The general name for the quadratic differential forms
of the surface given in coordinates on the surface
and satisfying the usual transformation laws under transformations of these
coordinates. The fundamental forms of a surface characterize the basic intrinsic
properties of the surface and the way it is located in space
in a neighbourhood of a given point; one usually singles
out the so-called first, second and third fundamental forms.
The
first fundamental form
characterizes the
interior geometry
of the surface in a neighbourhood of a given point. This
means that measurements on the surface can be carried out by means
of it. Suppose that the surface is given by the equation
where
and
are coordinates on the surface; and
is the differential of the radius vector
along a chosen direction from a point
to an infinitesimally close point
(see
Fig. a).
Figure: f042200a
The principal linear part of growth of the arc length
is expressed by the square of
:
where
The form
is the
first fundamental form of the surface.
See also
first fundamental form
of a surface.
The
second fundamental form
characterizes the local structure of the surface in
a neighbourhood of a regular point. Thus, choose
a unit normal vector to the surface at
,
where
if the triple of vectors
has a right-hand orientation and
in the opposite case. The doubled principal linear part
of the deviation of the point
on the surface (see
Fig. b) from the tangent plane at the point
is given by
where
Figure: f042200b
The form
is called the
second fundamental form of the surface.
See also
Second fundamental form.
The first and second fundamental forms define
two important common scalar quantities which are invariant
under a transformation of the coordinates on the surface. Namely,
the determinant of the ratio of the second with respect to the first one is the
Gaussian curvature
of the surface at the point:
while the trace of this ratio,
defines the
mean curvature
of the surface at the point.
Specifying the first (positive definite) and second fundamental forms
defines the surface up to a motion (the
Bonnet theorem).
The
third fundamental form of the surface
is the square of the differential of the unit normal vector
to the surface at the point
(see
Fig. c):
Figure: f042200c
The third fundamental form of a surface is equal to the
principal linear part of growth of the angle between the vectors
and
under displacement along the surface from
to
;
it is the first fundamental form of the spherical image of the surface (cf.
Spherical map).
The three fundamental forms are related by the linear dependence
In addition to the fundamental forms listed above, other
fundamental forms are sometimes encountered (see, for example,
[3]).