159 research outputs found
Knot Tightening By Constrained Gradient Descent
We present new computations of approximately length-minimizing polygons with
fixed thickness. These curves model the centerlines of "tight" knotted tubes
with minimal length and fixed circular cross-section. Our curves approximately
minimize the ropelength (or quotient of length and thickness) for polygons in
their knot types. While previous authors have minimized ropelength for polygons
using simulated annealing, the new idea in our code is to minimize length over
the set of polygons of thickness at least one using a version of constrained
gradient descent.
We rewrite the problem in terms of minimizing the length of the polygon
subject to an infinite family of differentiable constraint functions. We prove
that the polyhedral cone of variations of a polygon of thickness one which do
not decrease thickness to first order is finitely generated, and give an
explicit set of generators. Using this cone we give a first-order minimization
procedure and a Karush-Kuhn-Tucker criterion for polygonal ropelength
criticality.
Our main numerical contribution is a set of 379 almost-critical prime knots
and links, covering all prime knots with no more than 10 crossings and all
prime links with no more than 9 crossings. For links, these are the first
published ropelength figures, and for knots they improve on existing figures.
We give new maps of the self-contacts of these knots and links, and discover
some highly symmetric tight knots with particularly simple looking self-contact
maps.Comment: 45 pages, 16 figures, includes table of data with upper bounds on
ropelength for all prime knots with no more than 10 crossings and all prime
links with no more than 9 crossing
The Shapes of Tight Composite Knots
We present new computations of tight shapes obtained using the constrained
gradient descent code RIDGERUNNER for 544 composite knots with 12 and fewer
crossings, expanding our dataset to 943 knots and links. We use the new data
set to analyze two outstanding conjectures about tight knots, namely that the
ropelengths of composite knots are at least 4\pi-4 less than the sums of the
prime factors and that the writhes of composite knots are the sums of the
writhes of the prime factors.Comment: Summary text file of tight knot lengths and writhing numbers stored
in anc/ropelength_data.txt. All other data freely available at
http:://www.jasoncantarella.com/ and through Data Conservanc
Symmetric Criticality for Tight Knots
We prove a version of symmetric criticality for ropelength-critical knots.
Our theorem implies that a knot or link with a symmetric representative has a
ropelength-critical configuration with the same symmetry. We use this to
construct new examples of ropelength critical configurations for knots and
links which are different from the ropelength minima for these knot and link
types.Comment: This version adds references, and most importantly an
acknowledgements section which should have been in the original postin
Fast GPU-Based Two-Way Continuous Collision Handling
Step-and-project is a popular way to simulate non-penetrated deformable
bodies in physically-based animation. First integrating the system in time
regardless of contacts and post resolving potential intersections practically
strike a good balance between plausibility and efficiency. However, existing
methods could be defective and unsafe when the time step is large, taking risks
of failures or demands of repetitive collision testing and resolving that
severely degrade performance. In this paper, we propose a novel two-way method
for fast and reliable continuous collision handling. Our method launches the
optimization at both ends of the intermediate time-integrated state and the
previous intersection-free state, progressively generating a piecewise-linear
path and finally reaching a feasible solution for the next time step.
Technically, our method interleaves between a forward step and a backward step
at a low cost, until the result is conditionally converged. Due to a set of
unified volume-based contact constraints, our method can flexibly and reliably
handle a variety of codimensional deformable bodies, including volumetric
bodies, cloth, hair and sand. The experiments show that our method is safe,
robust, physically faithful and numerically efficient, especially suitable for
large deformations or large time steps
- …