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Figure 4, we can see that the coordinates for
b
in
ʓ
are
(
b
1
,b
2
,b
3
)=(
x
1
+
ʴ
3
(
xz
)+
ʲ
1
,z
2
+
ʴ
1
(
yz
)+
ʲ
2
,y
3
+
ʴ
2
(
xy
)+
ʲ
3
)
=(
x
1
,z
2
,y
3
)+(
ʴ
3
(
xz
)
,ʴ
1
(
yz
)
,ʴ
2
(
xy
)) + (
ʲ
1
,ʲ
2
,ʲ
3
)
.
(2)
We now analyze how the coordinates of vertices change from
ʓ
to
ʓ
.Weuse
w
C
to denote the weight of faces inside
C
, i.e.,
w
C
=
f∈F
(
T|
C
)
w
(
f
).
Lemma 8.
(proof in the long version) For each
b
∈
V
(
T
)
,
⊧
⊨
(
b
1
,b
2
−
(
ʴ
1
(
yz
)+
w
C
)
,b
3
+
ʴ
1
(
yz
)+
w
C
)
if
b
∈
D
1
(
x
)
(
b
1
+
ʴ
2
(
xy
)+
w
C
,b
2
,b
3
−
D
2
(
z
)
(
b
1
−
(
ʴ
3
(
xz
)+
w
C
)
,b
2
+
ʴ
3
(
xz
)+
w
C
,b
3
)
if
b ∈ D
3
(
y
)
(
x
1
,z
2
,y
3
)+(
ʴ
2
(
xy
)
,ʴ
3
(
xz
)
,ʴ
1
(
yz
)) + (
ʲ
3
,ʲ
1
,ʲ
2
)
if
b
(
ʴ
2
(
xy
)+
w
C
))
if
b
∈
(
b
1
,b
2
,b
3
)=
⊩
∈I
(
b
1
,b
2
,b
3
)
otherwise
where
I
is the set of interior vertices of
T
|
C
.
a
1
a
1
a
1
a
1
ʔ
1
(
yz
)
ʔ
1
(
yz
)
S
S
y
z
ʔ
3
(
xz
)
ʔ
2
(
xy
)
ʔ
3
(
xz
)
ʔ
2
(
xy
)
y
z
b
b
x
x
a
3
a
3
a
2
a
2
a
3
a
3
a
2
a
2
Fig. 4. A flip of a counter-clockwise oriented separating triangle
xyz
We now examine what happens during a linear morph from
ʓ
to
ʓ
.We
first deal with faces strictly interior to
C
. The following two lemmas are proved
formally in the long version.
Lemma 9.
For an arbitrary weight distribution no face formed by interior ver-
tices of
T|
C
collapses in the morph
ʓ,ʓ
.
Proof sketch.
Consider a face inside
C
formed by internal vertices
b,c,e
whose
coordinates with respect to
T
|
C
are
ʲ,ʳ,ʵ
, respectively. Examining (2) and
Lemma 8 we see that the coordinates of
b,c,e
in
ʓ
and
ʓ
depend in exactly
the same way on the parameters from
T
C
and differ only in the parameters
ʲ,ʳ,ʵ
. Therefore triangle
bce
collapses during the morph if and only if it col-
lapses during the linear transformation on
ʲ,ʳ,ʵ
where we perform a cyclic shift
of coordinates, viz., (
ʲ
1
,ʲ
2
,ʲ
3
) becomes (
ʲ
3
,ʲ
1
,ʲ
2
), etc. No triangle collapses
during this transformation because it corresponds to moving each of the three
outer vertices
x, y, z
in a straight line to its clockwise neighbour.
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