Kevin Walker’s post of thesis says error in proof of 3.3, see Hausmann,

sur la topologie des bras articules ’89 (I have a photocopy)

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https://www.researchgate.net/publication/

265130442_Computational_Methods_for_Coordinating_Multiple_Construction_Cranes/figures?lo=1

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https://www.geometrictools.com/index.html by David Eberly

https://www.geometrictools.com/Documentation/DistanceLine3Line3.pdf

Let the endpoints of the first and second segments be P0 and P1, and Q0 and Q1. The segments can be parameterized by s and t along line. The squared distance between two points on the segments is the quadratic function $R(s, t) = as^2-2bst+ct^2+2ds-2et+f$ where a and c are squared lengths of segments. For nondegenerate segments, a > 0 and c > 0. note $ac-b^2\geq 0$. the implementation will allow point-segment pair or point-point pair. The segments are not parallel when ac – b^2 > 0, so that R(s,t) is a paraboloid and the level sets are ellipses. The segments are parallel when ac-b^2 = 0, in which case the graph of R(s,t) is a parabolic cylinder and the level sets are lines. define = ac – b^2. In calculus terms, the goal is to minimize R(s; t) over the unit square [0; 1]^2. Because R is a continuously differentiable function, the minimum occurs either at an interior point of the square where the gradient is 0 or at a point on the boundary of the square. There are nine candidates for the minimum: the four corners, four edge points, and one interior point. A simple algorithm is to compute all 9 critical points, the last one only when ac-b^2 > 0, evaluate R at those points, and select the one that produces the minimum squared distance. However, this is a slow algorithm. A smarter search for the correct critical point is called for.

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LaValle RobotPlan.pdf recommended by robert ghrist. Here’s the link! http://planning.cs.uiuc.edu/

p58 chains links 3.3

p66 tree links 3.4

p67 loops ends on p68

p81 configuation spaces, 4.2

p86 config.obstacles

p92 4.4

p113 collisions

p164 chap.7

p177 planning closed chains 7.4

p182 folding problems 7.5

p183 knots

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http://www.colab.sfu.ca/KnotPlot/sticknumbers/

http://www.knotplot.com/cat/ms-symm.html

http://www.knotplot.com/thesis/ Interactive Topological Drawing, Robert Glenn Scharein

https://mathoverflow.net/questions/79488/is-there-a-knot-theory-for-graphs

https://math.stackexchange.com/questions/311431/knots-and-graphs

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ribbon diagrams, conceived by Jane S. Richardson for 1980 article by Jane S. Richardson, at Duke,

The Anatomy and Taxonomy of Protein Structure,in Advances in Protein Chemistry. Influenced by

earlier individual illustrations, her hand-drawn ribbon diagrams were the first schematics of 3D

protein structure to be produced systematically to illustrate a classification of structures.