Hyper-Dimensional Deformation Simulation

Alvin Shi, Haomiao Wu, and Theodore Kim

Two 4D armadillo prisms brush past each other. In the top left, a 3D slice where the armadillos are most visible. Bottom left, a different 3D slice reveals the extrusion direction. In the middle are two stills from a four-dimensional rotation about their collision at frame 800. On the right, they have failed to hug and instead have passed through each other.

Abstract

We present a method for simulating deformable bodies in four spatial dimensions. To accomplish this, we generalize several pieces of the traditional simulation pipeline. Starting from the meshing stage, we propose a simple method for generating a pentachoral mesh, the 4D analog of a tetrahedral mesh. Next, we show how to generalize the deformation invariants, allowing us to construct 4D hyperelastic energies that lead directly to hyper-dimensional deformation forces. Finally, we formulate collision detection and response in 4D. Our eigenanalyses of the resulting deformation and collision energies generalize to arbitrarily higher dimensions. The resulting simulations display a variety of previously unseen visual phenomena.

Supplement Video (YouTube) PDF(Compressed) Supplement(Compressed) Coding Resources

Raw Videos

Bunny Rotations - a bunny rotates on several planes
Bunny Rotation,
Armadillo Hugging - two armadillo cylinders brush past eachother
Hug Phasing, Hug Cylinder View, Frame 800 4D Turnaround,
Twist Compression - various cylinders get twisted on out-of-slice caps
Alien Twist, Alien Twist Cylinder View, Octopus Twist, Octopus Twist Cylinder View,
Thin/Thick Wringing - wringing a thin and thick octopus cylinder yields different visual results
Thin Wring, Thin Wring Cylinder View, Thick Wring, Thick Wring Cylinder View,
4D Noodles - spherical cylinders fall into a hyperspherical bowl with different deformation energies
SNH, SNH(Alternate View), SNH(Slice Slide), ARAP, STVK(Inverts),
Cantilevers - cantilever beams of different stiffness parameters droop and phase through the viewing slice
E275, E550, E1010, E2000, E4000,

Acknowledgemenets

This work was supported by The Teng and Han Family Fund and NSF IIS-2132280. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.