5.1 PARAMETRIC REPRESENTATION OF SURFACES

Parametric representation of a surface means that the position vector of a current surface point is associated with two variable parameters u and ? and is represented by the following vector equation:

Here, i, j, and k are the unit vectors of the coordinate axes; functions

determine the Cartesian coordinates of the surface points if u and ? are given.

We may imagine a rectangle G in the plane of parameters ( u, ?) (Fig. 5.1.1). Vector equation (5.1.1) sets the correspondence between the given point of the rectangle G and the single point r( u, ?) of the surface. Generally, one-to-one correspondence is not guaranteed; it may happen that the given surface point r( u, ?) corresponds to more than one point of the rectangle G. A surface with one-to-one correspondence between the set of parameters ( u, ?) and the position vector r( u, ?) is called a simple surface. Such a surface does not have points of self-intersection.

Figure 5.1.1: Mapping of a surface.

5.2 CURVILINEAR COORDINATES

Parameters ( u, ?) are called curvilinear coordinates (Gaussian coordinates) on the surface. Consider that one of the curvilinear coordinates is fixed, for instance ? = ? ₀, and the other one ( u) is varied. Then, equation

represents a line on the surface that is called the u line. Similarly,...