Graphics Reference
In-Depth Information
Figure 15.12.
Dropping frames on surfaces.
Figure 15.13.
Projecting a curve onto a
surface.
Then g : [0,1] Æ
S
defined by g(t) = p(s(t)) is a curve on the surface. How should s be
defined from the
q
i
? If we are not careful then, depending on the parameterization
p(u,v), g may wiggle in undesired ways on the surface. The point is that the user was
outlining a curve on the
surface
not in its parameter space. Most likely, the curve
should go along a geodesic from one point to the next. Therefore, one way to try to
generate the curve from
q
i
to
q
i+1
is to follow the geodesic which starts at
q
i
in the
“direction from
q
i
to
q
i+1
.” As an approximation to this direction we can take the vector
q
i
q
i+1
. Now we can apply the methods discussed in Section 15.3.1. Unfortunately, the
complete curve may now have corners at the dropped points.
In the case of polygonal surfaces one wants to find a polygonal curve that pass
through the dropped points. Connect the dropped points by discrete geodesics.
A related problem is to project a given curve
C
orthogonally to a surface
S
. What
this means is that each point
p
on the curve
C
should project to the point
q
on the
surface
S
that is closest to it. In the smooth case a necessary condition is that
(
)
¥=,
pq n
-
0
q
where
n
q
is a normal to
S
at
q
. See Figure 15.13. The problem of finding the point on
a smooth surface that is closest to a single point was already considered in Section
14.2. In our case here we could pick points
p
i
on the curve and find their closest points
q
i
on the surface by solving equations like equations (14.4) or (14.5). Connecting the
points
q
i
by paths on the surface would give us a curve that is an approximation to
the projection curve. Alternatively, if the curve
C
and surface
S
are parameterized by
functions g(t) and j(u,v), respectively, then the equation above is equivalent to the
equations
