhipingmiandefangchengheerciqumianfangcheng.html

二、二次曲面

球面：$(x-a)^2 + (y-b)^2 + (z-c)^2 = R^2$文章源自公式库网-https://www.gongshiku.com/html/201808/gaodengshuxuegongshizhipingmiandefangchengheerciqumianfangcheng.html

椭球面：$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1$文章源自公式库网-https://www.gongshiku.com/html/201808/gaodengshuxuegongshizhipingmiandefangchengheerciqumianfangcheng.html

单叶双曲面：$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} - \dfrac{z^2}{c^2} = 1$文章源自公式库网-https://www.gongshiku.com/html/201808/gaodengshuxuegongshizhipingmiandefangchengheerciqumianfangcheng.html

双叶双曲面：$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} - \dfrac{z^2}{c^2} = -1$文章源自公式库网-https://www.gongshiku.com/html/201808/gaodengshuxuegongshizhipingmiandefangchengheerciqumianfangcheng.html

椭圆抛物面：$z = \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}$文章源自公式库网-https://www.gongshiku.com/html/201808/gaodengshuxuegongshizhipingmiandefangchengheerciqumianfangcheng.html

双曲抛物面（马鞍面）：$z = \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2}$文章源自公式库网-https://www.gongshiku.com/html/201808/gaodengshuxuegongshizhipingmiandefangchengheerciqumianfangcheng.html

锥面：$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} - \dfrac{z^2}{c^2} = 0$