Graphics Reference
In-Depth Information
each of the columns of
R
(
t
)(
Eq. 7.23
). By defining a special matrix to represent cross-products
(
Eq.7.24
), one can represent
Equation 7.23
by matrix multiplication (
Eq. 7.25
)
.
RðÞ¼ R
1
½
ðÞ R
2
ðÞ R
3
ðÞ
(7.23)
_
RðÞ¼ o R
1
½
ðÞ o R
2
ðÞ o R
3
ðÞ
2
4
3
5
¼
2
4
3
5
2
4
3
5
A
B
A
y
B
z
A
z
B
y
A
z
B
x
A
x
B
z
A
x
B
y
A
y
B
x
0
A
z
A
y
B
x
B
y
B
z
A B ¼
A
z
0
A
x
(7.24)
A
y
A
x
0
_
RðÞ¼o ðÞ
RðÞ
(7.25)
Consider a point,
Q
, on a rigid object. Its position in the local coordinate space of the object is
q
;
q
does
not change. Its position in world space,
q
(
t
), is given by
Equation 7.26
.
The position of the body in space is
given by
x
(
t
), and the orientation of the body is given by
R
(
t
). The velocity of the particle is given by
differentiating
Equation 7.26
. The relative position of the particle in the rigid body is constant. The change
in orientation is given by
Equation 7.25
,
while the change in position is represented by a velocity vector.
These are combined to produce
Equation 7.27
.
The world space position of the point
Q
in the object, taking
into consideration the object's orientation, is given by rearranging
Equation 7.26
to give
R
(
t
)
q ¼ q
(
t
)
x
(
t
).
Substituting this into
Equation 7.27
and distributing the cross-product produces
Equation 7.28
.
qðÞ¼RðÞq þ xðÞ
(7.26)
q_ ðÞ¼o ðÞ
RðÞq þ vðÞ
(7.27)
q_ ðÞ¼o ðÞðqðÞxðÞÞþvðÞ
(7.28)
Center of mass
The center of mass of an object is defined by the integration of the differential mass times its position in
the object. In computer graphics, the mass distribution of an object is typically modeled by individual
points, which is usually implemented by assigning a mass value to each of the object's vertices. If the
individual masses are given by m
i
, then the total mass of the object is represented by
Equation 7.29
.
Using an object coordinate system that is located at the center of mass is often useful when modeling
the dynamics of a rigid object. For the current discussion, it is assumed that x(t) is the center of mass of
an object. If the location of each mass point in world space is given by
q
i
(
t
), then the center of mass is
represented by
Equation 7.30
.
M ¼ Sm
i
(7.29)
Sm
i
q
i
ðÞ
x
ðÞ¼
(7.30)
M
Forces
A linear force (a force along a straight line),
, applied to a mass,
m
, gives rise to a linear acceleration, a,
by means of the relationship shown in
Equations 7.31 and 7.32
. This fact provides a way to calculate
acceleration from the application of forces. Examples of such forces are gravity, viscosity, friction,
impulse forces due to collisions, and forces due to spring attachments. See
Appendix B.7
for the basic
equations from physics that give rise to such forces.
f
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