抛物面

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

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

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

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

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

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

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

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

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

乘法表

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

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

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

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

$\ z = x y$

$z_1 (x,y) = {x^2 - y^2 \over 2}$

$\ z_2 (x,y) = x y$

$f(z) = {1\over 2} z^2 = f(x + i y) = z_1 (x,y) + i z_2 (x,y)$

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.