Given a point on every branch of an
algebraic curve, we are able to trace the curve using differential curve
properties. The idea is to find increments
and
such that
, when we have
.
Figure 5.19:
A zoomed view of an algebraic curve near a point
Let us Taylor expand
(5.85)
When
and
are not both zero or
, in
order to have
and
to the first
order approximation, we must have
(5.86)
or
(5.87)
assuming
. However, as illustrated in
Fig. 5.19
leads to a point
which may
be far from the curve
. Newton's method on
with initial approximation
may be used to
compute
with high accuracy and in an efficient manner. For
vertical branches, i.e. when
is very small, we may use
.
To avoid these special stepping procedures,
(5.86) may be rewritten as
(5.88)
where
,
are considered as functions of a parameter
. The
solution to the differential equation is given by
(5.89)
(5.90)
where
is an arbitrary nonzero factor. For example,
can be chosen
to provide arc length parametrization using the first fundamental form
(3.13) of the surface as a normalization condition
(5.91)
where
,
and
are first fundamental form coefficients of the
parametric surface evaluated at
,
. Equations
(5.89) and (5.90) form a system
of two first order nonlinear differential equations which can be
solved by the Runge-Kutta or other methods with adaptive step size
[69,126]. For the tracing method to work properly, we
must provide all the starting points of all branches in advance. Step
size selection is complex and too large a step size may lead to
straying or looping [124], as in Fig. 5.20, in
the presence of constrictions where
is very small.
Tracing through singularities (
) is also
problematic. When
is small then the right
hand sides of (5.89) and (5.90) are
small and step size needs to be reduced for topologically reliable
tracing of the curve.
Figure 5.20:
Step size problems in tracing method
Next: 5.8.1.3 Characteristic points
Up: 5.8.1 Rational polynomial parametric/implicit
Previous: 5.8.1.1 Formulation
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December 2009