Biparametric Surface of Revolution:
A point on this surface is indicated as:
Q (u, φ) = [x(u) y(u) cos φ y(u) sin φ]
Rotating plane curves also produce surfaces of revolution. A sphere is produced by rotating semicircle in the xy plane around the x-axis or y-axis whose center is at origin where the parametric equation of the circle is as p (r, θ).
Px = x = r cos θ 0 < θ < π
Py = y = r sin θ
and the parametric equation of the sphere is as Q (θ, φ)
Q (θ, φ) = [x(θ) y(θ) cos θ y(θ) sin φ]
= [r cos θ r sin θ cos φ r sin θ sin φ] 0 ≤ θ ≤π
0 ≤ φ ≤ 2π
An ellipsoid is attained through rotating an origin centered semi-ellipse in the xy-plane either
x or y-axis. The parametric equation of the semi-ellipse is following
x = a cos θ 0 ≤ θ ≤ π
y = b sin θ
The parametric equation for a point on the ellipsoid of revolution Q (θ, φ) is
Q (θ, φ) = [a cos θ b sin θ cos φ b sin θ sin φ] 0≤θ ≤ π
0 ≤φ ≤ 2π