Graphics Reference
In-Depth Information
Figure 15.5.
Shortest curves on nonconvex surfaces.
Figure 15.6.
The angle at a vertex.
Proof.
See [MiMP87]. The fact that the curve intersects the edges in the edge
sequence in their interior is needed here.
If our surface is convex, then one can show that, except for possibly its first
and last point, a discrete geodesic will not go through any vertices of the surface.
Unfortunately, things get more complicated in the nonconvex case. See Figure 15.5.
However, a discrete geodesic that passes through a vertex has to satisfy an interest-
ing geometric condition there.
Definition.
The
angle
of a face at one of its vertices
v
is the angle between the two
edges of the face that meet in
v
.
Suppose that a simple pwl curve goes through a vertex
v
. Let
e
and
e
¢ be the two
distinct adjacent edges of the curve that meet in
v
. By momentarily dividing a face
into two, if necessary, we may assume that both
e
and
e
¢ lie in edges of faces that
meet in
v
. Clearly, we can now divide the faces that meet in
v
into two edge-adjacent
sequences whose first and last faces meet in edges containing
e
and
e
¢. For each of
these two sequences add up the angles of their faces at
v
.
Definition.
The smaller of the two sums is called the
angle
that the curve makes at
v
.
In Figure 15.6, q is the angle of the curve at
v
. The reader should not assume that
the faces adjacent to a vertex can be flattened out in the plane. Figure 15.5(a) shows
an example where this would not be possible.
