Graphics Reference
In-Depth Information
A spring is a common tool for modeling flexible objects, to keep two objects at a prescribed dis-
tance, or to insert temporary control forces into an environment. Notice that a spring may correspond to
a geometric element, such as a thread of a cloth, or it may be defined simply to maintain a desired
relationship between two points (sometimes called a
virtual spring
), such as a distance between the
centers of two spheres. When attached to an object, a spring imparts a force on that object depending
on the object's location relative to the other end of the spring. The magnitude of the spring force (
f
s
)is
proportional to the difference between the spring's rest length (
L
r
) and its current length (
L
c
) and is in
the direction of the spring. The constant of proportionality (
k
s
), also called the
spring constant
, deter-
mines how much the spring reacts to a change in length—its
stiffness
(see
Eq. 7.7
)
. The force is applied
to both ends of the spring in opposite directions.
p
2
p
1
kp
2
p
1
k
f
s
¼k
s
ðL
c
L
r
Þ
(7.7)
The spring described so far is a
linear spring
because there is a linear relationship between how
much the length of the spring deviates from the rest length and the resulting force. In some cases,
it is useful to use something other than a linear spring. A
biphasic spring
is a spring that is essentially
a linear spring that changes its spring constant at a certain length. Usually a biphasic spring is used to
make the spring stiffer after it has stretched past some threshold and has found uses, for example, in
facial animation and cloth animation. Other types of
nonlinear springs
have spring “constants” that are
more complex functions of spring length.
A
damper
, like a spring, is attached to two points. However, the damper works against its relative
velocity. The force of a damper (
f
d
) is negatively proportional to, and in the direction of, the velocity of
the spring length (
v
s
). The constant of proportionality (
k
d
), or damper constant, determines how much
resistance there is to a change in spring length. So the force at p
2
is computed as in
Equation 7.8
. The
force on p
1
would be the negative of the force on p
2
.
p
2
p
1
kp
2
p
1
k
p
2
p
1
kp
2
p
1
k
f
d
¼k
d
p
2
p
1
(7.8)
Viscosity
, similar to a damper, resists an object traveling at a velocity. However, viscosity is resist-
ing movement through a medium, such as air. A force due to viscosity (
f
v
) is negatively proportional to
f
v
¼k
v
v
(7.9)
velocity. In a closed system (i.e., a system in which energy is not changing),
total momentum is conserved. This can be used to solve for unknowns in the system. For a system
consisting of multiple objects in which there are no externally applied forces, the total momentum
is constant (
Eq. 7.10
).
Momentum
is mass
X
m
i
v
i
¼
c
(7.10)
The rotational equivalent to force is
torque
. Think of torque,
t
, as rotational force. Analogous to (linear)
velocity and (linear) acceleration are
angular velocity
,
o
,and
angular acceleration
,
a
. The mass of an
object is a measure of its resistance to movement. Analogously, the
moment of inertia
is a measure of
an object's resistance to change in orientation. The moment of inertia of an object is a 3
3 matrix
of values,
I
, describing an object's distribution of mass about its center of mass. The relationship to torque,
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