Computer graphics processing and selective visual display system – Computer graphics processing – Animation
Reexamination Certificate
2000-05-11
2003-06-03
Vo, Cliff N. (Department: 2671)
Computer graphics processing and selective visual display system
Computer graphics processing
Animation
Reexamination Certificate
active
06573897
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention generally relates to real-time, interactive animation of objects, and, more specifically, to real-time, interactive animation of two- and three-dimensional deformable objects.
2. Background
A mass-spring system is a useful way to model a deformable two- or three-dimensional body in an environment in which the body is subject to external forces. According to a technique known as explicit Euler integration, the forces on the mass points of the object at the beginning of a time period &Dgr;t are used to estimate the velocities and locations of the mass points at the end of the time period &Dgr;t. The explicit Euler integration step may be used to estimate the locations of the mass points of the object over successive time periods. Deformation of the object resulting from the external forces can be animated by displaying the object as represented by these updated mass points over successive time periods.
A problem with the explicit Euler technique, however, is that, in order to achieve stability, the duration of the time period over which the explicit integration is performed should be inversely proportional to the stiffness of the mass-spring system used to model the object. Since most real-world objects will require relatively stiff springs for modeling purposes, the duration of the time period over which the integration must be performed will be relatively small. The smaller the time period over which the integration is performed, the slower the resulting animation of the object. The result is that, using the explicit Euler integration technique, the majority of real-world objects cannot be animated in real-time. Moreover, all known techniques which have been attempted to overcome this problem yield results which lack realism.
For further background on the subject, the reader is referred to “Interactive animation of structured deformable objects,” Mathieu Desbrun et al., Graphics Interface '99 Proceedings, pp. 1-8, June, 1999, Kingston, Ontario; “Interactive Animation of Cloth-like Objects for Virtual Reality,” Mathieu Desbrun et al., published in “Cloth Modeling and Animation,” A. K Peters, Ltd., Publisher, Donald H. House & David E. Breen, Editors, Kathryn Maier, Associate Editor, Apr. 30, 2000; and “Interactive Animation of Cloth-like Objects in Virtual Reality,” Mark Meyer et al., submitted for publication in Journal of Visualization and Computer Graphics. All of these papers or publications are hereby incorporated by reference herein as through set forth in full.
Accordingly, there is a need for a system of and method for animating two- and three-dimensional objects which overcomes one or more disadvantages of the prior art.
There is also a need for a system of and method for animating two- and three-dimensional objects in real-time while maintaining realistic animation quality.
A further need is for a system of and method for animating two- and three-dimensional objects which fulfills any of the foregoing needs, singly or in combination.
SUMMARY OF THE INVENTION
In accordance with the purpose of the invention as broadly described herein, there is provided a method for updating the locations and velocities of the mass points of a two- or three-dimensional object to reflect passage of a time period having a duration &Dgr;t. The object is represented by a mass-spring system comprising a plurality of mass points interconnected with springs. In one embodiment, the method comprises the steps of: performing a modified implicit integration step to determine new locations of the mass points of the object after a time period &Dgr;t, optionally correcting the new locations of the mass points of the object after the time period &Dgr;t in order to preserve angular momentum, and optionally further correcting the new locations of the mass points of the object after the time period &Dgr;t to correct for high stiffness or long time step conditions.
In one embodiment, the method further comprises displaying the object as represented by the corrected locations of the mass points, and successively repeating the foregoing steps over at least one successive time period.
In one configuration, the modified implicit integration step comprises the substeps of calculating the internal force on each mass point of the object at the beginning of the time period &Dgr;t due to the action of springs; adding an artificial viscosity term to each of these internal forces; filtering out high frequencies from the resultant forces; calculating new velocities for each of the mass points of the object using the filtered forces and any external force applied to the mass points; and calculating new positions for each of the mass points of the object from the updated velocities.
In one implementation, the modified implicit integration step is preceded by a precomputation step in which an n×n matrix H is precomputed, where n is the number of mass points in the object. The matrix H is defined such that H
ij
, for i≠j, is equal to k
ij
, where k
ij
is the spring constant for the spring between the ith and jth mass points of the object, and H
ij
is equal to
-
∑
j
≠
i
k
ij
.
This latter term is the negative of the sum of the spring constants for all mass points connected to mass point i with a spring.
Once the matrix H has been precomputed, the matrix W is precomputed in accordance with the following expression:
W
=
(
I
n
-
dt
2
m
H
)
-
1
where I
n
is the n×n identity matrix, dt is the duration time period over which the implicit integration is being performed, m is the mass of a single mass point, and H is the hessian matrix previously discussed.
The precomputation step in this implementation is next followed by a series of steps in which, for each mass point i, the internal force F
i
on the mass point due to the action of springs i is first determined, and then an artificial viscosity term is computed and added to this internal force. This series of steps can be represented by the following pseudo-code:
For each mass point i
F
i
=0
For each mass point j linked to mass point i with a spring
F
i
=
F
i
+
k
ij
(
&LeftBracketingBar;
x
i
-
x
j
&RightBracketingBar;
-
l
ij
0
)
x
i
-
x
j
&LeftBracketingBar;
x
i
-
x
j
&RightBracketingBar;
F
i
=F
i
+k
ij
dt
(
v
j
−v
i
)
In the foregoing, F
i
is the force on mass point i due to springs and viscosity, k
ij
is the spring constant for the spring between mass points i and j, x
i
is the vector representing the current location of mass point i, x
j
is the vector representing the current location of mass point j, l
ij
0
is the rest length of the spring between mass points i and j, dt is the duration of the time step over which the implied integration step is being performed, v
i
is the current velocity of mass point i, and v
j
is the current velocity of mass point j.
This series of steps is then followed in this implementation by a series of steps to filter out high frequencies from the force field determined in the previous series of steps, and to determine updated velocities and locations for each of the object mass points. This series of steps can be represented by the following pseudo-code:
For each mass point i
F
i
filtered
=
∑
j
F
j
W
ij
v
i
n
+
1
=
v
i
n
+
[
F
i
filtered
+
F
i
ext
]
dt
m
x
i
new
=x
i
+v
i
n+1
dt
In the foregoing, F
j
is the force on mass point j computed in the previous loop of pseudo-code, W
ij
is the element of matrix W located at the intersection of the ith row and jth column, F
i
ext
is the external force exerted on mass point i, m is the mass of a single mass point, v
i
n
is the velocity of mass point i at the inception of the time period of duration dt, x
i
is the location of mass point i at the inception of the time period of duration dt, v
i
n+1
is the updated velocity of a mass point, and x
i
new
is the updated location of the mass point.
The addition of an artificial viscosity term to the internal forces due
Barr Alan H.
Desbrun Mathieu
Meyer Mark
Schroeder Peter
Howrey Simon Arnold & White , LLP
Vo Cliff N.
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