Hello,

I want to compute Alpha shapes of weighted points in 2D (and later 3D).

When I compare my own implementation with CGAL, I get very different

results for the alpha values of the simplices.

I tried this basic example:

point 0: 4 5.7 weight 16

point 1: 2.5 5 weight 4

point 2: 3 3

point 3: 6 4 weight -9

point 4: 8 3

We get the triangles 012 and 024, which is fine.

They should have squared radius 6.06 and 11.79, but CGAL gives me 2.15

and 6.62. I don't know where these numbers come from.

Just to be clear, my definition for the weighted Alpha shape is the

following:

A simplex is contained in the Alpha shape of radius r if the smallest

sphere orthogonal to all its points (AB orthogonal if

d(A,B)^2-weight_A-weight_B=0) has weight <= r^2.

For triangle 012 I compute

solve (4 - x)^2 + (5.7 - y)^2 - 16 - w = 0, (2.5 - x)^2 + (5 - y)^2 - 4

- w = 0, (3 - x)^2 + (3 - y)^2 - 0 - w = 0

to get the center (x,y) and weight/radius^2 w of this weighted

circumsphere, which indeed gives w=6.06...

Why is the alpha value in CGAL 2.15? The correct value does not even

appear in the list of alpha values (which are 0.685 1.0625 2.15052

5.8225 6.25 6.6212).

Did I understand something wrong about how to use alpha shapes in CGAL?

Thanks for your help,

Kathi

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