Michael Corral: Vector Calculus

1.6 Surfaces 43
The equations of spheres and cylinders are examples of second-degree equations in
3 , i.e. equations of the form
Ax 2 + By 2 +Cz 2 +Dxy+Exz+Fyz+Gx+Hy+Iz+ J= 0 (1.33)
for some constants A, B,..., J. If the above equation is not that of a sphere, cylinder,
plane, line or point, then the resulting surface is called a quadric surface.
One type of quadric surface is the ellipsoid,
given by an equation of the form:
c
z
x 2
a 2+y2 b 2+z2 c2= 1 (1.34)
Inthecasewherea=b=c,thisisjustasphere.
In general, an ellipsoid is egg-shaped (think of
an ellipse rotated around its major axis). Its
traces in the coordinate planes are ellipses.
0
a
x
Figure 1.6.4 Ellipsoid
b
y
Two other types of quadric surfaces are the hyperboloid of one sheet, given by
an equation of the form:
x 2
a 2+y2 b 2−z2 c2= 1 (1.35)
and the hyperboloid of two sheets, whose equation has the form:
x 2
a 2−y2 b 2−z2 c2= 1 (1.36)
z
z
0
y
0
y
x
Figure 1.6.5 Hyperboloid of one sheet
x
Figure 1.6.6 Hyperboloid of two sheets