抛物面的曲率的范围？

hbghlyj

圆锥$z^2=x^2+y^2$的曲率为$0$

椭球$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$($a≥b≥c>0$)曲率$$K(x,y,z)=\frac{1}{a^2 b^2 c^2\left(\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}\right)^2}$$
在顶点处取最值
$$
K_{\max}=K(\pm a,0,0)=\frac{a^2}{b^2c^2}
\qquad
K_\min=K(0,0,\pm c)=\frac{c^2}{a^2b^2}
$$

hbghlyj

抛物面$z=x^2+y^2$

\( f(x, y) = x^2 + y^2 \):
\[
K = \frac{f_{xx}f_{yy} - f_{xy}^2}{(1 + f_x^2 + f_y^2)^2} = \frac{4}{(1 + 4x^2 + 4y^2)^2}\in(0,4]\]

Aluminiumor

参数化曲面方程后，计算第一基本形式和第二基本形式，然后代公式就可以计算曲率了吧
当然，这是通法，计算起来可能比较麻烦

hbghlyj

双曲抛物面 $z = x^2 - y^2$

\( f(x, y) = x^2 - y^2 \):
\[
K = \frac{f_{xx}f_{yy} - f_{xy}^2}{(1 + f_x^2 + f_y^2)^2} = \frac{-4}{(1 + 4x^2 + 4y^2)^2}\in[-4, 0)
\]
椭圆抛物面 $z = x^2 + \frac{y^2}{b^2}$

\( f(x, y) = x^2 + \frac{y^2}{b^2} \):
\[
K = \frac{f_{xx}f_{yy} - f_{xy}^2}{(1 + f_x^2 + f_y^2)^2} = \frac{4}{b^2(1 + 4x^2 + \frac{4y^2}{b^4})^2} \in(0,\frac4{b^2}]
\]
单叶双曲面 $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$

\[K=-\frac{c^6}{(c^4+a^2z^2+c^2z^2)^2}\in[-\frac1{c^2},0)\]
双叶双曲面 $\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$

\[
K =\frac{c^6}{[c^4-(a^2+c^2)z^2]^2}\in(0,\frac{c^2}{a^4}]
\]