Smooth constraints for Spline Variational ModelingJulien Lenoir(1), Laurent Grisoni(1),

Philippe Meseure(1,2), Yannick Rémion(3), Christophe Chaillou(1)

(1) Alcove Project - INRIA Futurs - LIFL - University of Lille (France)(2) SIC, University of Poitiers (France)

(3) LERI, University of Reims champagne-Ardenne (France)

Outlines Introduction, Previous work & Objectives

A continuous model

Smooth constraints

Results

Introduction Context of variational modelling

Geometrical constraints Energy minimization

Relation with physically based modelling Lagrange formalism (energy minimization) Static vs dynamic

What we call « smooth constraint » Example: sliding point constraint

Previous work [Welch and Witkin 92] Variational surface

modeling Lagrange formalism – static simulation Lagrange multipliers Ponctual constraints & global constraints

[Witkin et al 87] Multiple object definition: parametric, implicit Energy function to minimize

Only parametric: Fixed point, surface attachment… Parametric and Implicit: Floating attachment

Objectives Propose

Dynamic solution for variational modeling Class of smooth constraint for parametric object

[Terzopoulos and Qin 94] D-NURBS for sculpting[Remion et al. 99] Dynamic spline Lagrange dynamics formalism Lagrange multipliers Baumgarte stabilization scheme

Outlines Introduction, Previous work & Objectives

A continuous model

Smooth constraints

Results

A continuous model (1D) [Lenoir et al. 2002]

Geometry defined as a spline:

Apply Physical Properties Homogeneous mass : m Kinetic energy :

External forces : Deformation energy : E Gravity :

n

kkk sbtts

0

)()(),,( qqP

s

n

kkk dssbtmtK

2

0

)()(21),( qq

)()(/ sbFsQ ii PF

s

n

kk

yg dssbtqmgtE

k0

)()(),( q

qk=(qkx,qky,qkz) position of the control pointsbk are the spline base functionst is the time, s the parametric abscissa

Resolution Simulation using the Lagrange formalism

We obtain the following system:

where :

M is band, symmetric and constant over time

i

i

iqEQ

qK

dtd

)(

BA Μ

MM

M

000000

Μ dssbsbm jsiij )()(Μ

izyx q ),,( AAAA

Including constraints Let g be a constraint:

Baumgarte technique [Baumgarte 72]

Overall equation includes the Lagrange multipliers

Each scalar equation requires a Lagrange multiplier

0),,( qtsg

0122

g

tgt

g

EB

λA

0LLM T

EA Lwritten as

Outlines Introduction, Previous work & Objectives

A continuous model

Smooth constraints

Results

We transform the constraint equation:0qg ),,( ts

Smooth constraints

Authorizing s to depend on time0qg ),),(( tts

Baumgarte scheme:

=> s dynamics is needed=> s is considered as a new unknown:

A Free Variable

0122

g

tgt

g

A new equation is needed to control the value of s Principle of Virtual Power:

A constraint must not work, so we get [Remion03]:

Smooth constraints

0gλ s

.

Force which ensuresthe constraint

ss

Pg

P(s(t),t)=P0

λ

Example on a sliding point constraint

Smooth constraints The dynamic system becomes:

Resolution with decomposition of the accelerations[Remion03] : Tendancy (without constraints) Usual constraints correction Smooth constraints correction

Time consuming method

E

B

λ

A0

0000

sLL

LLM

s

Ts

T

Smooth constraints v2

Force which ensuresthe constraint

ss

Pg

P(s(t),t)=P0

λ

0gλ s

.Normalcase :

ss

Pg

0gλ s

.

P(s(t),t)=P0

λ

Generalcase :

Example of sliding point constraint:

We simplify the equation by allowing the constraint to “work”. It enforces s to reach the solution: 0gλ

s

s ..

The overall system becomes:

Resolution by decomposition :

Same complexity but less stage (50%) of computation

E

B

λ

A0

00

0s

LLLLM

s

Ts

T

ct AAA

tsc

Ts

Tc

t

LsLLLsLM

M

AEAλλABA

tTss

T

Ts

Tc

t

LLLLLMLsLM

M

AEλλλABA

111

1

1

1

Smooth constraint v2

Smooth constraint v2 Examples of smooth constraint:

sliding point sliding tangent sliding curvature

Possibility to define multiple constraints relative to one free variable Example: Sliding point constraint with tangent control

Sliding point constrained to a point links to an object

Results Correct re-parametrization of the spline:

Results A shoelace:

Results A hang rope:

Results Sliding point constraint on a 2D spline:

Conclusion & Perspectives Proposition of smooth constraint class

sliding point constraint sliding tangent constraint sliding curvature constraint

Dynamic simulation => control of the end user

Correct re-parametrization of the curve

Use to introduce local friction