3.3 Second fundamental form

(3.22) |

where is the

where is called the normal curvature of the surface at in the direction . In other words, is the magnitude of the projection of onto the surface normal at , with a sign determined by the orientation of the surface normal at .

By differentiating
along the curve
with respect to
we obtain

thus

where

Since and are perpendicular to , we have and , and hence we have an alternative expression for , and

Computation of curvatures at points where the surface representation is degenerate (see Sect. 1.3.6) is given in [453].

The numerator of (3.26) is the *second
fundamental form*
, i.e.

and , , are called second fundamental form coefficients. Therefore the normal curvature is given by

where is the direction of the tangent line to at . We can observe that at a given point on the surface depends only on which leads to the following theorem due to Meusnier.

**Theorem 3.3.1.**
*All curves lying on a surface
passing
through a given point
with the same tangent line have the
same normal curvature at this point.*

Using this theorem we can say that the normal curvature is positive
when the center of the curvature of the normal section curve, which is
a curve through
cut out by a plane that contains
and
is on the same side of the surface normal (see Fig.
3.7 (a)). Sometimes the positive normal
curvature is defined in the opposite direction, i.e. the center of
curvature of the normal section curve is on the opposite side of the
surface normal as illustrated in Fig.
3.7 (b).
In such cases (3.23) (3.30) become

The latter convention is often used in the area of offset curves and surfaces in the context of NC machining. Throughout this book we refer to the first convention as convention (a) and to the second one as convention (b). We have listed all the equations, which involve changes due to this convention in the last page of this chapter.

Suppose
is a point on a surface and
is a point in the
neighborhood of
and
is the surface
containing
and
, as in Fig. 3.8. Now suppose
and
are the points
and
,
then Taylor's expansion gives

(3.32) |

Therefore

(3.33) |

Thus using (3.28), (3.29), the projection of onto is

(3.34) |

where the higher order terms are neglected and since , we get

Thus
is equal to twice the distance from
to the
tangent plane of the surface at
within second order terms.
We want to observe in which situation
is positive and negative or
in other words we want to examine
in which side of the tangent plane
lies.
When
, (3.35) becomes
, which can be considered as a quadratic
equation in terms of
or
.
If we solve for
, assuming
, we obtain

(3.36) |

which leads us to the following four cases:

- If
, there is no real root. This means
there is no intersection between the surface and its tangent plane
except at point
. Point
is called
*elliptic point*(Fig. 3.9(a)). For example, an ellipsoid consists entirely of elliptic points. - If
and
, there are
double roots. The surface
intersects its tangent plane with one line
,
which passes through point
. Point
is called
*parabolic point*(Fig. 3.9(b)). For example, a circular cylinder consists entirely of parabolic points. - If
, there are two roots. The surface
intersects its tangent plane with two lines
, which intersect at point
Point
is called
*hyperbolic point*(Fig. 3.9(c)). For example, a hyperboloid of revolution consists entirely of hyperbolic points. - If
, the surface and the tangent plane have
a contact of higher order than in the preceding cases. Point
is
called a
*flat*or*planar point*.