Intersection of polytope and hyperplane

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Determining the intersection of polytope and hyperplane in arbitrary dimension is of central interest in computational geometry. In some paper(this one in particular: https://pdfs.semanticscholar.org/f4a6/4713dbf19883878e1357a6dc2bdfc2a04f30.pdf) author mentioned the "Naive Algorithm" of finding all the intersection between edges and hyperplane.

However, I couldn't find a proof that the intersection of polytope and hyperplane is determined by the intersection of the hyperplane and polytope's edges; is this unproven? If it is can someone post a source?

discrete-mathematics computational-geometry open-problem

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edited Jul 31 at 3:23

asked Jul 30 at 22:22

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Determining the intersection of polytope and hyperplane in arbitrary dimension is of central interest in computational geometry. In some paper(this one in particular: https://pdfs.semanticscholar.org/f4a6/4713dbf19883878e1357a6dc2bdfc2a04f30.pdf) author mentioned the "Naive Algorithm" of finding all the intersection between edges and hyperplane.

However, I couldn't find a proof that the intersection of polytope and hyperplane is determined by the intersection of the hyperplane and polytope's edges; is this unproven? If it is can someone post a source?

discrete-mathematics computational-geometry open-problem

share|cite|improve this question

edited Jul 31 at 3:23

asked Jul 30 at 22:22

404 n found

13

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Determining the intersection of polytope and hyperplane in arbitrary dimension is of central interest in computational geometry. In some paper(this one in particular: https://pdfs.semanticscholar.org/f4a6/4713dbf19883878e1357a6dc2bdfc2a04f30.pdf) author mentioned the "Naive Algorithm" of finding all the intersection between edges and hyperplane.

However, I couldn't find a proof that the intersection of polytope and hyperplane is determined by the intersection of the hyperplane and polytope's edges; is this unproven? If it is can someone post a source?

discrete-mathematics computational-geometry open-problem

share|cite|improve this question

edited Jul 31 at 3:23

asked Jul 30 at 22:22

404 n found

13

Determining the intersection of polytope and hyperplane in arbitrary dimension is of central interest in computational geometry. In some paper(this one in particular: https://pdfs.semanticscholar.org/f4a6/4713dbf19883878e1357a6dc2bdfc2a04f30.pdf) author mentioned the "Naive Algorithm" of finding all the intersection between edges and hyperplane.

However, I couldn't find a proof that the intersection of polytope and hyperplane is determined by the intersection of the hyperplane and polytope's edges; is this unproven? If it is can someone post a source?

discrete-mathematics computational-geometry open-problem

share|cite|improve this question

edited Jul 31 at 3:23

asked Jul 30 at 22:22

404 n found

13

share|cite|improve this question

share|cite|improve this question

edited Jul 31 at 3:23

edited Jul 31 at 3:23

edited Jul 31 at 3:23

asked Jul 30 at 22:22

404 n found

13

asked Jul 30 at 22:22

404 n found

13

asked Jul 30 at 22:22

404 n found

13

13

add a comment | 

add a comment | 

1 Answer
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The polygon lies in a plane. The intersection of a hyperplane and a plane is either the entire plane or a line in the plane. Thus, if a hyperplane intersects the polygon in some point, it intersect the polygon's plane in a line through that point. This line necessarily crosses the edges of the polygon. Thus you can find the polygon's intersection with a hyperplane by finding the intersections of its edges with the hyperplane.

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answered Jul 31 at 3:20

joriki

164k10179328

• 
  
  
  

  
  

  
  
  
  
  
  

  
  sorry, by polygon I mean polytope
  – 404 n found
  Jul 31 at 3:22
  

  
  
  
  

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1 Answer
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1 Answer
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The polygon lies in a plane. The intersection of a hyperplane and a plane is either the entire plane or a line in the plane. Thus, if a hyperplane intersects the polygon in some point, it intersect the polygon's plane in a line through that point. This line necessarily crosses the edges of the polygon. Thus you can find the polygon's intersection with a hyperplane by finding the intersections of its edges with the hyperplane.

share|cite|improve this answer

answered Jul 31 at 3:20

joriki

164k10179328

• 
  
  
  

  
  

  
  
  
  
  
  

  
  sorry, by polygon I mean polytope
  – 404 n found
  Jul 31 at 3:22
  

  
  
  
  

add a comment | 

up vote
0
down vote

The polygon lies in a plane. The intersection of a hyperplane and a plane is either the entire plane or a line in the plane. Thus, if a hyperplane intersects the polygon in some point, it intersect the polygon's plane in a line through that point. This line necessarily crosses the edges of the polygon. Thus you can find the polygon's intersection with a hyperplane by finding the intersections of its edges with the hyperplane.

share|cite|improve this answer

answered Jul 31 at 3:20

joriki

164k10179328

• 
  
  
  

  
  

  
  
  
  
  
  

  
  sorry, by polygon I mean polytope
  – 404 n found
  Jul 31 at 3:22
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

The polygon lies in a plane. The intersection of a hyperplane and a plane is either the entire plane or a line in the plane. Thus, if a hyperplane intersects the polygon in some point, it intersect the polygon's plane in a line through that point. This line necessarily crosses the edges of the polygon. Thus you can find the polygon's intersection with a hyperplane by finding the intersections of its edges with the hyperplane.

share|cite|improve this answer

answered Jul 31 at 3:20

joriki

164k10179328

The polygon lies in a plane. The intersection of a hyperplane and a plane is either the entire plane or a line in the plane. Thus, if a hyperplane intersects the polygon in some point, it intersect the polygon's plane in a line through that point. This line necessarily crosses the edges of the polygon. Thus you can find the polygon's intersection with a hyperplane by finding the intersections of its edges with the hyperplane.

share|cite|improve this answer

answered Jul 31 at 3:20

joriki

164k10179328

share|cite|improve this answer

share|cite|improve this answer

answered Jul 31 at 3:20

joriki

164k10179328

answered Jul 31 at 3:20

joriki

164k10179328

answered Jul 31 at 3:20

joriki

164k10179328

164k10179328

• 
  
  
  

  
  

  
  
  
  
  
  

  
  sorry, by polygon I mean polytope
  – 404 n found
  Jul 31 at 3:22
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  sorry, by polygon I mean polytope
  – 404 n found
  Jul 31 at 3:22
  

  
  
  
  

sorry, by polygon I mean polytope
– 404 n found
Jul 31 at 3:22

sorry, by polygon I mean polytope
– 404 n found
Jul 31 at 3:22

add a comment | 

 

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