# 抛物面

${\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.}$

${\displaystyle z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}.}$

## 性质

a = b时，曲面称为旋转抛物面，它可以由抛物线绕着它的轴旋转而成。它是抛物面反射器的形状，把光源放在焦点上，经镜面反射后，会形成一束平行的光线。反过来也成立，一束平行的光线照向镜面后，会聚集在焦点上。

## 曲率

${\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{u^{2} \over a^{2}}+{v^{2} \over b^{2}}\right)}$

${\displaystyle K(u,v)={4 \over a^{2}b^{2}\left(1+{4u^{2} \over a^{4}}+{4v^{2} \over b^{4}}\right)^{2}}}$

${\displaystyle H(u,v)={a^{2}+b^{2}+{4u^{2} \over a^{2}}+{4v^{2} \over b^{2}} \over a^{2}b^{2}\left(1+{4u^{2} \over a^{4}}+{4v^{2} \over b^{4}}\right)^{\frac {3}{2}}}}$

${\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{u^{2} \over a^{2}}-{v^{2} \over b^{2}}\right)}$

${\displaystyle K(u,v)={-4 \over a^{2}b^{2}\left(1+{4u^{2} \over a^{4}}+{4v^{2} \over b^{4}}\right)^{2}}}$

${\displaystyle H(u,v)={-a^{2}+b^{2}-{4u^{2} \over a^{2}}+{4v^{2} \over b^{2}} \over a^{2}b^{2}\left(1+{4u^{2} \over a^{4}}+{4v^{2} \over b^{4}}\right)^{\frac {3}{2}}}.}$

## 乘法表

${\displaystyle z={x^{2} \over a^{2}}-{y^{2} \over b^{2}}}$

${\displaystyle z={1 \over 2}\left(x^{2}+y^{2}\right)\left({1 \over a^{2}}-{1 \over b^{2}}\right)+xy\left({1 \over a^{2}}+{1 \over b^{2}}\right)}$

${\displaystyle z={2 \over a^{2}}xy}$ .

${\displaystyle z={x^{2}-y^{2} \over 2}}$ .

${\displaystyle \ z=xy}$

${\displaystyle z_{1}(x,y)={x^{2}-y^{2} \over 2}}$

${\displaystyle \ z_{2}(x,y)=xy}$

${\displaystyle f(z)={1 \over 2}z^{2}=f(x+iy)=z_{1}(x,y)+iz_{2}(x,y)}$

## 参考文献

• Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 133, 1987.
• Gray, A. "The Paraboloid." §13.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 307-308, 1997.
• Harris, J. W. and Stocker, H. "Paraboloid of Revolution." §4.10.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 112, 1998.
• Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, pp. 10-11, 1999.
• Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.