Contour curves for constant curvature satisfy the following equation
(8.127)
where
is a curvature at the given point (
). The
procedure for tracing contour curves is same as that of tracing method
in the intersection problem that we discussed in
Sect. 5.8.1. We now consider a space curve which
lies on the surface represented by the parametric form
. Differentiating (8.127) with
respect to
yields
(8.128)
where
,
are the first derivatives with respect to
, and (
,
) gives the direction of the contour line in
parameter space.
The solutions to (8.128) can be written as
(8.129)
where
is an arbitrary nonzero factor. When the curvature map
is constructed in the
parameter space,
can be chosen to
provide arc-length parametrization in the parameter domain as follows
(8.130)
or when it is constructed on the surface itself,
can be chosen
to provide arc length parametrization using first fundamental form
(3.13) of the surface as a normalization condition
as in (5.91)
(8.131)
where
and
are given in (8.35),
(8.48) and (8.60) for Gaussian,
mean and principal curvatures respectively.
The points of the contour
curves are computed successively by integrating the initial value problem
for a system of coupled nonlinear differential equations
(8.129) using, for example the Runge-Kutta method or a more
sophisticated
variable stepsize and variable order Adams method [69].
Starting points are computed by the method described in Sect.
8.5.2. Accuracy of the contour curve depends on the number of
points used to represent the contour curve by straight line
segments. Note that for principal curvatures,
and
become
singular at an umbilical point, therefore we avoid the contour
level which is equivalent to the curvature value at the umbilics.
To generate a color curvature map, graphics packages require
closed polygonal geometry. Therefore, for display purposes and also
for theoretical interest, it is necessary to decompose a parameter
sub-domain further into closed polygonal regions. We may use the trip
algorithm introduced by Preusser [337] to polygonize the
area between contour curves.
A closed polygon is defined by contour curves and the border lines of
the domain as illustrated in Fig. 8.10.
Let
and
be the starting and
ending point of a contour curve respectively, and
be the
vertices of the rectangular sub-domain. The algorithm is given
as follows:
Trip algorithm [337]
Start the first round trip from
.
At
continue in a counter-clockwise direction on the
rectangular boundary to the next starting point
or vertex.
Search a way back to the starting point
by following
a contour curve from
, or a rectangular boundary
from a vertex. As a result, the first polygon is given by
-
-
-
-
-
.
A second polygon is started at the next unused starting point in
counter-clockwise direction and is completed in the
same way which results in
-
-
-
-
-
.
When there are no unused points
, we start the remaining
polygons at points
that have served only as the end of a contour curve,
i.e.
and
.
Figure 8.10:
Trip algorithm
Next: 8.5.4 Examples
Up: 8.5 Contouring constant curvature
Previous: 8.5.2 Finding starting points
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December 2009