1. Elastically Deformable Models
Demetri Terzopoulos
John Platt
Kurt Fleischer
1987

2. Outline Dynamics of Deformable Models Energies of Deformation
Applied Forces
Implementation of Deformable Models
Simulation Examples

3. Dynamics of Deformable Models
Lagrange’s form:
a: a point in a body
r(a,t): position of a at time t
μ(a): mass density of the body at a
γ(a): damping density of the body at a
ε(r): a function, potential energy of deformation
external force
Inertial force
damping force
elastic force

4. Energies of Deformation
Develop potential energies of deformation ε(r) associated with the elastically deformable models.
Analysis of Deformation
Energies for Curves, Surfaces, and Solids

5. Analysis of Deformation
Distance between two point in Euclidean 3-space:
Metric tensor G:
Curvature tensor B:
For space curve:
Arc length: s(r(a)) Curvature: κ(r(a)) Torsion: τ(r)
n: unit surface normal

6. Energies for Curves, Surfaces, and Solids
resistance: α-streching, β-bending , γ-twisting
Matrix norm
!! For rigid motion, ε(r) = 0.

7. Applied Forces Gravitational force: Spring force:
Force on the surface of body:
Net external force:
f(r,t) = fgravity + fspring + fviscous + fcollision
g: gravitational field
k: spring constant
c: strength of the fluid force n(a): unit normal on the surface
v(a,t): velocity of the surface relative to some constant stream velocity

8. Implementation of Deformable Models
A Simplified Elastic Force
Discretization
Numerical Integration Through Time

9. A Simplified Elastic Force
Simplified deformation energy for a surface: 7
first variational derivative
, are weighting function.
: tension
, : resistance
: rigidity

10. Discretization1/2 continuous → discrete
Forward first differnece oprators:
Backward first differnece oprators:
Forward and backward cross difference operators:
Central second difference operators:

11. Discretization2/2
elastic force:
discrete form equations (1):

12. Numerical Integration Through Time
t = 0 to t = T is subdivided into equal time steps △t

13. Simulation Examples1
Two different static behaviors of an elastic surface.
a: simulates a thin plate. ( = 0, = positive constant)
b: simulates a membrane. ( > 0, = 0)
(a)
(b)

14. Simulation Examples2 A ball resting on a supporting elastic solid.
The solid has a metric tensor.
The internal elastic force interacts with the collision force to deform the solid.

15. Simulation Examples3 A shrink wrap effect. a: a model of a rigid jack.
b: a spherical membrane is stretched to surround the jack.
(a)
(b)

16. Simulation Examples4
Simulation of a flag waving in the wind.

17. Simulation Examples5
Simulation of a carpet falling onto two rigid bodies in a gravitational field.
Modeled as a membrane. ( = 0, = positive constant)
The carpets slides off the bodies due to the interaction between gravity and repulsive collision force.