The Hyperbolic Paraboloid
The hyperbolic paraboloid centered at the the point (x 0 , y 0 , z 0 ) is described by
the equation
z−z 0
c
=−(x−x^0 )
2
a^2
+(y−y^0 )
2
b^2
, (7 .38)
This surface is saddle shaped and can also be described using the parametric
equations
x−x 0 =u, y −y 0 =v, z −z 0 =c
(
−u
2
a^2
+v
2
b^2
)
(7 .39)
where −h≤u≤hand −k≤v≤kfor selected constants hand k. These parametric
equations can be used to construct the two-parameter surface r (u, v )from which the
coordinate curves and normal vector can be constructed.
It is left as an exercise to show that under a rotation of axes and scaling using
the equations
x−x 0
a = ̄xcos θ− ̄ysin θ,
y−y 0
b = ̄xsin θ+ ̄ycos θ,
z−z 0
c = ̄z
with θ=π/ 4 , the hyperbolic paraboloid can be represented ̄z= ̄xy ̄.
Figure 7-9. Hyperbolic paraboloid.
Surfaces of Revolution
Any surface which can be created by rotating a curve about a fixed line is called
a surface of revolution. The fixed line about which the curve is rotated is called the
axis of revolution. Some examples of surfaces of revolution are the sphere which is
created by rotating the semi-circle x^2 +y^2 =r^2 , −r≤x≤rand y≥ 0 about the y= 0
