Contour lines in the parameter space of a bivariate function can be
separated into three categories:
Local maxima and minima of the function are encircled by
closed contour curves [210].
At the precise level of
a saddle point, the contour curves cross (self-intersect) or exhibit more
complex behavior (eg.
contour lines of monkey saddle
, dog saddle
etc,
[205]).
Contour curves start from a domain
boundary point and end at a domain boundary point.
If the surface is subdivided along the iso-parametric lines which
contain the local maxima and minima of curvature inside the domain
and the contouring levels of curvature are chosen such that the
contour curves avoid saddle points, as shown in Figs.
A.3, A.4, A.5, A.6, each
sub-patch will contain simple contour branches without loops or
singularities. Therefore we can find all the starting points of the
various levels of contour curves along the parameter domain boundary of
each sub-patch by finding the roots of the following equations.
Starting with the Gaussian curvature
(8.111)
(8.112)
where
is the constant Gaussian curvature value. These equations can be rewritten as follows:
(8.113)
(8.114)
Equations (8.113), (8.114) are univariate polynomials of degree
-
and
-
, respectively.
Similarly for the mean curvature
(8.115)
(8.116)
where
is the constant mean curvature value. These equations can
be rewritten as follows
(8.117)
(8.118)
where
.
Equations (8.117), (8.118) are the univariate irrational
functions involving polynomials and square roots of polynomials which
come from the normalization of the normal vector of the surface (see
(3.3)).
is a polynomial of degree
and
is a
polynomial of degree
.
Finally for the principal curvatures
(8.119)
(8.120)
(8.121)
(8.122)
where
, the
signs correspond to the
maximum and minimum principal curvatures, and
is the
constant value of principal curvature and
is a polynomial
function defined in (8.63). Equations
(8.119) through (8.122)
can be rewritten as follows:
(8.123)
(8.124)
(8.125)
(8.126)
Equations (8.123) through (8.126)
are the univariate irrational functions involving polynomials and two
square roots of polynomials which come from the analytical expression
of the principal curvature and normalization of the normal vector of
the surface. The starting points for contour curves of curvature occur
in pairs, since non-loop contour lines must start from a domain
boundary and must end at a domain boundary point.
Cuspidal edges are loci of points on the surface which have tangent
discontinuity and appear as sharp edges on the surface. Cuspidal edges
of an offset surface correspond to points on the progenitor where one
of the principal curvatures is equal to
(
is the
offset distance)
[94].
Therefore we can use (8.123) through
(8.126) to compute the starting points for tracing
cuspidal edges on the offset, if the surface is subdivided along the
iso-parametric lines which contain the local maxima and minima of
principal curvatures, such that each sub-patch will not contain loops
of cuspidal edges in its interior.