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/