Graphics Reference
In-Depth Information
z
i
1
z
i
z
i
1
z
i
x
i
x
i
origin
FIGURE B.32
Determining the origin and
x
-axis of the
i
th frame.
chosen as the intersection of the
x
-axis of the previous frame and the joint axis when the displacement
is zero.
The following procedure can be used to construct the frames for intermediate joints.
1.
For each joint, identify the axis of rotation for revolute joints and the axis of displacement for
prismatic joints. Refer to this axis as the
z
-axis of the joint's frame.
2.
For each adjacent pair of joints, the
i
th
1 and
i
th for
i
from 1 to
n
, construct the common
perpendicular between the
z
-axes or, if they intersect, the perpendicular to the plane that contains
them. Refer to the intersection of the perpendicular and the
i
th frame's
z
-axis (or the point of
intersection of the two axes) as the origin of the
i
th frame. Refer to the perpendicular as the
x
-axis of
3.
Construct the
y
-axis of each frame to be consistent with the right-hand rule (assuming right-hand
space).
B.5
Interpolating and approximating curves
This section covers many of the basic terms and concepts needed to interpolate values in computer
animation. It is not a complete treatise of curves but an overview of the important ones. While many
of the terms and concepts discussed are applicable to functions in general, they are presented as they
relate to functions having to do with the practical interpolation of points in Euclidean space as typically
used in computer animation applications. For more complete discussions of the topics contained
here, see, for example, Mortenson [
14
], Rogers and Adams [
18
]
, Farin [
4
], and Bartels, Beatty, and
Barsky [
1
].
B.5.1
Equations: some basic terms
For present purposes, there are three types of equations:
explicit
,
implicit
, and
parametric
.
Explicit
equations
are of the form
y ΒΌ f
(
x
). The explicit form is good for generating points because it generates
a value of
y
for any value of
x
put into the function. The drawback of the explicit form is that it is
dependent on the choice of coordinate axes, and it is ambiguous if there is more than one
y
for a given
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