Alpha Shapes Used for Shape Modelling Creates shapes out of point sets Gives a hierarchy of shapes. Has been used for detecting pockets in proteins. For reverse engineering Convexity A set S in Euclidean space is said to be convex if every straight line segment
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A set S in Euclidean space is said to be convex if every straight line segment
having its two end points in S lies entirely in S.
The smallest convex set that contains the entire point set.
This set is a convex polyhedra since it is an intersection
of half spaces. These polyhedra define a decomposition
of Rd. The voronoi complex V(P) of P is the collection
of all voronoi objects.
Delaunay complex is the dual of the voronoi complex.
Delaunay triangulations are simplicial complexes.
The space generated by point pairs that can be touched by an
empty disc of radius alpha.
Alpha Controls the desired level of detail.
(To get to General Position)
Lot of Algorithms available!!!
( I got confused )
“Creativity is the art of hiding Sources!”
“If the facts don't fit the theory, change the facts.”
Weighted Voronoi: Seems not so tough yet
Edelsbrunner: Union of balls and alpha shapes are
homotopy equivalent for all alpha.
Courtesy Dey, Giesen and John 04.
The Dual Complex: Assuming General position, at most
3 Voronoi Cells meet at a point.
For fixed weights, alpha, It’s a alpha complex!
The subset of delaunay tesselation in d-dimensions that has simplices having
Circumradius greater than Alpha.
It’s a Simplicial Complex all the way
( for a topologist )
A Filter!!!! (an order on the simplices)
A Filtration??? (sequence of complexes)
As the Balls grow(Alpha becomes bigger) on the input point set, the dual marches thru the Filteration, defining a set of shapes.
That’s it!! Wasn’t it a cute idea for 1983!
(And get papers accepted too, That’s tough)