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9.2 Lines of curvature near umbilics
It is easily verified from (9.1) that
,
and
simultaneously vanish at
the umbilics. Therefore for all
,
,
(3.41) is satisfied, and hence we cannot determine
the direction of the lines of curvature which pass through the
umbilic. In this section we investigate the pattern of the lines of
curvature near generic umbilics. Generic
umbilics are stable with respect to small
perturbations of the function representing the surface, while
non-generic umbilics are unstable
[24,364,397]. Darboux [71] has
described three generic
features of lines of curvature in the vicinity of an umbilic. The
three generic features are called star, (le) monstar and
lemon based on the pattern of the net of lines of curvature.
Color Plate A.7 illustrates these three patterns
of the net of lines of curvature at the umbilic. The red solid line
corresponds to the maximum principal curvature lines and the dotted
blue line corresponds to minimum principal curvature lines, where
convention (b) of sign of normal curvature is used
(see Fig. 3.7 (b) and Table
3.2). Three
lines of curvature pass through the umbilic for monstar and star,
while only one passes for the lemon. The criterion distinguishing
monstar from star is that all three directions of lines of curvature
through an umbilic are contained in a right angle, whereas in the star
case they are not contained in a right angle. There are no other
patterns except for non-generic cases. An example of a non-generic
umbilic can be offered by the two poles of a convex closed surface of
revolution [165]. Figure 9.1
(a) shows the non-generic umbilic of a paraboloid of revolution
which has an umbilic that infinite number of lines of
curvature pass through. If we perturb a coefficient in the function
representing the surface slightly to
corresponding to an elliptic paraboloid then the non-generic umbilic
splits into two lemon-type generic umbilics as shown in Fig.
9.1 (b).
Consider a surface in Monge form
(9.2)
where
is a
smooth
function, i.e. it has continuous derivatives up
to order three.
We can Taylor
expand the z component of the surface as in (8.73).
Suppose the
surface
has an umbilic at the origin and its tangent plane
coincides with the
plane, then it is apparent that
. Evaluating the coefficients of the first and
second fundamental forms of the explicit surface at the origin, which
are given in (3.63) through
(3.65), we obtain
(9.3)
(9.4)
Then it is apparent from (9.1) that
and
.
Consequently we can rewrite (8.73) into a
simpler form:
(9.5)
From (9.5), we can observe that the equation of
the surface near the umbilic is governed by the cubic form
(9.6)
where
(9.7)
Note that
,
,
and
vanish for a
paraboloid of revolution
at
.
Figure 9.1:
Lines of curvature of paraboloid of revolution
and elliptic paraboloid. Solid lines and dotted lines represent
maximum and minimum principal curvature lines respectively: (a)
paraboloid of revolution (
) has a non-generic umbilic
at (0,0,0), (b) elliptic paraboloid (
) has
two lemon-type umbilics at (0,
0.1890,0.0357) (adapted from
[257])
To study the behavior of the umbilics we can express
(9.6) in polar coordinates
and
for a fixed radius
[24]:
(9.8)
It can be easily verified that
.
Therefore the cubic function is an antisymmetric function of
.
The roots of
will give the angles where local maxima
and minima of
may occur around the umbilic, depending on
the multiplicity of the roots. When there are three distinct roots,
each of the roots gives the local extremum. When there are two equal
roots, the double roots will provide neither maxima nor minima, however the
single root gives an extremum. When there are three equal roots, the
triple roots give an extremum.
Since it is an
antisymmetric function, maxima and minima of
occur on the same
straight line which passes through the umbilic. Differentiating
(9.8) with respect to
and setting the equation
equal to zero yields
(9.9)
When one of the roots of (9.9) is
or
then
must be zero, and when
or
then
must be zero. Conversely we can say
that when
one of the roots is
or
, and
when
one of the roots is
or
.
Consequently when
, we can divide
(9.9) by
resulting in
(9.10)
where
. Similarly when
, we can divide (9.9) by
resulting in
(9.11)
where
.
These cubic equations may be reduced by the substitution
(9.12)
to the normal form [403]
(9.13)
where
(9.14)
(9.15)
(9.16)
(9.17)
The solutions to the cubic equation are given by:
When
; there are three distinct roots,
one is real root and the other two are
conjugate complex roots. The real root gives a function
extremum and is given by
(9.18)
When
; there are three real roots at least
two of which are equal and are given by
(9.19)
where the upper sign is to be used if
is positive and the lower
sign if
is negative. Therefore there is at most one root which
will provide a function extremum. This is a non-generic case, since small
perturbation will yield the case either above or below.
When
; there are three unequal real roots,
which provide three function extrema, and are given by
(9.20)
where
and the upper sign is to be used if
is positive and the lower if
is negative.
Consequently there is either one single angle (lemon) or three
different angles (star, monstar) corresponding to one maximum
opposite one minimum or three maxima opposite three minima for generic
case. Corresponding to these angles there are lines of curvature
either one or three passing through the umbilics.
Another way of classifying an umbilic is to compute the index
around it [24,364].
The lemon and
monstar have the same index
, while the star has the
index
. The index is defined as an amount of rotation
that a straight line tangent to lines of curvature experiences when
rotating in the counterclockwise direction around a small closed path
around the umbilic. To compute the index of the umbilic, we can
evaluate the angle
, which is the angle of principal
direction, at
points along a boundary curve which surrounds the
umbilic. The angle
is obtained by using the first of
(3.41) (see Table 3.2 for sign
convention) as
(9.21)
where
. Since
can also be obtained from the second equation of (3.41)
we also get
(9.22)
If both
and
are zero or small
in absolute value, we use
(9.22) otherwise we use (9.21), or
if
we can invert
(9.21) and solve for
.
Consequently the index can be computed by
(9.23)
where
(9.24)
and
is the modulo operator. It is used to account for the
first point which is also the last point at which the direction field
is evaluated. For the examples in this book, 20 points per
boundary curve were adequate for estimation of the index. Fig.
9.2
illustrates the direction field of the maximum principal curvature around
the star, monstar and lemon type umbilics.
Figure 9.2:
Direction field near umbilics: (a) star-type,
(b) monstar-type, (c) lemon-type (adapted from [257])
Next: 9.3 Conversion to Monge
Up: 9. Umbilics and Lines
Previous: 9.1 Introduction
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December 2009