Curves in Computer Graphics - Computer Graphics and Geometric Modeling - page 450

Suppose that we have two C k curves

[ Æ

m

() = ()

p q ,:, 01 R with p 1 q 0 . (11.130)

How “smooth” is

(

[]

) » (

[]

) X = p 01 , q 01 ,

at b? How can we control the smoothness, or lack of it, there? These are the type of
questions considered in this section. We might be tempted to say that the composi-
tion of these two curves will certainly be continuous because of the hypothesis that
p(1) = q(0), but this does not make much sense unless we know what is meant by the
composite map .

The parametric curve g: [0,1] Æ R m defined by

Definition.

Π[ ]

() = ()

g u p 2 u for u 0 1 2 ,

( ) Π[

]

= q 21 u - for u 121 ,

is called the composite of the curves p(u) and q(u) in expression (11.130).

Note that by the chain rule, the speed of the curve g(u) is twice that of the curves
p(u) and q(u). This change was forced on us because we wanted the domain of the
composite to be [0,1]. On the other hand, the tangent lines of g(u) agree with those
of p(u) and q(u). This is what is important to us and not the fact that the speed
changed by a common multiple. If we wanted the velocity of the new curve to be the
same as the velocities of the old ones then we could have defined a curve with domain
[0,2], which agrees with p(u) on [0,1] and with q(u - 1) on [1,2].
Now, when two regular parametric curves meet at a point where they have a
common tangent line, they can be reparameterized (for example, using the arc-length
parameterization) so that their tangent vectors match where they meet to make the
composite differentiable. Mathematically, therefore, it is unimportant whether or not
the tangent vectors match exactly because we always get a differentiable manifold.
However, there are practical reasons for allowing parametric curves to meet with a
common tangent line but distinct tangent vectors.
Returning to the curves in (11.130), suppose that p([0,1]) has the same tangent
line at p(1) as q([0,1]) at q(0), that is, p¢(1) = a q¢(0) , where a π 0. The curve will look
smooth but it will not be differentiable unless a = 1. On the other hand, if we had
chosen different parameterizations, then this composite might be differentiable.
Because it is convenient to allow for “wrong” parameterizations, the general question
is whether one can tell from p and q alone whether X will be smooth.
Assume that p(u) and q(u) are regular curves.

Definition. We say that the regular curves p(u) and q(u) in (11.130) meet with kth
order geometric continuity , or G k continuity , if there is an equivalent parameterization
r(u) for p(u) with respect to an orientation-preserving change of parameters so that
r(u) and q(u) meet with C k continuity, that is, the composite curve is also C k at

p(1) = q(0). A regular curve is called a G k continuous or simply G k

curve if it admits