1. Smooth constraints for Spline Variational Modeling Julien 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)

2. Outlines Introduction, Previous work & Objectives A continuous model Smooth constraints Results

3. 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

4. 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

5. 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

6. Outlines Introduction, Previous work & Objectives A continuous model Smooth constraints Results

7. 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 : q k =(q k x,q k y, q k z ) position of the control points b k are the spline base functions t is the time, s the parametric abscissa

8. Resolution Simulation using the Lagrange formalism We obtain the following system: where : M is band, symmetric and constant over time

9. Including constraints Let g be a constraint: Baumgarte technique [Baumgarte 72] Overall equation includes the Lagrange multipliers Each scalar equation requires a Lagrange multiplier written as

10. Outlines Introduction, Previous work & Objectives A continuous model Smooth constraints Results

11. We transform the constraint equation: Smooth constraints Authorizing s to depend on time Baumgarte scheme: => s dynamics is needed => s is considered as a new unknown: A Free Variable

12. 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 Force which ensures the constraint P(s(t),t)=P 0 Example on a sliding point constraint

13. 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

14. Smooth constraints v2 Force which ensures the constraint P(s(t),t)=P 0 Normal case : P(s(t),t)=P 0 General case : Example of sliding point constraint: We simplify the equation by allowing the constraint to “work”. It enforces s to reach the solution:

15. The overall system becomes: Resolution by decomposition : Same complexity but less stage (50%) of computation Smooth constraint v2

16. 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

17. Results Correct re-parametrization of the spline:

18. Results A shoelace:

19. Results A hang rope:

20. Results Sliding point constraint on a 2D spline:

21. 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