Nonlinear Projections - Transformations and Projections in Computer Graphics - page 198

May I repeat what I told you here: treat nature by means of the cylinder, the sphere,
the cone, everything brought into proper perspective so that each side of an object or
a plane is directed towards a central point.
—Paul Cezanne to Emile Bernard, 15 April 1904
Developable surfaces . A developable surface is one that can be opened or un-
rolled to become flat without introducing any distortions or deforming it. A plane is
developable, as are the cone and the cylinder. As a result, most methods for projecting
a globe start by projecting it on a cone or a cylinder (while introducing distortions) and
then unfolding this projection to become flat.
A developable surface is constructed by rolling or twisting a flat sheet of material
without stretching or shrinking it. A ruled (or lofted) surface is linear in one direction.
The parametric expression of such a surface is of the form P ( u, w ) and it is linear either
in u or in w . Such surfaces are simple but are not always developable.
Exercise 4.16: Are there any other developable surfaces in addition to the cylinder,
cone, and plane?
(a)
(b)
Figure 4.47: Principles of Projection.
Figure 4.47 illustrates the principle of employing developable surfaces for sphere
projection. The cylinder, cone, or plane can either be tangent to the sphere [part (a) of
the figure] or secant to it [part (b)]. In the latter case, the cylinder and cone intersect
the sphere in two circles and the plane intersects it in a single circle. The areas of
contact between the sphere and the developable surface are called the standard parallel