user_guide:tutorials:regular_subdivisions

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user_guide:tutorials:regular_subdivisions [2019/01/25 13:40] – ↷ Page moved from user_guide:regular_subdivisions to user_guide:tutorials:regular_subdivisions oroehriguser_guide:tutorials:regular_subdivisions [2019/01/28 17:40] – ↷ Links adapted because of a move operation oroehrig
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 If you use javaview for visualization, then this might look similar to the following: If you use javaview for visualization, then this might look similar to the following:
  
-{{:tutorial:subdiv_of_points_1.gif?300|}}+{{user_guide:subdiv_of_points_1.gif?300|}}
  
 Note that the quadrilateral contains point 2 in its interior and that this point is colored black. This corresponds to the fact that the lifted point 2 lies above the convex hull of the lifted points 0,1,3 and 4. Therefore the maximal cell describing the quadrilateral does not contain the point 2. We may change the lifting function by giving point 2 height ''0'' as well: Note that the quadrilateral contains point 2 in its interior and that this point is colored black. This corresponds to the fact that the lifted point 2 lies above the convex hull of the lifted points 0,1,3 and 4. Therefore the maximal cell describing the quadrilateral does not contain the point 2. We may change the lifting function by giving point 2 height ''0'' as well:
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 </code> </code>
  
-{{:tutorial:subdiv_of_points_2.gif?300|}}+{{user_guide:subdiv_of_points_2.gif?300|}}
  
 This new height function generates the same polyhedral complex, yet a different subdivision of the point set. Note that the maximal cell that describes the quadrilateral now contains the point 2, but it is not a vertex of that cell. This is the reason for the yellow coloring. This corresponds to the fact that its lifting lies in the convex hull of the lifted quadrilateral, whereat it is not a vertex of the lifted polytope. Changing the height of point 2 again to ''-1'' the subdivision should become a triangulation of the point set.  This new height function generates the same polyhedral complex, yet a different subdivision of the point set. Note that the maximal cell that describes the quadrilateral now contains the point 2, but it is not a vertex of that cell. This is the reason for the yellow coloring. This corresponds to the fact that its lifting lies in the convex hull of the lifted quadrilateral, whereat it is not a vertex of the lifted polytope. Changing the height of point 2 again to ''-1'' the subdivision should become a triangulation of the point set. 
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 </code> </code>
  
-{{:tutorial:subdiv_of_points_3.gif?300|}}+{{user_guide:subdiv_of_points_3.gif?300|}}
  
 Indeed, the regular subdivision arising from this height function is a triangulation. Since in this case point 2 is a vertex of the subdivision it is colored red as well. Indeed, the regular subdivision arising from this height function is a triangulation. Since in this case point 2 is a vertex of the subdivision it is colored red as well.
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 polytope > $PC->VISUAL; polytope > $PC->VISUAL;
 </code> </code>
-{{:tutorial:reg_subdiv_polytopal_subdiv.gif?300}}+{{user_guide:reg_subdiv_polytopal_subdiv.gif?300}}
  
 Check the interactive help to get more information. Check the interactive help to get more information.
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 polytope > $S->VISUAL; polytope > $S->VISUAL;
 </code> </code>
-{{:tutorial:square_reg_subdiv.png?400|}}+{{user_guide:square_reg_subdiv.png?400|}}
  
 As one can see from the picture this subdivision should be regular, since we can easily find a weight vector which induces this subdivision. Just lift all the points in the inner square to 0 and the points on the outer square to 1. But now we want to take a look at all vectors which induce this subdivision. This can be achieved by using the method ''secondary_cone''. As one can see from the picture this subdivision should be regular, since we can easily find a weight vector which induces this subdivision. Just lift all the points in the inner square to 0 and the points on the outer square to 1. But now we want to take a look at all vectors which induce this subdivision. This can be achieved by using the method ''secondary_cone''.
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 polytope > $nreg_pc->VISUAL; polytope > $nreg_pc->VISUAL;
 </code> </code>
-{{:tutorial:square_nreg_subdiv.png?400|}}+{{user_guide:square_nreg_subdiv.png?400|}}
  
 This is quite similar to the ''mother of all examples'' (See: "Triangulations" from DeLoera, Rambau and Santos). So with a similar argument we can see that this subdivision is indeed not regular. How does polymake feel about this? This is quite similar to the ''mother of all examples'' (See: "Triangulations" from DeLoera, Rambau and Santos). So with a similar argument we can see that this subdivision is indeed not regular. How does polymake feel about this?
  • user_guide/tutorials/regular_subdivisions.txt
  • Last modified: 2019/02/04 22:55
  • by 127.0.0.1