For a parametric space curve
, a focal curve
or an evolute, shown
in Fig. 8.5, is defined as
(8.15)
where
is a unit
principal normal vector and
is a
nonzero curvature of the curve. Thus the focal curve or evolute
of a curve is the locus
of its centers of curvature.
Focal surfaces [146,145] can be constructed
in a similar way by using the principal
curvature functions of the given surface. For a given parametric
surface patch
the two associated focal surfaces are
defined as
(8.16)
where
is a unit surface normal and
is a
nonzero principal curvature (
or
). Here sign convention (a) (see Fig.
3.7 (a)) is employed for both
(8.15) and (8.16). When sign
convention (b) is assumed, we simply replace the plus sign by the minus sign.
Figure 8.5:
Focal curve. The thin curve is the focal curve of the
thick curve
Focal curves and surfaces provide another method for
visualizing curvature.
An important application of focal curves is in testing the curvature
continuity of surfaces across a common boundary. Curves on the focal
surfaces of each surface, which are at the common boundary,
,
where
is the common boundary curve (or
linkage curve) on the progenitor surface
, can
be compared to determine the curvature continuity of the surfaces.
According to the linkage curve theorem by Pegna and Wolter
[305] two surfaces joined with first-order or tangent plane
continuity along a first-order continuous linkage curve can be shown
to be second-order smooth on the linkage curve
if the normal curvatures along the
linkage curve on each surface agree in one direction other than the
tangent direction to the linkage curve. Comparison of focal surfaces
allows for a visual assessment of the accuracy for the second-order contact
of two patches along their linkage curve. Namely if the two focal
curve point sets (defined via
and
) of
one surface patch agree along the linkage curve with the two
corresponding focal curve point sets of the adjacent patch then these
surface patches have curvature continuous surface contact along the
linkage curve. The deviation of the corresponding focal curves
indicates the amount of discontinuity of curvature of both adjacent
patches along the linkage curve.