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Curved Paths
Piecewise circular curves are used in manufacturing, robotics, and highway engineering, but I haven’t found many online references for them. As with circular arcs, piecewise circular curves can handle offsets, distances, and interpolation. Here are some papers I used to learn about circular arcs, biarcs, and piecewise circular curves:
game-design  mathematics  constraint-satisfaction  engineering-design  rather-interesting  aesthetics  the-mangle-in-practice  representation  plane-geometry  parametrization 
yesterday by Vaguery
🙆‍♀️ *Excitement in non-binary*
representation  from twitter_favs
7 days ago by daisyk
[1704.01166] Regenerative random permutations of integers
Motivated by recent studies of large Mallows(q) permutations, we propose a class of random permutations of ℕ+ and of ℤ, called regenerative permutations. Many previous results of the limiting Mallows(q) permutations are recovered and extended. Three special examples: blocked permutations, p-shifted permutations and p-biased permutations are studied.
combinatorics  permutations  infinite-things  dynamical-systems  probability-theory  to-understand  representation 
13 days ago by Vaguery
This Australian fashion model is in high demand. She also has Down syndrome.
Free, C. (2019, February 11). his Australian fashion model is in high demand. She also has Down syndrome. The Washington Post [MSN].
Disability  representation 
14 days ago by maw
[1805.02222] A Computer Approach to Determine the Densest Translative Tetrahedron Packings
In 1900, as a part of his 18th problem, Hilbert proposed the question to determine the densest congruent (or translative) packings of a given solid, such as the unit ball or the regular tetrahedron of unit edges. Up to now, our knowledge about this problem is still very limited, excepting the ball case. It is conjectured that, for some particular solids such as tetrahedra, cuboctahedra and octahedra, their maximal translative packing densities and their maximal lattice packing densities are identical. To attack this conjecture, this paper suggests a computer approach to determine the maximal local translative packing density of a given polytope, by studying associated color graphs and applying optimization. In particular, all the tetrahedral case, the cuboctahedral case and the octahedral case of the conjecture have been reduced into finite numbers of manageable optimization problems.
packing  open-questions  geometry  looking-to-see  representation  feature-construction  to-write-about 
16 days ago by Vaguery
[1707.06708] Local-global principles in circle packings
We generalize work of Bourgain-Kontorovich and Zhang, proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group ≤PSL2(K) satisfying certain conditions, where K is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof of this is that  possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in PSL2(K) containing a Zariski dense subgroup of PSL2(ℤ).
plane-geometry  number-theory  purdy-pitchers  packing  to-understand  representation  to-write-about 
17 days ago by Vaguery

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