Graphics Reference
In-Depth Information
(1) Fitting a quadratic surface to the given data near
p
and using its normal and
Gauss curvature as approximations.
(2) Approximating the surface at
p
by a set of triangles incident to
p
and using
various types of averages of their normals.
(3) Approximating the Gauss curvature at
p
by discretizing its definition based
on the Gauss map where one thinks of it as a limit of the quotient of small
areas containing
p
and the area of their images on the unit sphere. See equa-
tion (9.42) in Chapter 9 in [AgoM05].
(4) Approximating the Gauss curvature at
p
by means of the angle deficit
method.
One surprising conclusion was that the popular method (4) is not always very
accurate.
Andersson ([Ande93]) discusses how one could design a surface by modifying its
curvature. This is carried out in terms of solutions to boundary value problems for
partial differential equations.
Ye ([Ye96]) points out how a color-coded Gauss curvature map can be used to
judge the fairness of a surface. A smoothly varying map is good and rapidly varying
ones are bad. Ye answers the following question about the fairness of a surface where
two patches meet:
Question:
Can the curvature continuity between the patches be visualized by means of the
Gauss curvature? Alternatively, if two patches are tangent-plane continuous along their
common edge and they have the same Gaussian curvature along the common edge, are
they curvature continuous there?
There is a similar question for mean curvature. Let k
n
(
C
,
S
,
p
) denote the normal cur-
vature at a point
p
of a curve
C
lying in surface
S
.
Definition.
Let
S
1
and
S
2
be surfaces that are tangent-plane continuous along a
curve
C
. The surfaces are said to be
curvature continuous
along
C
if for all curves
C
1
and
C
2
on
S
1
and
S
2
, respectively, that meet and are tangent at a point
p
on
C
we have
that k
n
(
C
1
,
S
1
,
p
) = k
n
(
C
2
,
S
2
,
p
).
Ye gives an answer to the question in terms of Dupin indicatrices, principal cur-
vatures, and Gauss and mean curvatures. Mean curvatures turn out to be a better way
to measure curvature continuity.
Kaklis and Ginnis ([KakG96]) address the problem of constructing shape-
preserving C
2
surfaces that interpolate point sets lying on parallel planes. Call the
planar curves p
i
(u) interpolating the data of a given plane a
skeletal line
. We are basi-
cally looking for a skinning surface p(u,v) for the skeletal lines p
i
(u). Kaklis and Ginnis
describe how one can get a skinning surface that has the property that if the curva-
ture of adjacent curves p
i
(u) and p
i+1
(u) has the same sign over an interval [u
j
,u
j+1
],
then the curvature of all the curves p(u,v), v Π[v
i
,v
i+1
] also has the same sign over the
interval [u
j
,u
j+1
].
Wolter and Tuohy ([WolT92]) describe how to compute curvatures for degenerate
surface patches.
Other aspects of surfaces that are sometimes interesting are their lines of curva-
ture. Analyzing these involves solving differential equations. See [BeFH86]. Lines of
curvature are used to define principal patches.
