Graphics Reference
In-Depth Information
g
max
min
Figure 34.16: As we traverse the line of curvature
γ
, the directional curvature of the surface
S in the direction of
γ
varies from point to point. We've marked a local maximum and
minimum; these are ridge and valley points, respectively. Shown in red (valley) and blue
(ridge) lines are the rest of the ridge and valley points of the surface.
changes from point to point. Local minima and maxima of the principal curva-
ture along a line of curvature are called
ridges
and
valleys,
respectively (see Fig-
ure 34.16). These curves have been used to help communicate shape, but they
suffer from two problems. The first is that, in practice, algorithms for computing
ridges and valleys tend to be “noisy,” that is, they tend to produce lots of short
segments, which are distracting rather than informative. The second is that ridges
and valleys often occur in pairs, so we end up with two lines where an artist would
draw only one (see Figure 34.18 for an example).
Instead of using the principal curvature directions on a surface to define curves
along which to find local maxima and minima, as we did for ridges and valleys,
we can use a different vector field, one that depends on our
view
of the surface
rather than being intrinsic to the surface itself, independent of view, as are ridges
and valleys. This is the approach taken by DeCarlo et al. [DFRS03]. We'll follow
their development.
We l e t
v
(
P
)=
E
P
denote the view vector at each point of the surface.
Notice that this vector points
from P
toward the eye
E
.Andwelet
w
(
P
)
denote
the projection of
v
(
P
)
onto the tangent plane at
P
; omitting the argument
P
,
w
is
defined by
−
w
=
v
−
(
v
·
n
)
v
,
(34.7)
where
n
=
n
(
P
)
is the unit normal to the surface at the point
P
. The curvature in
the direction
w
is called the
radial curvature
κ
r
.
Note that the radial curvature
depends both on the point
P and
on the location of the eye: It's not an intrinsic
property of the surface.
