Can you triangulate a position without knowledge of distance?

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So, here's the set up. You've got 3 points on a 2D plane, and one object to be located. The object is never within the triangle that 3 points make. You are given knowledge of which point is closest to the object.



Is there a way to find distance to the object using the information presented? If not, how many points would be required, would you never be able to get the distance, without more information?



I imagine that the more points you add, the smaller the 'cone' pointing to the object is, but I don't think you can get the actual distance.



(This is just something I thought about while playing a game. It seemed like a mathy type of question, so I asked it here.)




geometry




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  • Given any finite set of points, the "closest" relation will give you finite many criterion to determine position, you can never use them to resolve all points because there are infinitely many of them!
    – achille hui
    Jul 24 at 11:41










  • You can try a Voronoi tessellation of the plane based on the known points, but unless the "closest known point" is strictly inside the convex hull of the known points (it is not with a triangle) then there is no limit on the possible distance of the unknown point. Triangulation is slightly different as it involves measuring angles
    – Henry
    Jul 24 at 11:42











  • @Henry Yeah. I meant to air quote the "triangulate".
    – Sango Dragon
    Jul 24 at 12:05














up vote
1
down vote

favorite












So, here's the set up. You've got 3 points on a 2D plane, and one object to be located. The object is never within the triangle that 3 points make. You are given knowledge of which point is closest to the object.



Is there a way to find distance to the object using the information presented? If not, how many points would be required, would you never be able to get the distance, without more information?



I imagine that the more points you add, the smaller the 'cone' pointing to the object is, but I don't think you can get the actual distance.



(This is just something I thought about while playing a game. It seemed like a mathy type of question, so I asked it here.)




geometry




share|cite|improve this question



















  • Given any finite set of points, the "closest" relation will give you finite many criterion to determine position, you can never use them to resolve all points because there are infinitely many of them!
    – achille hui
    Jul 24 at 11:41










  • You can try a Voronoi tessellation of the plane based on the known points, but unless the "closest known point" is strictly inside the convex hull of the known points (it is not with a triangle) then there is no limit on the possible distance of the unknown point. Triangulation is slightly different as it involves measuring angles
    – Henry
    Jul 24 at 11:42











  • @Henry Yeah. I meant to air quote the "triangulate".
    – Sango Dragon
    Jul 24 at 12:05












up vote
1
down vote

favorite









up vote
1
down vote

favorite











So, here's the set up. You've got 3 points on a 2D plane, and one object to be located. The object is never within the triangle that 3 points make. You are given knowledge of which point is closest to the object.



Is there a way to find distance to the object using the information presented? If not, how many points would be required, would you never be able to get the distance, without more information?



I imagine that the more points you add, the smaller the 'cone' pointing to the object is, but I don't think you can get the actual distance.



(This is just something I thought about while playing a game. It seemed like a mathy type of question, so I asked it here.)




geometry




share|cite|improve this question











So, here's the set up. You've got 3 points on a 2D plane, and one object to be located. The object is never within the triangle that 3 points make. You are given knowledge of which point is closest to the object.



Is there a way to find distance to the object using the information presented? If not, how many points would be required, would you never be able to get the distance, without more information?



I imagine that the more points you add, the smaller the 'cone' pointing to the object is, but I don't think you can get the actual distance.



(This is just something I thought about while playing a game. It seemed like a mathy type of question, so I asked it here.)





geometry





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share|cite|improve this question




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asked Jul 24 at 11:25









Sango Dragon

82




82











  • Given any finite set of points, the "closest" relation will give you finite many criterion to determine position, you can never use them to resolve all points because there are infinitely many of them!
    – achille hui
    Jul 24 at 11:41










  • You can try a Voronoi tessellation of the plane based on the known points, but unless the "closest known point" is strictly inside the convex hull of the known points (it is not with a triangle) then there is no limit on the possible distance of the unknown point. Triangulation is slightly different as it involves measuring angles
    – Henry
    Jul 24 at 11:42











  • @Henry Yeah. I meant to air quote the "triangulate".
    – Sango Dragon
    Jul 24 at 12:05
















  • Given any finite set of points, the "closest" relation will give you finite many criterion to determine position, you can never use them to resolve all points because there are infinitely many of them!
    – achille hui
    Jul 24 at 11:41










  • You can try a Voronoi tessellation of the plane based on the known points, but unless the "closest known point" is strictly inside the convex hull of the known points (it is not with a triangle) then there is no limit on the possible distance of the unknown point. Triangulation is slightly different as it involves measuring angles
    – Henry
    Jul 24 at 11:42











  • @Henry Yeah. I meant to air quote the "triangulate".
    – Sango Dragon
    Jul 24 at 12:05















Given any finite set of points, the "closest" relation will give you finite many criterion to determine position, you can never use them to resolve all points because there are infinitely many of them!
– achille hui
Jul 24 at 11:41




Given any finite set of points, the "closest" relation will give you finite many criterion to determine position, you can never use them to resolve all points because there are infinitely many of them!
– achille hui
Jul 24 at 11:41












You can try a Voronoi tessellation of the plane based on the known points, but unless the "closest known point" is strictly inside the convex hull of the known points (it is not with a triangle) then there is no limit on the possible distance of the unknown point. Triangulation is slightly different as it involves measuring angles
– Henry
Jul 24 at 11:42





You can try a Voronoi tessellation of the plane based on the known points, but unless the "closest known point" is strictly inside the convex hull of the known points (it is not with a triangle) then there is no limit on the possible distance of the unknown point. Triangulation is slightly different as it involves measuring angles
– Henry
Jul 24 at 11:42













@Henry Yeah. I meant to air quote the "triangulate".
– Sango Dragon
Jul 24 at 12:05




@Henry Yeah. I meant to air quote the "triangulate".
– Sango Dragon
Jul 24 at 12:05










1 Answer
1






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accepted










The picture shows the Voronoi domains of a set of points, i.e. the loci of points that are closer to a given point of the set than to any other (cells separated by gray edges).



If you exclude all points that are inside a red triangle, what remains is a set of infinite domains, and obviously the distance is indeterminate.



enter image description here






share|cite|improve this answer





















  • That's essentially what I expected. I've got to read up on this Voronoi tessellation. Looks cool.
    – Sango Dragon
    Jul 24 at 12:06










  • There are three little areas outside the red triangles in your diagram which are part of finite grey cells
    – Henry
    Jul 24 at 12:33










  • @Henry: yes that can happen.
    – Yves Daoust
    Jul 24 at 13:05










  • This is what I was hinting at when I said the "closest known point" is strictly inside the convex hull of the known points in my comment on the question
    – Henry
    Jul 24 at 16:20










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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote



accepted










The picture shows the Voronoi domains of a set of points, i.e. the loci of points that are closer to a given point of the set than to any other (cells separated by gray edges).



If you exclude all points that are inside a red triangle, what remains is a set of infinite domains, and obviously the distance is indeterminate.



enter image description here






share|cite|improve this answer





















  • That's essentially what I expected. I've got to read up on this Voronoi tessellation. Looks cool.
    – Sango Dragon
    Jul 24 at 12:06










  • There are three little areas outside the red triangles in your diagram which are part of finite grey cells
    – Henry
    Jul 24 at 12:33










  • @Henry: yes that can happen.
    – Yves Daoust
    Jul 24 at 13:05










  • This is what I was hinting at when I said the "closest known point" is strictly inside the convex hull of the known points in my comment on the question
    – Henry
    Jul 24 at 16:20














up vote
1
down vote



accepted










The picture shows the Voronoi domains of a set of points, i.e. the loci of points that are closer to a given point of the set than to any other (cells separated by gray edges).



If you exclude all points that are inside a red triangle, what remains is a set of infinite domains, and obviously the distance is indeterminate.



enter image description here






share|cite|improve this answer





















  • That's essentially what I expected. I've got to read up on this Voronoi tessellation. Looks cool.
    – Sango Dragon
    Jul 24 at 12:06










  • There are three little areas outside the red triangles in your diagram which are part of finite grey cells
    – Henry
    Jul 24 at 12:33










  • @Henry: yes that can happen.
    – Yves Daoust
    Jul 24 at 13:05










  • This is what I was hinting at when I said the "closest known point" is strictly inside the convex hull of the known points in my comment on the question
    – Henry
    Jul 24 at 16:20












up vote
1
down vote



accepted







up vote
1
down vote



accepted






The picture shows the Voronoi domains of a set of points, i.e. the loci of points that are closer to a given point of the set than to any other (cells separated by gray edges).



If you exclude all points that are inside a red triangle, what remains is a set of infinite domains, and obviously the distance is indeterminate.



enter image description here






share|cite|improve this answer













The picture shows the Voronoi domains of a set of points, i.e. the loci of points that are closer to a given point of the set than to any other (cells separated by gray edges).



If you exclude all points that are inside a red triangle, what remains is a set of infinite domains, and obviously the distance is indeterminate.



enter image description here







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 24 at 11:57









Yves Daoust

111k665203




111k665203











  • That's essentially what I expected. I've got to read up on this Voronoi tessellation. Looks cool.
    – Sango Dragon
    Jul 24 at 12:06










  • There are three little areas outside the red triangles in your diagram which are part of finite grey cells
    – Henry
    Jul 24 at 12:33










  • @Henry: yes that can happen.
    – Yves Daoust
    Jul 24 at 13:05










  • This is what I was hinting at when I said the "closest known point" is strictly inside the convex hull of the known points in my comment on the question
    – Henry
    Jul 24 at 16:20
















  • That's essentially what I expected. I've got to read up on this Voronoi tessellation. Looks cool.
    – Sango Dragon
    Jul 24 at 12:06










  • There are three little areas outside the red triangles in your diagram which are part of finite grey cells
    – Henry
    Jul 24 at 12:33










  • @Henry: yes that can happen.
    – Yves Daoust
    Jul 24 at 13:05










  • This is what I was hinting at when I said the "closest known point" is strictly inside the convex hull of the known points in my comment on the question
    – Henry
    Jul 24 at 16:20















That's essentially what I expected. I've got to read up on this Voronoi tessellation. Looks cool.
– Sango Dragon
Jul 24 at 12:06




That's essentially what I expected. I've got to read up on this Voronoi tessellation. Looks cool.
– Sango Dragon
Jul 24 at 12:06












There are three little areas outside the red triangles in your diagram which are part of finite grey cells
– Henry
Jul 24 at 12:33




There are three little areas outside the red triangles in your diagram which are part of finite grey cells
– Henry
Jul 24 at 12:33












@Henry: yes that can happen.
– Yves Daoust
Jul 24 at 13:05




@Henry: yes that can happen.
– Yves Daoust
Jul 24 at 13:05












This is what I was hinting at when I said the "closest known point" is strictly inside the convex hull of the known points in my comment on the question
– Henry
Jul 24 at 16:20




This is what I was hinting at when I said the "closest known point" is strictly inside the convex hull of the known points in my comment on the question
– Henry
Jul 24 at 16:20












 

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