3.1 Tangent plane and surface normal

Let us consider a curve , in the parametric domain of a parametric surface as shown in Fig. 3.1. Then is a parametric curve lying on the surface . The tangent vector to the curve on the surface is evaluated by differentiating with respect to the parameter using the chain rule and is given by

where subscripts and denote partial differentiation with respect to and , respectively.

Figure 3.1: The mapping of a curve in 2-D parametric space onto a 3-D parametric surface

The tangent plane at point can be considered as a union of the tangent vectors of the form (3.1) for all through as illustrated in Fig. 3.2. Point corresponds to parameters , . Since the tangent vector (3.1) consists of a linear combination of two surface tangents along iso-parametric curves and , the equation of the tangent plane at in parametric form with parameters , is given by

Figure 3.2: The tangent plane at a point on a surface

The surface normal vector is perpendicular to the tangent plane (see Fig. 3.3) and hence the unit normal vector is given by

By using (3.3), the equation of the tangent plane at can be written in the implicit form as

where is a point on the tangent plane.

Figure 3.3: The normal to the point on a surface

Definition 3.1.1. A regular (ordinary) point on a parametric surface is defined as a point where . A point which is not a regular point is called a singular point.

The condition requires that at point the vectors and do not vanish and have different directions, i.e. and are linearly independent. As we discussed in Sect. 1.3.6, in some design problems we need to employ triangular patches defined by parametrization over a rectangular domain. Such a degenerated patch can be generated by collapsing one boundary curve into a single point or by arranging for two partial derivatives and at one of the corners of a quadrilateral patch to be collinear. In both cases has zero magnitude at the degenerate corner point and (3.3) cannot be used. Conditions for the existence of surface normals at these degenerate corner points have been discussed in [116,92,453,457]. The concept of a regular surface requires additional conditions beyond the existence of a tangent plane everywhere on the surface, such as absence of self-intersections. This concept is presented fully in do Carmo [76].

There are essential and artificial singularities [444]. The essential singularities arise from specific features of the surface geometry such as the apex of a cone. The artificial singularities arise from the choice of parametrization.

Example 3.1.1. The elliptic cone can be described in a parametric form , where , and , , are constants. We have

thus

We can easily observe that the surface becomes singular only at , which corresponds to the apex of the cone.

The unit normal vector for an implicit surface can be derived by considering two parametric curves , lying on an implicit surface , and intersecting at point on the surface with different tangent directions. Thus we have the relations:

Total differentiation of (3.5) with respect to and , respectively, yields

Now if we multiply (3.6) by and subtract (3.7) multiplied by , and if we multiply (3.6) by and subtract (3.7) multiplied by we can deduce the following relation

which indicates that vector (also known as gradient of ) is in the direction of the cross product of the two tangent vectors at , i.e. in the normal direction. Thus the unit normal vector of the implicit surface is given by

provided that .

Alternatively, we can derive (3.9) by considering an arbitrary parametric curve on an implicit surface , leading to the relation . Since is arbitrary, must be perpendicular to the tangent plane, and hence it is a normal vector.

The tangent plane of an implicit surface at point with coordinates can be obtained by replacing the normal vector of parametric surface in (3.4) with (3.9), which leads to

where and , in (3.10) are evaluated at .

Example 3.1.2. The elliptic cone of Example 3.1.1 has also the following implicit representation . The magnitude of the normal vector , where , becomes 0 only when = = =0 corresponding to the apex of the cone as also derived in Example 3.1.1.