latitude on the sphere given by λ=π 2 −θand when φis held constant, one obtains
a coordinate curve representing some line of longitude on the sphere.
The representation r =r (φ, θ) = rsin θcos φˆe 1 +rsin θsin φˆe 2 +rcos θˆe 3 is a vector
representation for points on the sphere of radius rwith r (φ 0 , θ)a curve of longitude
and r (φ, θ 0 ) a line of latitude and these curves are called coordinate curves on the
surface of the sphere. The vectors ∂r
∂φ
and ∂r
∂θ
are tangent vectors to the coordinate
curves. The cross product ∂r
∂φ
×∂r
∂θ
produces a normal vector to the surface of the
sphere.
The Ellipsoid
The ellipsoid centered at the point (x 0 , y 0 , z 0 )is represented by the equation
(x−x 0 )^2
a^2
+(y−y^0 )
2
b^2
+(z−z^0 )
2
c^2
= 1 (7 .29)
and if
a=b > c it is called an oblate spheroid.
a=b < c it is called a prolate spheroid.
a=b=c it is called a sphere of radius a.
Figure 7-5. Oblate and prolate spheroids.
The ellipsoid can also be represented by the parametric equations
x−x 0 =acos θcos φ, y −y 0 =bcos θsin φ, z −z 0 =csin θ (7 .30)
where −π 2 ≤θ≤π 2 and −π≤φ≤π. The figure 7-5 illustrates the oblate and prolate
spheroids centered at the origin.
