8 research outputs found
Bounding Helly numbers via Betti numbers
We show that very weak topological assumptions are enough to ensure the
existence of a Helly-type theorem. More precisely, we show that for any
non-negative integers and there exists an integer such that
the following holds. If is a finite family of subsets of such that for any
and every
then has Helly number at most . Here
denotes the reduced -Betti numbers (with singular homology). These
topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number.
Our proofs combine homological non-embeddability results with a Ramsey-based
approach to build, given an arbitrary simplicial complex , some well-behaved
chain map .Comment: 29 pages, 8 figure
Development and validation of a generic 3D model of the distal femur
The development and validation of a virtual generic 3D model of the distal femur using computer graphical methods is presented. The synthesis of the generic model requires the following steps: acquisition of bony 3D morphology using standard CT (computed tomography) imaging; alignment of 3D models reconstructed from CT images with a common coordinate system; computer graphical sectioning of the models; extraction of bone contours from the image sections; combining and averaging of extracted contours; and 3D reconstruction of the averaged contours.\ud
The generic models reconstructed from the averaged contours of six cadaver femora were validated by comparing their surface geometry on a point to point basis with that of the CT reconstructed reference models. The mean errors ranged from 0.99 - 2.5 mm and were in agreement with the qualitative assessment of the models