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