Polyhedra Walter Whiteley July 2015 Start with spherical block and hole polyhedra Block Hole Expanding Expanding Contracting Contracting a b c d Recent Extension If triangulated sphere has one added crossbeam ID: 279030
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Slide1
Flat Faces in Block and Hole Polyhedra
Walter Whiteley
July 2015Slide2
Start with spherical block and hole polyhedra
Block
Hole
Expanding
Expanding
Contracting
Contracting
(a)
(b)
(c)
(d)Slide3
Recent Extension
If triangulated sphere has one added cross-beam
a
nd resulting graph is 4 connectedthen redundantly rigid (Wendy
Finbow-Singh, WW)Question is it generically globally rigid?Slide4
Flattening Extension
Ask that ‘faces’ are kept as triangulated
planes,
with natural
vertices, or vertices on natural edges.Specialized geometry – is this still ‘generically’ rigid?needs modification and extension of proof.
Can be done (with Wendy Finbow-Singh).Slide5
Two Operations
Add vertices as necessary along edges;
“selected” an
edge across faceSplit face along edge to create two faces. Need capacity to place faces on distinct planes without warping any of the other faces;Tool is the Steinitz sequence for (convex) spherical polyhedra
. Analyzed in: How to design or describe a polyhedron, J. of Intelligent and Robotic Systems 11 (1994), 135-160 Slide6
Face SplitSlide7
Comments:
This works for spherical
polyhedra
- 3 connected planar graphDoes not change the selection of blocks and holes and connectivity criteriaWhat about
toroidal polyhedra?
Is there a connectivity assumption that is sufficient? (e.g. 6 connected?)