Figure 7-6. Elliptic paraboloid
The Elliptic Paraboloid
The elliptic paraboloid centered at the point (x 0 , y 0 , z 0 )is described by the equa-
tion
(x−x 0 )^2
a^2
+(y−y^0 )
2
b^2
=z−z^0
c
(7 .31)
It can also be represented by the parametric equations
x−x 0 =a
√
ucos v, y −y 0 =b
√
usin v, z −z 0 =cu (7 .32)
where 0 ≤v ≤ 2 π and 0 ≤u≤h. The elliptic paraboloid centered at the origin is
illustrated in the figure 7-6.
The Elliptic Cone
The elliptic cone centered at the point (x 0 , y 0 , z 0 )is represented by an equation
having the form
(x−x 0 )^2
a^2 +
(y−y 0 )^2
b^2 =
(z−z 0 )^2
c^2 (7 .33)
A parametric representation for the elliptic cone is given by
x−x 0 =au cos v, y −y 0 =bu sin v, z −z 0 =cu
for 0 ≤v≤ 2 πand −h≤u≤h. The elliptic cone centered at the origin is illustrated
in the figure 7-7.
