Graphics Reference
In-Depth Information
Definition.
A set of points is said to be
fair
if there exists a fair curve that interpo-
lates these points.
This leads to various solutions to the curve-fitting problem. Two well-known
methods are the
least squares method
and the
energy function approach
. The latter is
based on the idea that nature will naturally produce fair curves and their shapes are
determined by minimizing certain bending energy integrals. A variety of these integrals
are described in [HosL93]. Another approach to finding a fair interpolating curve is to
start with the obvious polygonal curve that connects the points and then to improve its
shape using
interpolatory refinement schemes
. See [Kobb96] for an overview of several of
such schemes and how they can be described in a more systematic way.
Now if the control points of a curve are chosen badly, then there is not much one
can do to improve its shape. Therefore, part of the task of coming up with fair curves
is picking good data to interpolate or approximate. We want to eliminate the hope-
fully few bad points.
For more about fairing and curvature see Section 15.2. See also [SéCM95] where
fairing is obtained by minimizing various functionals. An adaptive approach to fairing
digitized point data is described in [LWZL02].
11.13
Parallel Transport Frames
The Frenet frames to a curve provide a moving coordinate system along the points of
a curve that often comes in handy. They are, for example, useful in dealing with gen-
eralized cylinders. The problem is that Frenet frames are not defined at points of a
curve where the second derivative vanishes and the curve is locally flat. This section
briefly describes a more general way to define a moving coordinate system that applies
to all
regular
curves, whether or not they have a vanishing second derivative. For
additional information see [Bish75], [ShaB84], [Bloo90], or [HaMa95].
Given a regular space curve p(t), let T(t) be the unit tangent vector of the curve
at p(t), that is,
1
()
=
¢
()
Tt
pt
pt
.
¢
()
Our specific problem is to find parameterized unit vectors n
1
(t) and n
2
(t) so that the
triple F(t) = (T(t),n
1
(t),n
2
(t)) defines a continuously varying orthonormal basis for
R
3
at p(t). If the second derivative of p(t) does not vanish, then one solution is the Frenet
frame (T(t),N(t),B(t)), where N(t) and B(t) are the principal normal and binormal to
p(t) at t, respectively. Alternatively, we can pick an arbitrary orthonormal basis
(T(t
0
),n
1
(t
0
),n
2
(t
0
)) of vectors at a start parameter t
0
and try to propagate this basis
along p(t) by rotating it by a rotation based on the way that T(t) rotates as it moves,
independent of the curvature. Such frames, defined more precisely shortly, are called
parallel transport frames.
An Algorithm for Computing Parallel Transport Frames F(t
i
) at Points p(t
i
).
Assume that we already have the frame F(t
i-1
) = (T(t
i-1
),n
1
(t
i-1
),n
2
(t
i-1
)) at p(t
i-1
). Let
