Graphics Reference
In-Depth Information
1.3 Definitions
Basically, each hair strand is a poly line. We will use the terms
local
and
global
often when we explain local shape constraints. Global is in world frame and local
is in local frame, which is attached to the starting point of a line segment.
The index of vertices starts from the root of a hair strand that is attached to
a scalp.
P
i
is the position of vertex
i
in the current time step. The zeroth time
step is the rest state, and we use a right superscript to express it explicitly (i.e.,
P
i
). Here, we focus only on vertices in one strand when we explain algorithms.
Therefore, vertex index
i
is always unique.
In case we need to explicitly clarify which coordinate system we are using, we
specify it using a left superscript (i.e.,
i−
1
P
i
means the position of vertex
i
in the
current time step defined in the local frame
i
1). When the position is defined
in the world coordinate system, we can drop the frame index (i.e.,
w
P
i
=
P
i
).
In terms of transforms, we define
i−
1
T
i
as a full transformation containing
rotation
i−
1
R
i
and translation
i−
1
L
i
. It transforms
i
P
i
+1
to
i−
1
P
i
+1
such that
i
−
i
P
i
+1
. Because of careful indexing of vertices in the strand, the
following equation holds:
−
1
P
i
+1
=
i−
1
T
i
·
w
T
i
=
w
T
0
·
0
T
1
·
1
T
2
...
i
−
2
T
i−
1
·
i
−
1
T
i
.
·
In this chapter, we call
i−
1
T
i
a local transform and
w
T
i
a global transform.
In the case of vertex 0, local transform and global transform are the same such
that
−
1
T
0
=
w
T
0
.
In Figure 1.2, local frames are defined at each vertex. Vectors
x
i
,
y
i
,and
z
i
are basis vectors of the local frame of vertex
i
in the current time step.
x
i
is simply defined as a normalized vector of
P
i
− P
i−
1
. As an exception,
x
0
=
(
P
1
− P
0
)
/ P
1
− P
0
. In Figure 1.3, basis vectors are shown in red, yellow, and
blue.
To describe the head transform, we use
w
T
H
, which transforms the head
from the rest state to the current state and is an input from user or predefined
animations.
y
i
+1
i
+ 1
x
i
+1
y
i
x
i
i
z
i
+1
z
i
x
i
-1
y
w
i
- 1
y
i
-1
z
i
-1
x
w
w
z
w
Figure 1.2.
Local frames in a hair strand.
