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Existing of non-intersecting rays

Constructing a circle through a point in the interior of an angleHow many rays can made from $4$ collinear points?Angle between different rays (3d line segments) and computing their angular relationshipsIntersecting three rays and a sphere of known radiusDesigning a distance function between raysLines between point and sphere surface intersecting a planeIntersecting planes stereometry problemIs a single line, line segment, or ray a valid angle?Coxeter, Introduction to Geometry, ordered geometry, parallelism of rays and linesNon-congruent angle of an isosceles triangle

2

1

$begingroup$

Given $n$ points on a plane, it seems intuitive that it’s possible to draw a ray (half-line) from each point s. t. the $n$ rays do not intersect.

But how to prove this?

geometry

share|cite|improve this question

asked Mar 24 at 2:19

athosathos

1,03911440

$endgroup$

add a comment | 

2

1

$begingroup$

Given $n$ points on a plane, it seems intuitive that it’s possible to draw a ray (half-line) from each point s. t. the $n$ rays do not intersect.

But how to prove this?

geometry

share|cite|improve this question

asked Mar 24 at 2:19

athosathos

1,03911440

$endgroup$

add a comment | 

$begingroup$

Given $n$ points on a plane, it seems intuitive that it’s possible to draw a ray (half-line) from each point s. t. the $n$ rays do not intersect.

But how to prove this?

geometry

share|cite|improve this question

asked Mar 24 at 2:19

athosathos

1,03911440

$endgroup$

share|cite|improve this question

asked Mar 24 at 2:19

athosathos

1,03911440

share|cite|improve this question

asked Mar 24 at 2:19

athosathos

1,03911440

asked Mar 24 at 2:19

athosathos

1,03911440

asked Mar 24 at 2:19

athosathos

1,03911440

1 Answer
1

active

oldest

votes

2

$begingroup$

Pick any point $P$ in the plane that is not on a line containing two or more of the given $n$ points. At each point, draw the ray in the direction away from $P$.

One can in fact do better: It is possible to draw lines through all $n$ points that do not intersect. Choose an orientation that is not parallel to any of the lines between any two of the given points, and draw parallel lines in that orientation through each point.

share|cite|improve this answer

edited Mar 24 at 2:37

answered Mar 24 at 2:30

FredHFredH

3,7321024

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  +1 for being slightly faster than me :)
  $endgroup$
  – Severin Schraven
  Mar 24 at 2:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thx ! This is a “oh of course “ moment of me
  $endgroup$
  – athos
  Mar 24 at 2:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It is almost trivial but you need to prove the existence of a point $P$ not collinear to the rest, for example consider the case of points all lying on the same line where this property fails. Luckily the orientation method works in that case too and the existence of such an orientation is guaranteed since picking an arbitary axis the angle of the line between any two points form a finite set.
  $endgroup$
  – Μάρκος Καραμέρης
  Mar 24 at 2:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @ΜάρκοςΚαραμέρης $P$ is not one of the given points. Since a finite set of lines does not exhaust the plane, there are plenty of possible choices.
  $endgroup$
  – FredH
  Mar 24 at 2:48
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @FredH Ah ok makes perfect sense now!
  $endgroup$
  – Μάρκος Καραμέρης
  Mar 24 at 2:52
  
  

  
  
  
  

add a comment | 

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2

$begingroup$

Pick any point $P$ in the plane that is not on a line containing two or more of the given $n$ points. At each point, draw the ray in the direction away from $P$.

One can in fact do better: It is possible to draw lines through all $n$ points that do not intersect. Choose an orientation that is not parallel to any of the lines between any two of the given points, and draw parallel lines in that orientation through each point.

share|cite|improve this answer

edited Mar 24 at 2:37

answered Mar 24 at 2:30

FredHFredH

3,7321024

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  +1 for being slightly faster than me :)
  $endgroup$
  – Severin Schraven
  Mar 24 at 2:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thx ! This is a “oh of course “ moment of me
  $endgroup$
  – athos
  Mar 24 at 2:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It is almost trivial but you need to prove the existence of a point $P$ not collinear to the rest, for example consider the case of points all lying on the same line where this property fails. Luckily the orientation method works in that case too and the existence of such an orientation is guaranteed since picking an arbitary axis the angle of the line between any two points form a finite set.
  $endgroup$
  – Μάρκος Καραμέρης
  Mar 24 at 2:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @ΜάρκοςΚαραμέρης $P$ is not one of the given points. Since a finite set of lines does not exhaust the plane, there are plenty of possible choices.
  $endgroup$
  – FredH
  Mar 24 at 2:48
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @FredH Ah ok makes perfect sense now!
  $endgroup$
  – Μάρκος Καραμέρης
  Mar 24 at 2:52
  
  

  
  
  
  

add a comment | 

2

$begingroup$

Pick any point $P$ in the plane that is not on a line containing two or more of the given $n$ points. At each point, draw the ray in the direction away from $P$.

One can in fact do better: It is possible to draw lines through all $n$ points that do not intersect. Choose an orientation that is not parallel to any of the lines between any two of the given points, and draw parallel lines in that orientation through each point.

share|cite|improve this answer

edited Mar 24 at 2:37

answered Mar 24 at 2:30

FredHFredH

3,7321024

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  +1 for being slightly faster than me :)
  $endgroup$
  – Severin Schraven
  Mar 24 at 2:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thx ! This is a “oh of course “ moment of me
  $endgroup$
  – athos
  Mar 24 at 2:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It is almost trivial but you need to prove the existence of a point $P$ not collinear to the rest, for example consider the case of points all lying on the same line where this property fails. Luckily the orientation method works in that case too and the existence of such an orientation is guaranteed since picking an arbitary axis the angle of the line between any two points form a finite set.
  $endgroup$
  – Μάρκος Καραμέρης
  Mar 24 at 2:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @ΜάρκοςΚαραμέρης $P$ is not one of the given points. Since a finite set of lines does not exhaust the plane, there are plenty of possible choices.
  $endgroup$
  – FredH
  Mar 24 at 2:48
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @FredH Ah ok makes perfect sense now!
  $endgroup$
  – Μάρκος Καραμέρης
  Mar 24 at 2:52
  
  

  
  
  
  

add a comment | 

$begingroup$

Pick any point $P$ in the plane that is not on a line containing two or more of the given $n$ points. At each point, draw the ray in the direction away from $P$.

One can in fact do better: It is possible to draw lines through all $n$ points that do not intersect. Choose an orientation that is not parallel to any of the lines between any two of the given points, and draw parallel lines in that orientation through each point.

share|cite|improve this answer

edited Mar 24 at 2:37

answered Mar 24 at 2:30

FredHFredH

3,7321024

$endgroup$

share|cite|improve this answer

edited Mar 24 at 2:37

answered Mar 24 at 2:30

FredHFredH

3,7321024

answered Mar 24 at 2:30

FredHFredH

3,7321024

answered Mar 24 at 2:30

FredHFredH

3,7321024