SIGGRAPH 2020 Technical Paper: n-Dimensional Rigid Body Dynamics

# SIGGRAPH 2020 Technical Paper: n-Dimensional Rigid Body Dynamics

Jan 06, 2021
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Abstract: I present a formulation for Rigid Body Dynamics that is independent of the dimension of the space. I describe the state and equations of motion of rigid bodies using geometric algebra. Using collision detection algorithms extended to nD I resolve collisions and contact between bodies. My implementation is 4D, but the techniques described here apply to any number of dimensions. I display these four-dimensional rigid bodies by taking a three-dimensional slice through them. I allow the user to manipulate these bodies in real-time. Author: Marc ten Bosch (mtb design works, Inc., USA)

00:00 Introduction 01:15 Contributions 01:49 4D Toys Trailer 06:57 Background: Geometric Algebra 07:34 Background: Rigid Body Motion 07:49 Background: Angular Momentum 07:59 Background: Bivectors 08:55 Background: Rotors 10:05 Background: Instantaneous Velocity 10:25 Background: Angular Momentum 10:48 Background: Euler's Equation 10:59 n-Dimensional Generalization: Geometric Algebra Operator Matrices 11:48 n-Dimensional Generalization: Inertia Tensor 12:08 n-Dimensional Generalization: Operators and Inertia Tensor 12:39 n-Dimensional Generalization: Correcting Rotor Errors 13:18 Inertia Tensor of arbitrary nD mesh 13:34 Inertia Tensor of arbitrary nD mesh: Inertia Tensor is Linear 13:46 Inertia Tensor of arbitrary nD mesh: Compute Covariance Matrix 14:05 Inertia Tensor of arbitrary nD mesh: Transform Covariance into Inertia 14:30 Inertia Tensor of arbitrary nD mesh: Transforming Inertia Tensor 14:55 Inertia Tensor of arbitrary nD mesh: Diagonalizing Inertia Tensor 15:17 Collision Resolution / Detection 15:32 Collision Resolution / Detection: Hypersphere / Polytope 16:14 Collision Resolution / Detection: Polytope / Polytope 17:03 Dzhanibekov Effect: 3D Dzhanibekov Effect 17:27 Dzhanibekov Effect: 4D Dzhanibekov Effect 17:54 Dzhanibekov Effect: Reduces to 3D case 18:12 Dzhanibekov Effect: Stable states 18:30 Dzhanibekov Effect: Additional Oscillations 18:44 Future Work