Graphics Reference
In-Depth Information
(2) Two parametric curves meet with G
2
continuity if and only if they have the
same unit tangent and curvature vector at their common point, that is, the arc-length
parameterization is C
2
.
(3) In general, two parametric curves meet with G
k
continuity if and only if the
arc-length parameterization of the composite is C
k
at their common point.
Proof.
See [BarD89].
Because of Theorem 11.9.1(3), G
k
continuity is sometimes referred to as
C
k
arc-
length continuity
. Also, because of the relationship between curvature and the second
derivative, G
2
continuous curves are called
curvature continuous
.
Turning things around, one can show that the composite of the two curves in equa-
tion (11.130) meet with second-order geometric continuity if there exist two constants
b
1
> 0 and b
2
, so that equations (11.132) hold. Therefore, one can define a collection
of basis functions, called
Beta-splines
, satisfying (11.132) and parameterized by b
1
and
b
2
. These functions have all the basic properties as the regular B-spline basis func-
tions. Using them one now has additional control over the curvature and shape of a
curve because one can now alter the b
1
and b
2
values. Beta splines are the geometri-
cally continuous analog of ordinary B-splines.
There are other formulations of the geometric continuity problem. See [Fari97].
Each has its own advantages. As usual, it depends on the problem that one is trying
to solve as far as deciding on an approach.
A more geometric formulation of geometric continuity in the case of two cubic
Bézier curves is the following. Let
b
0
,
b
1
,
b
2
,
b
3
and
c
0
,
c
1
,
c
2
,
c
3
be the control points
of the two curves with
b
3
=
c
0
. See Figure 11.35. Let B
-
, B
+
, C
-
, and C
+
be the areas
of the triangles
b
1
b
2
b
3
,
b
2
b
3
d
,
db
3
c
1
, and
b
3
c
1
c
2
, respectively. Define r
-
= |
b
1
b
2
|/|
b
2
d
|,
r
+
= |
dc
1
|/|
c
1
,
c
2
|, and r = |
b
2
b
3
|/|
b
3
c
1
|. Then one can prove
We have G
2
continuity at
b
3
if r
2
11.9.2
Theorem.
= r
-
r
+
.
Proof.
See [Fari97].
The geometric constraint defined in Theorem 11.9.2 can be used to construct a
cubic G
2
spline from a set of given control points whose curvature a user can control
by specifying suitable tangents. Two interior Bézier points are added for each suc-
cessive pair of control points subject to the G
2
continuity constraints and the final
curve is a collection of Bézier curves. See [Fari97].
Figure 11.35.
Geometric continuity for Bézier
curves.
