Graphics Reference
In-Depth Information
For the cylinder, the two curvatures we've described are in fact the maximum
and minimum possible over all surface directions
u
at
P
. Because of this, they
are called the
principal curvatures,
typically denoted
u
κ
2
, with the associ-
ated directions being called the
principal directions.
(One approach to expressive
rendering for surfaces involves drawing strokes aligned with one or two principal
directions [Int97].) The two principal directions are
orthogonal;
this turns out to
be true at every point of every surface (except when the principal curvatures are
the same, in which case the principal directions are undefined; such points are
called
umbilic
). Furthermore, the principal directions
u
1
and
u
2
and their asso-
ciated curvatures
κ
1
and
(a)
(b)
Figure 34.14: (a) The two prin-
cipal curvatures on a cylinder:
Along the axis of the cylinder, the
curvature is zero; in the perpen-
dicular direction, the curvature
is
1
/
r, where r is the cylinder
radius. (b) To measure the cur-
vature in some direction
u
at P,
we intersect the surface with the
plane through P containing
u
and
n
, the normal to the surface. The
result is a curve (gray) in a plane,
whose curvature we can measure.
κ
1
and
κ
2
completely determine the curvatures in every other
direction; if
u
=cos(
θ
)
u
1
+sin(
θ
)
u
2
,
(34.5)
then the curvature in the direction
u
(or
directional curvature in direction u
)is
cos
2
(
κ
1
+sin
2
(
θ
)
θ
)
κ
2
.
(34.6)
Note that this formula does not depend on the orientation of
u
1
or
u
2
:Ifwe
negate
u
2
(giving an equally valid “principal direction”), for instance, the sign
of
)
, but leaves
sin
2
(
θ
changes, which alters the sign of
sin(
θ
θ
)
unchanged.
The principal direction
u
1
(
P
)
corresponding to the maximum directional curva-
ture at each point
P
is used to define the notion of a
ridge
or
valley:
The principal
directions can be joined together into a curve called a
line of curvature
1
(see
Figure 34.15). As we traverse a line of curvature, the principal curvature
κ
1
Figure 34.15: The curves in this diagram have tangents that are in the direction of
either greatest or least curvature. The “bends” near the center occur because the sur-
face is defined by two adjacent spline patches. (Courtesy of Nikola Guid and Borut
Zalik. Reprinted from
Computers & Graphics,
volume 19, issue 4, Nikola Guid, Crtomir
Oblonšek, Borut Žalik, “Surface Interrogation Methods,” pages 557-574, ©1995, with per-
mission from Elsevier.)
1. More explicitly: We can find a curve
t
→ γ
(
t
)
on the surface with the property that
γ
(
t
)=
u
1
(
γ
(
t
))
for every
t
,and
γ
(
0
)=
P
; this is the line of curvature through
P
.
