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Elastically Deformable Models Demetri Terzopoulos John Platt Kurt Fleischer 1987 1 Outline Dynamics of Deformable Models Energies of Deformation Applied Forces Implementation of Deformable Models Simulation Examples 2 Dynamics of Deformable Models Lagrange’s form: Inertial force external force damping force elastic force 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 3 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 4 Analysis of Deformation Distance between two point in Euclidean 3-space: Metric tensor G: Curvature tensor B: n: unit surface normal For space curve: Arc length: s(r(a)) Curvature: κ(r(a)) Torsion: τ(r) 5 Energies for Curves, Surfaces, and Solids Curve: resistance: α-streching, β-bending , γ-twisting Surface: Matrix norm Solid: !! For rigid motion, ε(r) = 0. 6 Applied Forces Gravitational force: g: gravitational field Spring force: k: spring constant Force on the surface of body: 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 Net external force: f(r,t) = fgravity + fspring + fviscous + fcollision 7 Implementation of Deformable Models A Simplified Elastic Force Discretization Numerical Integration Through Time 8 A Simplified Elastic Force Simplified deformation energy for a surface: first variational derivative ij , ij δεr δr 7 are weighting function. δε(r) δr ij : tension ij : rigidity ij, ij : resistance 9 Discretization 1/2 continuous → discrete Forward first differnece oprators: Backward first differnece oprators: Forward and backward cross difference operators: Central second difference operators: 10 Discretization 2/2 elastic force: discrete form equations (1): 11 Numerical Integration Through Time t = 0 to t = T is subdivided into equal time steps △t 12 Simulation Examples 1 Two different static behaviors of an elastic surface. a: simulates a thin plate. ( ij= 0, ij = positive constant) b: simulates a membrane. ( ij > 0, ij = 0) ij (a) (b) 13 Simulation Examples 2 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. 14 Simulation Examples 3 A shrink wrap effect. a: a model of a rigid jack. b: a spherical membrane is stretched to surround the jack. (a) (b) 15 Simulation Examples 4 Simulation of a flag waving in the wind. 16 Simulation Examples 5 Simulation of a carpet falling onto two rigid bodies in a gravitational field. Modeled as a membrane. ( ij= 0, ij = positive constant) The carpets slides off the bodies due to the interaction between gravity and repulsive collision force. 17