Parametric surface:

If a = b = r, then above equation decrease for a sphere.

where 0 < θ < 2π, 0 < φ < 2π. If a = b = r, Eq. (12) yields a torus along with a circular cross

section.

If a ≠ b, then a torus along with an elliptical cross section results. Figure 14 illustrated both a circular and an elliptical cross section torus.

A paraboloid of revolution is attained through rotating the parametric parabola

 x = at²              0 < t < T_max

y = 2at

about the x-axis. The parametric surface is specified by

Q (t, φ) = [at² 2at cos φ 2at sin φ]      0 ≤ t ≤ T_max

                                                            0 ≤ φ ≤2π

A hyperboloid of revolution is achieved by rotating the parametric hyperbola

x = a sec t        0 ≤ t ≤ t_max

y = b tan t

around the x-axis. The parametric surface is specified by

Q (t, φ) = [a sec t   b tan t cos φ   b tan t sin φ]         0 ≤ t ≤ tmax

                                                                                          0 ≤θ ≤ 2π