We assume that the given parametric surface
=
is a regular and non-periodic NURBS surface patch. Wolter
[448] shows that on a regular NURBS surface patch there
always exists a shortest path joining any two patch points. If the
surface patch is defined on a rectangular or even more generally on a
locally convex planar domain, then any shortest path in the patch
joining any two patch points must have a continuous tangent with
the path being arc length parametrized. If the shortest path (without its
end points) does not meet the patch boundary then this shortest path
is a geodesic in the sense of Definition 10.1.1 where this is
proven in [448] under very weak assumptions. We shall
henceforth assume throughout this chapter that the shortest path to be
computed will not meet the patch boundary except possibly at its
end points.
Let
be an arc length
parametrized regular curve on this surface which passes through point
as shown in Fig. 3.6 and denoted by
(10.1)
Let
be a unit tangent vector of
at
,
be a
unit normal vector of
at
,
be a unit surface normal
vector of
at
and
be a unit vector perpendicular to
in the tangent plane of the surface, defined by
. The
component of the curvature
vector
of
is the geodesic curvature vector
and is given by
(10.2)
The scalar function
(10.3)
is called the geodesic curvature of
at
, or equivalently
(10.4)
The unit tangent vector of the curve
can be obtained by
differentiating (10.1) with respect to the arc length
using the chain rule
(10.5)
Thus we have
(10.6)
and hence substituting (10.5) and (10.6) into
(10.4) yields
(10.7)
We can easily observe that the coefficients of
,
,
,
,
are all functions of the coefficients of the first fundamental form
,
and
and their derivatives,
,
,
,
,
,
. It is interesting to note that the normal curvature
depends on both the first and second fundamental forms,
while the geodesic curvature depends only on the first fundamental
form. Using the Christoffel symbols
(
) defined as follows [412]
(10.8)
geodesic curvature
reduces to
(10.9)
According to the definition, we can determine the differential
equation that any geodesic on a surface
must satisfy by simply setting
in
(10.9) and obtain
(10.10)
Alternatively, we can derive the differential equation for geodesics by
considering that the surface normal
has the direction of a
normal to the geodesic curve
(10.11)
Since
, (10.11) can
be rewritten as
(10.12)
By substituting (10.6) into equations
we have
(10.13)
(10.14)
By eliminating
from (10.13)
using (10.14), and eliminating
from (10.14) using
(10.13) and employing the Christoffel symbols, we obtain
[412]
(10.15)
(10.16)
Equations (10.15) and (10.16) are related by the
first fundamental form
and if we eliminate
from both equations, the equations reduce to
(10.10) with
taken as parameter.
These two second order differential equations can be rewritten as a
system of four first order differential equations [235]
(10.17)
(10.18)
(10.19)
(10.20)
We can also find this result by means of the general rules of the
calculus of variations [166]. We want to minimize
(10.21)
subject to the conditions
(10.22)
where
(10.23)
and
and
are given constants. It is well known from
calculus of variations that the solution of Euler's equation
[166]
(10.24)
gives an extreme value to the integral (10.21). When
(10.23) is substituted in Euler's equation
(10.24) we can derive the differential
equation for geodesics.
Example 10.2.1.
Let us obtain the geodesic equations for a parametric bilinear surface
(hyperbolic paraboloid)
(see Fig. 3.4). We have
thus, the Christoffel symbols become
Finally the geodesic equations for the bilinear surface are given by
Next: 10.2.2 Implicit surfaces
Up: 10.2 Geodesic equation
Previous: 10.2 Geodesic equation
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December 2009