5.7: Implicit Representation of a Curve; Contact of Curves

5.7 Implicit Representation of a Curve; Contact of Curves

Our work in Chapter 5 has centered around technical topics needed to derive equations describing certain natural objects. We now turn to some less technical but quite important questions that can further our understanding of differential geometry.

We have described a curve by the representation r= r (t). From this point of view, a curve is the image of some one-dimensional domain of the parameter t. Similarly, a surface was an image of a two-dimensional domain, given by a formula

(5.38)

However, we know that in Cartesian coordinates a sphere is described by the equation

and this cannot be represented in the form (5.38) for all points simultaneously. This is an example of the implicit form of description of a surface. We can describe any small part of a sphere in an explicit form of the type (5.38). This is a point from which the theory of manifolds arose: a surface such as a sphere can be divided into small portions, each of which can be described in the needed way.

However, implicit form descriptions of surfaces and curves are quite common in practice. Let us suppose that the equation of a surface is

In Cartesian coordinates this would look like

Earlier we described coordinate curves in space as sets given by the equation r= r( ? ¹, ? ², ? ³) when two of the three coordinate parameters ? ¹, ?