← ArchivePaper1987

Elastically Deformable Models

Demetri Terzopoulos, John Platt, Alan Barr, Kurt Fleischer

SIGGRAPHAcademic61 descendantsCFX Muscles

The paper that brought physically based elastic simulation to graphics, ancestor of all cloth and flesh solvers.

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Abstract

The theory of elasticity describes deformable materials such as rubber, cloth, paper, and flexible metals. We employ elasticity theory to construct differential equations that model the behavior of non-rigid curves, surfaces, and solids as a function of time. Elastically deformable models are active: they respond in a natural way to applied forces, constraints, ambient media, and impenetrable obstacles. The models are fundamentally dynamic and realistic animation is created by numerically solving their underlying differential equations. Thus, the description of shape and the description of motion are unified.

How to read this

Category: Method: physically based elastic deformation for graphics
Contributions: Differential equations from elasticity theory that model non-rigid curves, surfaces, and solids as a function of time
Active deformable models that respond naturally to forces, constraints, ambient media, and impenetrable obstacles
A unified, dynamic formulation where shape and motion are described together and animated by numerically solving the equations
Context: Standalone landmark work (no prior context given) that introduces continuum elasticity to graphics and is the ancestor of later cloth and flesh solvers.Builds on: The Synthesis of Cloth Objects
Correctness: Grounded in classical elasticity theory and demonstrated on deformable curves, surfaces, and solids; a reader should keep in mind the results depend on numerical integration choices and the era's compute, so stability and cost are practical concerns.
Clarity: Conceptually accessible but mathematically heavy; a first pass conveys the active-model idea, deeper passes are needed for the PDE and numerics.
How to read it: Read as the root of physics-based deformation: focus on the unification of shape and motion and on what makes the models active; budget a second and third pass for the elasticity equations and the numerical solution if you work in simulation.

Builds on

The Synthesis of Cloth Objects 1986

Built upon by

Related work

FEM Simulation of 3D Deformable Solids: A Practitioner's Guide to Theory, Discretization and Model Reduction 2012 / Course
Fine Wrinkling on Coarsely Meshed Thin Shells 2021 / TOG
Loki: A Unified Multiphysics Simulation Framework for Production 2022 / SIGGRAPH
The Synthesis of Cloth Objects 1986 / SIGGRAPH

Keywords

elastically deformable models elasticity theory physically based modeling deformation energy strain energy partial differential equations first fundamental form second fundamental form metric tensor curvature tensor finite difference discretization finite element method semi-implicit integration stiffness matrix collision force viscous force cloth deformable surfaces deformable solids

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