Some Classes of Surfaces

Abstract

In Chapter 20 we study three very important and popular classes of surfaces (algebraic, revolutionary and ruled) and the envelopes of surfaces. In Section 20.1 we use the resultant and the discriminant to deduce implicit polynomial equations of surfaces. In Section 20.2 we study surfaces of revolution, and give an example of a map on a torus that cannot be colored with six colors. In Section 20.3 we plot ruled surfaces of various types, and calculate their striction curves and distribution parameter. In Section 20.4 we use the notion of a tangent plane (see Section 19.4.1) to continue our studies from Sections 9.2–9.3. We plot some envelopes of surfaces and show that the envelope of a family of planes (as a particular case) is a ruled developable surface.