Graphics Reference
In-Depth Information
efficient in both space and time and much simpler to implement. Except at joints,
the representation automatically preserves volume.
The limitation of an articulated rigid body is that motion necessarily appears
mechanical because the individual pieces remain rigid. Were we to animate a char-
acter in this way, the character would appear robotic instead of humanoid. A more
general solution is the
articulated body.
This maintains a scene graph of refer-
ence frames, but performs some other kind of animation, such as keyframe vertex
interpolation, on the data within those reference frames. The particular combi-
nation of rigid frames with vertex-interpolated key frames was very popular for
real-time applications like games until fairly recently, when it was overtaken by
the more general scheme of skeletal animation.
The shape of the surface of a living creature under motion is dictated by the
dynamic position of its skeleton and the constraints of its musculature and other
internals. It is sensible to consider parameterizing virtual objects that model such
creatures in a similar way. (This is an example of the Wise Modeling principle
that says that you should decide what phenomena you want to model, and then be
sure your model is rich enough to represent those phenomena.) This is known as
a
skeletal animation
(a.k.a.
matrix skinning
) model.
A skeletal animation model defines a set of reference frames called
bones.
That is evocative but something of a misnomer, because reference frames cor-
respond most closely to the
joints
of the character, not the bones between those
joints. The bones may be in a common object space or arranged hierarchically into
a tree with parent-child relationships.
We are used to expressing a point as a weighted combination of axis vec-
tors in some reference frame. A single vertex in a skeletal animation mesh is
a weighted combination of points in multiple reference frames. For example, a
point near a human's elbow may be defined as halfway between a point defined
in the shoulder/upper arm's reference frame and a point defined in the elbow/
forearm's reference frame. This joint parameterization allows that vertex and its
similarly defined neighbors to define a smoothly deforming mesh near the elbow
as it bends. That is in contrast to the sharp intersection of surfaces defined in dif-
ferent reference frames observed under articulated rigid body animation.
We can think of a point
x
represented as a linear combination of specific points
P
b
transformed by corresponding bone transformation matrices
B
b
at given times
under various scalar weights
w
b
:
x
(
t
)=
b
∈
bones
(
B
b
(
t
)
P
b
)
w
b
,
(35.15)
where
w
b
=
1. However, since
P
b
and
w
b
are both part of the parameterization
that we will compute from an artist's input and all relevant operators are linear, in
practice we need only store 3-vectors
P
b
=
P
b
w
b
.
One typically constructs the representation by having an artist place a skeleton
inside a mesh that is in some standard pose. The artist then assigns bone weights
to each vertex and the system computes
P
b
for each vertex from them. A first
approximation might be that each vertex has a single nonzero weight, which is for
its position in the standard pose relative to the closest reference frame. The artist
then manipulates the skeleton to create a different pose and the vertices transform
