Smooth constraints for Spline Variational Modeling

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)

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Outlines

• Introduction, Previous work Objectives
• A continuous model
• Smooth constraints
• Results

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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

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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

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Objectives

• Propose
• Dynamic solution for variational modeling
• Class of smooth constraint for parametric object
• Terzopoulos and Qin 94 D-NURBS for
  sculptingRemion et al. 99 Dynamic spline
• Lagrange dynamics formalism
• Lagrange multipliers
• Baumgarte stabilization scheme

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Outlines

• Introduction, Previous work Objectives
• A continuous model
• Smooth constraints
• Results

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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

qk(qkx,qky,qkz) position of the control
points bk are the spline base functions t is the
time, s the parametric abscissa
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Resolution

• Simulation using the Lagrange formalism
• We obtain the following system
• where M is band, symmetric and constant over
  time

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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
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Outlines

• Introduction, Previous work Objectives
• A continuous model
• Smooth constraints
• Results

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Smooth constraints
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Smooth constraints

• A new equation is needed to control the value of
  s
• Principle of Virtual PowerA constraint must not
  work, so we get Remion03

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Smooth constraints

• The dynamic system becomes
• Resolution with decomposition of the
  accelerationsRemion03
• Tendancy (without constraints)
• Usual constraints correction
• Smooth constraints correction
• Time consuming method

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Smooth constraints v2

• Example of sliding point constraint

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Smooth constraint v2

• The overall system becomes
• Resolution by decomposition
• Same complexity but less stage (50) of
  computation

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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

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Results

• Correct re-parametrization of the spline

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Results

• A shoelace

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Results

• A hang rope

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Results

• Sliding point constraint on a 2D spline

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Conclusion Perspectives

• Proposition of smooth constraint class
• sliding point constraint
• sliding tangent constraint
• sliding curvature constraint
• Dynamic simulation gt control of the end user
• Correct re-parametrization of the curve
• Use to introduce local friction