A hemispherical dome is a surface of revolution.
Since slices parallel to its base form circles of various radii, one needs
to determine how the radius of these circles varies with the height of the
slice. In a hemispherical dome, the radius of the sphere is constant.
Call this radius *R*. The implicit equation of a sphere can be used
to derive the parametric equation of a hemisphere. The implicit equation
of a sphere is:

Taking those points on the sphere where *z* equals *v*, the
equation becomes

or

Notice that setting *r* so that

this equation becomes

which is the equation of a circle of radius *r* in the plane with
*z*-coordinate *v*. By solving for *r*, the equation now
becomes

This gives the radius as a function of height, so the parametric equation for the hemisphere is

where *u* is in the range [0, 2pi] and *v* is in the range [0,
*R*]. The *z*-coordinate*v* + 6.89*z* needs to begin at a
measured height of 6.89 cm.

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