The surface normal vector is perpendicular to the tangent plane (see
Fig. 3.3) and hence the unit normal vector is given by
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 .
There are essential and artificial singularities . 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.
The elliptic cone can be described in a parametric form
are constants. We have
The unit normal vector for an implicit surface can be derived by
considering two parametric curves
lying on an
, and intersecting at point
surface with different tangent directions. Thus we have the relations:
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
can be obtained by replacing the
normal vector of parametric surface in
(3.4) with (3.9),
which leads to
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.