Deformable Models
Marc Beck
CS525 2005
Physical vs. Non Physical models
Graphics models are separated into two general categories, physical and non-physical.
Most of us are more familiar with non-physical models which consist of geometric
shapes, curves and splines. Physical models use physically based principles animate and
modeling. This allows modeling of objects that would be difficult or impossible to model
using simple geometric methods.
Non-Physical Modeling Non-physical models are computationally efficient, but are limited by the skill of the
model designer. These models can be interactively deformed by moving vertices, or
control points.
FFD Free form deformation is a method of deforming a model with a higher degree of
control and power than simple control point manipulation. FFD works by
changing the space in which the object lies. For more information about FFD
read A. Barr’s Global and local deformations of solid primitives.
(http://portal.acm.org/citation.cfm?id=808573)
Physical models While non-physical models are computationally fast and mentally intuitive they are
lacking in certain respects. The models do not naturally deform within their
environment, each deformation must be done explicitly. Also, to make a complex model,
such as a human face, or an ocean wave, the user would have to be very talented in order
to deform the model so it looked natural as it was animated. These problems are
addressed by using physically based models
Discrete Physical models Discrete models break down the continuous nature of the model into discrete
points. This allows computation to be done quickly and easily. One such type of
discrete physical model is the mass spring system.
Mass Spring Systems
Mass spring systems consist of a collection of point masses which are connected
by springs. The springs are typically linear springs, making the math easier, but
other kinds of springs are perfectly acceptable as well.
In this diagram (taken from A Survey of Deformable Modeling in
Computer Graphics) point masses, m, are connected by springs with a spring
constant k. Each point mass therefore applies a force to a connected spring
relative to the amount of deformation and the spring constant of the spring
connecting them. Because k’s need not be constant within the model, mass spring
systems have a large flexibility in rigidity and deformability.
Mass spring systems have been very successfully used in modeling human skin
and facial animations. Spring systems are used with different spring coefficients
to model different layers of the face, such as the dermis, fatty tissue and muscle.
Forces are then applied to the muscles to cause reactions within the other layers to
deform the face. For more information about using spring systems to model facial
animations read Physically-based facial modeling, analysis, and animation by
Demetri Terzopoulos and K. Waters.
One drawback of the spring system are that it is an approximation of the
continuous physics in a continuous model. Another problem is that the models
are tweaked through defining of the interconnecting spring constants, which can
be difficult to derive.
Continuum Physical Models
Continuum physical models are solid bodies with mass and energies distributed
throughout. However computational methods of solving on a computer are
discrete. Therefore Continuum models must have a way of being discretized.
We will look at one continuous physical model using discretized deformation
energy. Objects are defined with a natural shape. Deformation of the object is a
function of the total elastic strain on the object. This makes ridged body motions
(which produce no strain on an object) have no deformation. The larger the total
strain on an object the higher its energy of deformation.
The strain on a curve, surface, or 3d object is calculated using elasticity theory,
and as the math is very complicated it will be left as an exercise.
Hint: http://portal.acm.org/citation.cfm?id=37427
Appling forces to these models yields realistic physical results.
There are also 2 constants in these models. A, which determines surface tensions.
With a positive A the surface wants to shrink into itself, with a negative A the
surface would like to grow. B controls the surface rigidity. With a positive B the
surface wants to be flatter, with a negative B the surface would like to be more
curved. To simulate a stretchy sheet of rubber, A would be quite low, to allow the
sheet to stretch, and B would be zero because the rubber has no surface rigidity.
Paper would have a relatively high A (as it does not stretch) and a moderate, non-
zero B.
Because these factors do not have to uniform over the model we can simulate
cracks or surface deviations on a model.
The lifting of elastic surfaces
Solid ball on a deformable cube
A rubber membrane shrinking around a jack
Flag using fluid flow as wind
Persian carpet falling over a static sphere and cylinder
References
Elastically Deformable Models: Demetri Terzopoulost, John Platt, Alan Barr,
Kurt Fleischert
http://portal.acm.org/citation.cfm?id=37427
A Survey of Deformable Modeling in Computer Graphics:
Sarah F. F. Gibson, Brian Mirtich
http://www.merl.com/publications/TR1997-019/
http://www.openfx.org/gallery/animations/
http://www.ugcs.caltech.edu/~rona/cloth/
http://www.cs.toronto.edu/~dt/
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