Graphics Reference
In-Depth Information
Finally, is there a notion of curvature in the case of polygonal objects? Such a
notion would be defined at vertices and would be a function involving the angle
between adjacent edges for curves and the sum of the angles of the faces meeting at
a vertex for surfaces.
15.3
Geodesics
15.3.1
Generating Smooth Geodesics
The mathematics of geodesics for surfaces in
R
3
is discussed in Section 9.10 in
[AgoM05]. The mathematical definition of a geodesic is that it is a
function
(a para-
meterized curve) defined by second-order differential equations. This is what we shall
mean by the term “geodesic,” but it is sometimes used more loosely, for example, to
refer to the underlying set that is traced out by a geodesic. A common statement is
that a straight line in the plane is a geodesic, but one needs to remember that a line
can be parameterized in many ways and only some of those parameterizations would
actually fulfill the mathematical definition of a geodesic. Two other definitions that
are given sometimes (and that consider geodesics as sets rather than maps) are:
The Kinematic Definition.
A
geodesic
is a curve traversed by a particle whose accel-
eration vector at a point lies in the plane spanned by the velocity vector and the normal
to the surface at that point. There is no “side-to-side” acceleration. Any acceleration
that there is, is used to keep the particle in the surface or to speed it up or slow it
down in the direction of the path.
The Static Force Definition.
On a convex surface, a curve is called a
geodesic
if a
thread stretched along the path it traces out on the surface is in static equilibrium
with respect to any sideways tension on it.
A true geodesic would satisfy both of these criteria. However, a geodesic in the
kinematic or static force sense would not necessarily be a real geodesic since its accel-
eration vector might not be orthogonal to its velocity vector. Nevertheless, by Theorem
9.10.11 in [AgoM05] it does trace out a geodesic path.
Note.
The boundary of a surface causes technical problems for the definition of a
geodesic because one often needs derivatives to be defined in open neighborhoods of
a point. To avoid such problems, we shall assume throughout this section that either
our surfaces have no boundary or that all the curves being defined are well away from
the boundary.
Consider a surface patch
S
in
R
3
parameterized by
[
]
¥
[
Æ
S
j :,
cd
ef
,
Any curve g(t) in
S
can be expressed in the form g(t) =j(a(t)), where
[
]
Æ
[
]
¥
[]
a :,
ab
cd
,
ef
,.
(If we were given g first we could define a=j
-1
°
g.) See Figure 15.1. Let
