In the sequel we assume
,
and
without loss of generality. According to this assumption the surface
is a hyperbolic paraboloid when
, an elliptic paraboloid
when
, a paraboloid of revolution when
, and a parabolic cylinder when
as illustrated in
Fig. 8.9. The paraboloid of revolution and the
parabolic cylinder can be considered as degenerate cases of the
elliptic paraboloid. When
, the principal direction is
not defined and the point
will become an umbilic. If
and
vanish at the same time, the surface is part of
a plane, and we do not investigate such cases.
In the case for offsets of explicit quadratic surfaces, there are
no self-intersections due to global distance function properties
[26], thus if
the maximum absolute value of the
negative minimum principal curvature determines the largest offset
without degeneracy. The largest magnitude of offset distance without
degeneracy is called the maximum offset distance
. In
Sect.
8.3 we discussed how to find the global
minimum of the minimum principal curvature of explicit quadratic
surfaces.
Due to Lemma 8.3.1 the
global minimum of the minimum principal curvature of the explicit
quadratic surface occurs at the origin, except for a parabolic cylinder
((11.63) with
=0) which has minima along the
-axis with curvature value
, and hence the maximum offset distance is determined to
be
. If the offset distance exceeds
, the offset starts to degenerate from the point
on the offset surface except for a parabolic
cylinder progenitor, where the offset starts to degenerate along the line
.
Substitution of the expression of the offset of the explicit quadratic
surface (11.63)
Theorem 11.3.2. The self-intersection curves of
offsets of the explicit quadratic surfaces
and their pre-images in the
-plane are
as follows [248]:
Proof:
Case (1): Since
is negative for the
hyperbolic paraboloid and we are assuming
and the left hand side of
(11.66) is always positive, this equation cannot be
used to derive the self-intersection curve. This
implies that the offset of a hyperbolic paraboloid does not
self-intersect in the
-direction (maximum principal
direction). However, we can use (11.67) to derive the
self-intersection curve in the
-direction (minimum
principal direction). Upon squaring and replacing
by
and
by
we obtain
Case (2): Since
is positive for the elliptic paraboloid, both
(11.66), (11.67) can be used to obtain
the self-intersection curves in the
-plane. This implies that the
offset of an elliptic paraboloid may self-intersect in both principal
directions. Since we have already
derived the equation from (11.67), we derive
another equation from (11.66). Upon
squaring and replacing
by
and
by
we obtain
Case (3): If we set
in (11.75) and
(11.77), both equations reduce to (11.72).
Also if we set
in (11.76),
the parabola reduces to the point
.
Case (4): Since
is zero for the parabolic cylinder,
(11.66) is not valid. Thus we set
in
(11.67) and replacing
by
we
obtain
, which is equivalent to
(11.74). The self-intersection curve in three dimensional
coordinates can
easily be obtained in a similar manner with Case (1).
Note that the self-intersection curve of the offset of an
elliptic paraboloid (when
) has a positive quadratic term,
while those of a hyperbolic paraboloid and an elliptic paraboloid (when
) have negative quadratic terms.
Example 11.3.4.
Consider an elliptic paraboloid
with offset distance
. Since
, the
offset surface self-intersects only in the
-direction. The
self-intersection curve is
(dashed line in Fig.
11.20) and its
pre-image in the
-plane is
(solid line in Fig.
11.20). The dot dashed line in this figure illustrates the
set of footpoints of the self-intersection curve on the progenitor
surface. A pair of thin solid straight lines emanating from two
distinct points on the surface
,
and
intersecting along the parabola are the pairs of vectors
and
.
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To illustrate Theorem 11.3.2, we plot pre-images of the
self-intersection curves along with cuspidal edges in the
-plane for the
hyperbolic paraboloid (
,
,
), the elliptic
paraboloids (
,
,
), (
,
,
) and (
,
,
), the
paraboloid (
,
) and the parabolic cylinder
(
,
,
) as depicted in Figs.
11.21 (a) to (f).
![]() |
It is interesting to note that when the progenitor surface is a
hyperbolic paraboloid or an elliptic paraboloid (see
Fig. 11.21 (a) to (d)), the pre-images of the
self-intersection curve of its offset which self-intersects in the
-direction and the cuspidal edge
always intersect tangentially at
. The pre-image of the
self-intersection curve of the offset of an elliptic paraboloid (see
Fig. 11.21 (c)), which self-intersects in
-direction, and the cuspidal edge
intersect tangentially at
, when the two cuspidal edges intersect with
the
-axis within the two umbilics. Whereas when the two cuspidal edges
intersect the
-axis outside the two umbilics (see Fig.
11.21 (d)), the pre-images of the self-intersection curve
and the cuspidal edge
intersect tangentially at
.
It is apparent from (11.21) that the direction of
the normal vector of the offset surface is opposite to that of the
progenitor surface inside the loop of
(dashed line) in the absence of the loop of
(see Figs.
11.21 (a), (b), (e)), and the regions between
outside the loop of
(dot dot dashed line) and
inside the loop of
(see Figs.
11.21 (c), (d)), while the direction is the same
within the loop of
(see Figs.
11.21 (c), (d)).
Figures 11.22, 11.23 and 11.24 show
self-intersecting offset surfaces, self-intersection curves and cuspidal
edges in 3-D space and the trimmed offset surface of a hyperbolic
paraboloid (
,
,
), an elliptic paraboloid
(
,
,
) and an elliptic paraboloid
(
,
,
), respectively.
![]() |
![]() |
![]() |
Figure 11.25 illustrates the self-intersections of the
offset of a bicubic Bézier surface patch. Figure 11.25
shows the pre-images of the self-intersection curve in the
parameter domain. The thick line represents the numerically traced
self-intersection curve, while the thin line represents the ellipses
of (11.69), (11.71), which are in quite
good agreement. The same bullet symbols are mapped to the same locations
on the offset surface. Figure
11.25
shows the mapping of the self-intersection curves in the parameter
domain onto the
progenitor surface. Finally, Fig.
11.25
shows the offset surface with its
self-intersections.
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