Game Development Reference
In-Depth Information
defined on the perpendicular plane located at distance
d
from the bone origin.
The support of
µ
d
is partitioned into three specific zones (
Z
d
,
in
,
Z
d
,
mid
and
Z
d
,
ext
) by
two concentric circles characterised by their respective radius
r
d
and
R
d
(Figure
10).
µ
d
is then defined as follows:
1
x
∈
Z
int
δ
(
x
,
Z
)
()
µ
x
=
f
ext
,
x
∈
Z
,
d
mid
(3)
R
−
r
d
d
0
x
∈
Z
ext
) is a user-
specified fall-off to be chosen among the following functions:
where
δ
(
x
,
Z
ext
) denotes the Euclidean distance from
x
to
Z
ext
and
f
(
⋅
π
3
2
x
,
x
,
x
,
sin(
x
),
x
and
3
x
. This set of functions allows a large choice for
2
designing the influence volume and ensures the generality of the model.
The affectedness measure
µ
) is defined in the same manner,
but using two half-spheres of radius
r
0
and
R
0
(respectively
r
l
and
R
l
) as
illustrated in Figure 10a.
The bone influence volume being defined, animating the virtual character
consists of deforming its mesh by translating its vertices according to the bone
transformation.
Here only affine transformations are applied to the bone controller. In virtual
character animation, the most widely used geometric transformation consists in
changing the orientation of the bone with respect to its parent in the skeleton
hierarchy. Thus, the bone can be rotated with respect to an arbitrary axis.
However, when special effects are needed, the bone can also be translated. For
instance, in cartoon-like animations, thinning and thickening the skin envelope are
frequently used. For such effects, the bone transformation must contain a scale
component specified with respect to a pre-defined direction.
The general form of the geometric transformation of a bone
b
is expressed as
a
µ
(respectively
−
+
0
l
4 ×
4
element matrix
T
obtained as follows:
T = TR
w
b
⋅
R
w
b
⋅
S
w
b
(4)
where
TR
w
b
,
R
w
b
,
S
w
b
give the bone translation, rotation and scale, respectively,
expressed in the world coordinate system.
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