Connection among the vertices on a plane or hyperplane

[Asad Akhlaq]

Hi,

I am solving a problem where I want know that whether certain vertices on a hyperplane are connected to each other or not (i.e there exists an edge between two vertices). Say in a 2D plane, if I have 3 or more vertices (points), how can I decide that if all or some of these vertices are connected to each other? Similar is the problem for a hyperplane. Do we have any function/class is sage that can do it? I heard about convex hull algorithms but I don't know how to apply them to this problem. OR can we use graphs to solve this problem?

Any help/hint in this regard will be highly appreciated.

Thanks

[Anton Sherwood]

On 2013-3-04 22:21, Asad Akhlaq wrote:
> I am solving a problem where I want know that whether certain
> vertices on a hyperplane are connected to each other or not (i.e
> there exists an edge between two vertices). Say in a 2D plane, if I
> have 3 or more vertices (points), how can I decide that if all or
> some of these vertices are connected to each other? Similar is the
> problem for a hyperplane. Do we have any function/class is sage that
> can do it? I heard about convex hull algorithms but I don't know how
> to apply them to this problem. OR can we use graphs to solve this
> problem?

Vertices and edges of what? Seems to me you need either a graph or a
convex hull for the question to have any meaning!

How does the problem arise?

--
*\\* Anton Sherwood *\\* www.bendwavy.org

[Asad Akhlaq]

Hi,

Thanks for your reply. Actually I am not a guy from mathematics and don't have a v.  good background either. Actually I have a simplex in 9D (it is a simplex of A8) with nine vertices. Each of these vertices are connected to each other (means each vertex is connected to eight other vertices of the simplex). I have a hyper plane that intersects this simplex. I found 17 intersection points on the plane that intersect with the simplex. Now I have two pieces of the simplex, one that includes the origin and the other that excludes the origin. My interest is in the part that excludes the origin. So the new piece has 17 vertices on the intersection plane and 4 vertices from the original simplex (in total I have 21 vertices in the new piece that is cut from the simplex). I want to know that whether those 17 intersection points are connected to each other or not. My ultimate goal is to find the volume of this new piece. To find the volume of this piece, can I just take a convex hull  of these 21 vertices and find the volume of that piece? Will it give me the correct result?

Thanks
Assad

[Anton Sherwood]

On 2013-3-05 16:04, Asad Akhlaq wrote:
> ... My ultimate goal is to find the volume of this new piece.

> To find the volume of this piece, can I just take a convex hull
> of these 21 vertices and find the volume of that piece? Will it
> give me the correct result?

Yes: since the original object (a simplex) is convex,
any piece sliced from it (by a planar cut) will also be convex.

[Asad Akhlaq]

Thanks for your reply. One more question please. In the original simplex I have 9 vertices and each vertex is connected to 8 others. In the new polytope, after cut by hyperplane, I have 5 vertices from the original simplex (that are connected to each other) plus 16 new vertices. Now how can I know if I have a one-to-one connection among all the  21 vertices in the new polytope? Is there any procedure to investigate that whether there is a connection among vertices?

Thanks
Assad

[Anton Sherwood]

On 2013-3-06 17:51, Asad Akhlaq wrote:
> Thanks for your reply. One more question please. In the original
> simplex I have 9 vertices and each vertex is connected to 8 others.
> In the new polytope, after cut by hyperplane, I have 5 vertices from
> the original simplex (that are connected to each other) plus 16 new
> vertices. Now how can I know if I have a one-to-one connection among
> all the 21 vertices in the new polytope? Is there any procedure to
> investigate that whether there is a connection among vertices?

The easiest way is probably a convex hull algorithm (as you suggested
before), assuming that it provides all the elements (edges etc.) and not
only the bounding hyperplanes (which you already have!).

[Asad Akhlaq]

OK sir. Thank you very much for your reply.