50 6. GLOBAL 2D SHAPE INTERPOLATION
where w
i
2 Œ0; 1. If we put a large w
i
, the rotation and scale of the triangle
i
would be
suppressed. We thus believe that our framework provides more user controllability over
previous approaches.
(iii) As is shown in [Baxter2008], we can symmetrize the interpolation by symmetrizing the
error function. Let E
i
.t/ be a global error function for a local homotopies A
i
.t/, and E
1
i
.t/
be that for A
1
i
.t/. en define a new error function by
E
0
i
.t/ WD E
i
.t/ C E
1
i
.1 t/:
is is symmetric in the sense that it is invariant under the substitution A
i
A
1
i
and
t 1 t . at means that the same minimizing solution is given if we swap the initial and
the terminal polygons and reversing time.
6.5 EXAMPLES OF CONSTRAINT FUNCTIONS
Now we give a concise list of the constraints we can incorporate into a constraint function
C.v
1
.t/; : : : ; v
n
.t//. See the demonstration video in [Kaji2012].
Some points must trace specified loci (for example, given by B-spline curves). is is realized
as follows: let u
k
.t/ be a user-specified locus of p
k
with u
k
.0/ D p
k
and u
k
.1/ D q
k
. en
add the term c
k
jjv
k
.t/ u
k
.t/jj
2
, where c
k
0 is a weight.
e directions of some edges must be fixed. is is realized by adding the term c
kl
jjv
k
.t/
v
l
.t/ e
kl
.t/jj
2
, where e
kl
.t/ 2 R
2
is a user-specified vector and c
kl
0 a weight. is
gives a simple way to control the global rotation.
e barycenter must trace a specified locus u
o
.t/. is is realized by adding the term
c
o
jj
1
n
P
n
kD1
v
k
.t/ u
o
.t/jj
2
, where c
o
> 0 is a weight. is gives a simple way to control
the global translation.
Likewise we can add as many constraints as we want.

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