The position vector r =r (u, v ) = au cos vˆe 1 +bu sin vˆe 2 +cu ˆe 3 describes a point on
the surface centered at the origin and the curves r (u 0 , v ), r (u, v 0 )define the coordinate
curves. The partial derivatives of r with respect to uand v are tangent vectors to
the coordinate curves and these vectors can be used to construct a normal vector to
the surface.
Figure 7-7. Elliptic cone
The Hyperboloid of One Sheet
The hyperboloid of one sheet centered at the point (x 0 , y 0 , z 0 ) and symmetric
about the z−axis is given by the equation
(x−x 0 )^2
a^2 +
(y−y 0 )^2
b^2 −
(z−z 0 )^2
c^2 = 1 (7 .34)
It can also be represented using the parametric equations
x−x 0 =acos ucosh v, y −y 0 =bsin ucosh v, z −z 0 =csinh v (7 .35)
where 0 ≤u≤ 2 π and −h < v < h. Here his usually selected as a small number, say
h= 1 as the selection of has a large number gives a scaling difference between the
parameters and distorts the final image.
The Hyperboloid of Two Sheets
The hyperboloid of two sheets centered at the point (x 0 , y 0 , z 0 ) and symmetric
about the z−axis is describe by the equation
−
(x−x 0 )^2
a^2 −
(y−y 0 )^2
b^2 +
(z−z 0 )^2
c^2 = 1 (7 .36)
