A surface of revolution is a surface generated by revolving a plane curve C about a line L lying in the same plane as the curve. The line L is called the axis of revolution [reference 2]. Some examples include hemispheres, cones and cylinders.

For the construction of this surface let L be the z-axis. In order to construct a surface of revolution using a parametric equation, it is important to first understand how a circle is constructed in the plane since the surface is made up of a series of circles at various heights. That is, if one slices the surface with a plane that is parallel to the xy-plane, the intersection is a circle. The parametric equation for a circle of radius 1 in the xy-plane is (x(u), y(u)) = (cosu, sinu) where 0 < u < 2pi. Hence, if one wants to construct a circle of radius r, the equation is Circle(u) = (rcosu, rsinu).

In a surface of revolution, the radius may be different at each height, so if the radius at height v is r(v), then the equation of the surface is

S(u,v) = (r(v) cosu, r(v) sinu, v)

where 0 < u< 2pi.

Optical Illusion & Projection in Domes: A Study of Guarino Guarini's Santissima Sindone