9.5 Local extrema of principal curvatures at umbilics
In this section we discuss a criterion which assures the existence of
local extrema of the principal curvature functions
and
at umbilical points of the surface
[257]. The problem of detecting local extrema of
principal curvature functions is motivated by engineering
applications. When a ball-end mill cutter is used for NC
machining, the cutter radius must be smaller than the smallest concave
radius of curvature of the surface to be machined to avoid local
overcut (gouging) (see Sect. 11.1.2). Gouging is the one
of the most critical problems in NC machining of free-form surfaces.
Therefore, we must determine the distribution of the principal
curvatures of the surface, which are upper and lower bounds of the
normal curvature at a given point, to select the cutter size. A
natural approach to locate local extrema of the functions
and
would in principle be to search for
zeros of the gradient vector field
and
and then use tools from differential calculus to
decide if at those zeros the principal curvature functions attain
extrema. The problem with this approach however is that the curvature
functions
and
are generally not
differentiable at the umbilics although those points may also be
candidates for local principal curvature extrema. We will present
a necessary and sufficient criterion, which always
detects the existence of a local extremum of the principal
curvature functions
and
at an umbilic,
except in presence of rare well defined and easily computable
conditions. Under such rare condition, the criterion will become only
sufficient. This criterion is practical because it is almost always
applicable and easily evaluated.
We discuss the local behavior of the functions
and
in the neighborhood of an umbilic. First let us consider
a Taylor expansion around an umbilic
for the function
defined in (9.25). We obtain
(9.53)
with
(9.54)
Note that (9.53) describes the remainder term in case of a
second order Taylor approximation of a
smooth function which is
guaranteed if the surface is
.
In the special case where all the second partial derivatives of
vanish,
the condition
implies that the third order partial
derivatives must also vanish. If we
consider the total number of possibilities where we have non-vanishing
partial derivatives up to third order, the case where all partial
derivatives vanish is statistically very rare.
Therefore, we focus our attention now on the
generic case where at least one of the second order partial
derivatives of
does not vanish.
Using (9.26), we obtain
and
at the umbilic, therefore
(9.53) reduces to
Next we Taylor expand the mean curvature
as follows:
(9.59)
where
(9.60)
Although the function
in the remainder terms are
different in (9.55), (9.57),
(9.59), we nonetheless use the same notation for
simplicity, since we are essentially interested in the common property
described in (9.54).
Consequently
in
equation
can
be expanded in the vicinity of an umbilic
as follows:
(9.61)
where
(9.62)
(9.63)
(9.64)
Therefore
can be considered as sum of the constant term
, the plane
, which is the tangent plane of
at
, and the elliptic cone
whose axis of symmetry is perpendicular to
-plane, since
,
.
First we assume
that
, in other words the tangent plane of
coincides with the
-plane. In this case
reduces to
. Figure
9.4 shows a positive elliptic cone
(maximum principal curvature) having a minimum at
.
When the elliptic cone is negative, minimum principal curvature has a maximum
at
. The condition
occurs when all the third order partial derivatives of the height
function in the Monge form are zero. This can be proved as follows
[257]:
Proof : The coefficients of first and second fundamental forms of
the surface in Monge form are given in (3.63) and
(3.65).
Their first order partial derivatives with respect to
are readily
evaluated:
(9.65)
Now we will differentiate (3.67)
9.3 with respect to
(9.66)
If the surface is in Monge form with an umbilic at the origin, we have
, which leads to
. Consequently if
, then
and
hence
. Similarly we can say that if
, then
. Since
and
can be written as
(9.67)
We can conclude that if
then
.
Note that in case
the term
is negligible for local extremum properties of the function
.
Consequently when
, or alternatively when
, the function
, hence
has a local
extremum at
, or more precisely,
has a
local minimum
and
has a local maximum at an umbilical point
.
Figure 9.5:
(a) The plane intersects the cone, (b) the plane does not
intersect the cone, (c) the plane and the cone are tangent to each other
(adapted from [257])
It is also possible that
may have a local extremum at
the umbilic when
. This is the situation when
the plane
is tilted against the
uv-plane.9.4 There are three possible cases, the plane
intersects the cone
transversally (see Fig.
9.5
), the plane
does not intersect the
cone
, apart from its apex, (see Fig.
9.5
) and the plane
and the cone
are tangent to each
other (see Fig. 9.5
). Figure
9.5
is the case when the plane intersects the
cone in two straight lines. In this case for some directions the plane
has a steeper slope than the cone, thus the sum
does not have an extremum at
, while in
case
the plane intersects the cone only at
, and the
cone always has a steeper slope than the plane, thus
has a local extremum at
. Consequently
we need to examine the equation
which upon squaring and using (9.60) and
(9.58) can be reduced to
so that we can view (9.68) as a quadratic
equation with unknown
or
. If
there exist
two distinct real roots, and thus there will be a real intersection
between the plane and the cone made up of two straight lines. If
there exist two identical real roots, and thus the cone
and the plane are tangent to each other, and additional evaluation of
higher order terms in the Taylor expansion is necessary to decide if
we have an extremum at the umbilic. If
there will be no
real root, and thus there is no intersection between the cone and the
plane. Consequently the criterion to have a local extremum of
principal curvatures, when
or
, is equivalent to the condition
. Hence the condition is
(9.70)
or equivalently upon using
(9.71)
Finally we can state the criterion as follows [257]:
Theorem 9.5.1.
(Criterion for extrema of principal curvature functions at umbilics):If we assume
that
is at least
smooth and at least one of the second
order partial derivatives of
does not vanish then:
If
at the umbilic, then
has a local minimum and
has a local
maximum.
If
at the umbilic, then
has a local minimum and
has a local
maximum if and only if
provided
. In
case
, additional evaluation of higher order terms in the Taylor
expansion is necessary.
occurs when the cone and plane are tangent to each other, which
is very rare. The condition
forces the criterion to be only
sufficient and not necessary. It is quite plausible that the
plane-cone tangential case (Fig. 9.5 (c)) is the
rare one, while cases plane and cone are intersecting (Fig.
9.5 (a)) or plane and cone are non-intersecting
(Fig. 9.5 (b)) are the generic ones.
When all the second order partial
derivatives of
vanish, we need to Taylor expand up to fourth
order in (9.53), since the condition
implies that the third order partial derivatives must vanish. Also
we need to Taylor expand up to second order in
(9.59). Consequently
can be expanded in the
neighborhood of an umbilic
as sum of constant, linear and
quadratic terms of mean curvature and the square root of fourth order
terms of
as
(9.72)
where
and
are the second order
partial derivatives terms of the Taylor expansion of the mean
curvature and the square root of fourth order partial derivatives
terms of the Taylor expansion of
. It is apparent that
and
have stationary point at
. It follows that the plane
(linear term) will
determine the local behavior of the function
. This
implies now that in case
,
cannot
have a local extremum at the umbilic due to strong monotonicity
behavior of the linear function. Therefore, if the second order
partial derivatives of
vanish at the umbilic, then
can only have an extremum in case
. However
is not sufficient to
guarantee a local extremum for
, since the point
can be a saddle point for the function
.
Note that here the Taylor expansion of the square root first yields
an approximation instead of the equal sign in (9.57).
However absorbing here the
error term of this square root Taylor expansion in the remainder of
(9.57) justifies the equality sign.
Note that we use the following observation
illustrated by Fig.
9.5
. The term
is
negligible for investigating the local extrema properties of the
function
at the umbilic
, provided the cone
and the plane
meet only at the point
. Namely in that case we have a positive number
such that
.
is related to the smallest possible slope between plane and
cone. Hence clearly
is negligible to
.