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:

x2 + y2 + z2 = R2.

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

x2 + y2 + v2 = R2


x2 + y2 = R2 - v2

Notice that setting r so that

r2 = R2 - v2

this equation becomes

x2 + y2 = r2

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

r = sqrt(R2 - v2)

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

Dome(u,v) = (sqrt(R2 - v2)cosu, sqrt(R2 - v2)sinu, v + 6.89)

where u is in the range [0, 2pi] and v is in the range [0, R]. The z-coordinate of this equation is v + 6.89 because the origin of the model is centrally placed at the base of the cone. Since the base of the hemispherical dome is 6.89 cm above the base of the cone, 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