Shortest maximum pairwise distance of points in a circle of radius R Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Mean distance between N equidistributed points in a circleA Sphere Containing Points of Pairwise Equal DistanceIs there any object/metric within all points are at same distance to each other?Determining the distance between 2 points based on their know distances to several other points.Given the pairwise distances between $n$ points, how can I find plausible coordinates for the points?Circle around three given circlesshowing that the maximum distance between any pair of points inside a circlemaximum distance between n points on unit circleGiven a pair of circle, Find 2 points on the perimeter of circle(one on each Circle) such that the Euclidean distance is K? Given Centre and Radius.How to localize points from an incomplete distance matrix in R?
Seeking colloquialism for “just because”
Why did the rest of the Eastern Bloc not invade Yugoslavia?
How to bypass password on Windows XP account?
How to deal with a team lead who never gives me credit?
Is it ethical to give a final exam after the professor has quit before teaching the remaining chapters of the course?
51k Euros annually for a family of 4 in Berlin: Is it enough?
porting install scripts : can rpm replace apt?
Bete Noir -- no dairy
How do pianists reach extremely loud dynamics?
How come Sam didn't become Lord of Horn Hill?
How to answer "Have you ever been terminated?"
The logistics of corpse disposal
How does the particle を relate to the verb 行く in the structure「A を + B に行く」?
What is a non-alternating simple group with big order, but relatively few conjugacy classes?
Identify plant with long narrow paired leaves and reddish stems
What is the logic behind the Maharil's explanation of why we don't say שעשה ניסים on Pesach?
What's the meaning of 間時肆拾貳 at a car parking sign
How to tell that you are a giant?
Why are Kinder Surprise Eggs illegal in the USA?
When do you get frequent flier miles - when you buy, or when you fly?
Is it true that "carbohydrates are of no use for the basal metabolic need"?
Why am I getting the error "non-boolean type specified in a context where a condition is expected" for this request?
Why are there no cargo aircraft with "flying wing" design?
How discoverable are IPv6 addresses and AAAA names by potential attackers?
Shortest maximum pairwise distance of points in a circle of radius R
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Mean distance between N equidistributed points in a circleA Sphere Containing Points of Pairwise Equal DistanceIs there any object/metric within all points are at same distance to each other?Determining the distance between 2 points based on their know distances to several other points.Given the pairwise distances between $n$ points, how can I find plausible coordinates for the points?Circle around three given circlesshowing that the maximum distance between any pair of points inside a circlemaximum distance between n points on unit circleGiven a pair of circle, Find 2 points on the perimeter of circle(one on each Circle) such that the Euclidean distance is K? Given Centre and Radius.How to localize points from an incomplete distance matrix in R?
$begingroup$
Given a positive real number $R$ and $n$ fixed points in a plane, find the largest possible value $M$ such that if the pairwise distance between the $n$ points is less or equal to $M$, there exists a circle of radius $R$ that contains all $n$ points.
I realize this is related to the smallest enclosing circle problem, but it is not the same. Basically, I have a set of points, and I know all the pairwise distances between them, and I am given a fixed radius. I want to be able to say that if the maximum pairwise distance between these points is less than $M$, then there is definitely a circle of radius $R$ that encloses all of them, where this M is as large as possible.
Obviously, for two points, the answer is $2R$. For three points, it gets more complicated, but using some geometry, I believe the answer is $sqrt2R$ (as in $sqrtR^2 + R^2$), but I haven't been able to prove it. I would like to generalize this to $n$ points, where I am fine with the answer being dependent on $n$.
My hypothesis is that $sqrt2R leq M leq 2R$ for all $n$, but I would be mainly interested in how large $M$ can be made as the number of points increases.
geometry euclidean-geometry
asked Mar 26 at 17:36
DennisDennis
111
$endgroup$
add a comment |
$begingroup$
Given a positive real number $R$ and $n$ fixed points in a plane, find the largest possible value $M$ such that if the pairwise distance between the $n$ points is less or equal to $M$, there exists a circle of radius $R$ that contains all $n$ points.
I realize this is related to the smallest enclosing circle problem, but it is not the same. Basically, I have a set of points, and I know all the pairwise distances between them, and I am given a fixed radius. I want to be able to say that if the maximum pairwise distance between these points is less than $M$, then there is definitely a circle of radius $R$ that encloses all of them, where this M is as large as possible.
Obviously, for two points, the answer is $2R$. For three points, it gets more complicated, but using some geometry, I believe the answer is $sqrt2R$ (as in $sqrtR^2 + R^2$), but I haven't been able to prove it. I would like to generalize this to $n$ points, where I am fine with the answer being dependent on $n$.
My hypothesis is that $sqrt2R leq M leq 2R$ for all $n$, but I would be mainly interested in how large $M$ can be made as the number of points increases.
geometry euclidean-geometry
asked Mar 26 at 17:36
DennisDennis
111
$endgroup$
add a comment |
$begingroup$
Given a positive real number $R$ and $n$ fixed points in a plane, find the largest possible value $M$ such that if the pairwise distance between the $n$ points is less or equal to $M$, there exists a circle of radius $R$ that contains all $n$ points.
I realize this is related to the smallest enclosing circle problem, but it is not the same. Basically, I have a set of points, and I know all the pairwise distances between them, and I am given a fixed radius. I want to be able to say that if the maximum pairwise distance between these points is less than $M$, then there is definitely a circle of radius $R$ that encloses all of them, where this M is as large as possible.
Obviously, for two points, the answer is $2R$. For three points, it gets more complicated, but using some geometry, I believe the answer is $sqrt2R$ (as in $sqrtR^2 + R^2$), but I haven't been able to prove it. I would like to generalize this to $n$ points, where I am fine with the answer being dependent on $n$.
My hypothesis is that $sqrt2R leq M leq 2R$ for all $n$, but I would be mainly interested in how large $M$ can be made as the number of points increases.
geometry euclidean-geometry
asked Mar 26 at 17:36
DennisDennis
111
$endgroup$
Given a positive real number $R$ and $n$ fixed points in a plane, find the largest possible value $M$ such that if the pairwise distance between the $n$ points is less or equal to $M$, there exists a circle of radius $R$ that contains all $n$ points.
I realize this is related to the smallest enclosing circle problem, but it is not the same. Basically, I have a set of points, and I know all the pairwise distances between them, and I am given a fixed radius. I want to be able to say that if the maximum pairwise distance between these points is less than $M$, then there is definitely a circle of radius $R$ that encloses all of them, where this M is as large as possible.
Obviously, for two points, the answer is $2R$. For three points, it gets more complicated, but using some geometry, I believe the answer is $sqrt2R$ (as in $sqrtR^2 + R^2$), but I haven't been able to prove it. I would like to generalize this to $n$ points, where I am fine with the answer being dependent on $n$.
My hypothesis is that $sqrt2R leq M leq 2R$ for all $n$, but I would be mainly interested in how large $M$ can be made as the number of points increases.
geometry euclidean-geometry
geometry euclidean-geometry
asked Mar 26 at 17:36
DennisDennis
111
asked Mar 26 at 17:36
DennisDennis
111
asked Mar 26 at 17:36
DennisDennis
111
asked Mar 26 at 17:36
DennisDennis
111
asked Mar 26 at 17:36
DennisDennis
111
111
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I don't have a complete solution but I can prove that your conclusion for $n=3$ points is wrong. Suppose that the maximum pairwise distance between 3 points is $M$. WLOG, suppose that the maximum distance $M$ is reached between points $A$ and $B$.
Where is the third point $C$? It has to be somewhere in the pink region. The borders of that region are segment $AB$ and two arcs, each encompassing $60^circ$ with centers at points $A$ and $B$. Wherever you put the third point ($C',C''$), you can always cover the whole triangle with the blue circle. But the blue circle cannot be any smaller because in that case it could not cover the triangle $ABC$. And in that case:
$$M=Rsqrt3$$
answered Mar 27 at 16:53
OldboyOldboy
9,53411138
$endgroup$
$begingroup$
Thanks! Just realized my flaw for the $n=3$ case. Theoretically, since points can be arbitrarily close together, I think $M$ cannot be any larger than $Rsqrt3$, and since adding more points only puts more restrictions on where the next one can go, I don't think $M$ gets any smaller, either. For my practical application, though, it would be interesting to see if $M$ could be made larger with a relatively high probability for points that are reasonably "spread out."
$endgroup$
– Dennis
Mar 27 at 21:07
$begingroup$
@Dennis If you find the answer useful, consider upvoting it, you don’t have to accept it.
$endgroup$
– Oldboy
Mar 27 at 22:19
$begingroup$
I did. Just didn't register because apparently I don't have enough "reputation points" or whatever.
$endgroup$
– Dennis
Mar 27 at 22:31
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3163510%2fshortest-maximum-pairwise-distance-of-points-in-a-circle-of-radius-r%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I don't have a complete solution but I can prove that your conclusion for $n=3$ points is wrong. Suppose that the maximum pairwise distance between 3 points is $M$. WLOG, suppose that the maximum distance $M$ is reached between points $A$ and $B$.
Where is the third point $C$? It has to be somewhere in the pink region. The borders of that region are segment $AB$ and two arcs, each encompassing $60^circ$ with centers at points $A$ and $B$. Wherever you put the third point ($C',C''$), you can always cover the whole triangle with the blue circle. But the blue circle cannot be any smaller because in that case it could not cover the triangle $ABC$. And in that case:
$$M=Rsqrt3$$
answered Mar 27 at 16:53
OldboyOldboy
9,53411138
$endgroup$
$begingroup$
Thanks! Just realized my flaw for the $n=3$ case. Theoretically, since points can be arbitrarily close together, I think $M$ cannot be any larger than $Rsqrt3$, and since adding more points only puts more restrictions on where the next one can go, I don't think $M$ gets any smaller, either. For my practical application, though, it would be interesting to see if $M$ could be made larger with a relatively high probability for points that are reasonably "spread out."
$endgroup$
– Dennis
Mar 27 at 21:07
$begingroup$
@Dennis If you find the answer useful, consider upvoting it, you don’t have to accept it.
$endgroup$
– Oldboy
Mar 27 at 22:19
$begingroup$
I did. Just didn't register because apparently I don't have enough "reputation points" or whatever.
$endgroup$
– Dennis
Mar 27 at 22:31
add a comment |
$begingroup$
I don't have a complete solution but I can prove that your conclusion for $n=3$ points is wrong. Suppose that the maximum pairwise distance between 3 points is $M$. WLOG, suppose that the maximum distance $M$ is reached between points $A$ and $B$.
Where is the third point $C$? It has to be somewhere in the pink region. The borders of that region are segment $AB$ and two arcs, each encompassing $60^circ$ with centers at points $A$ and $B$. Wherever you put the third point ($C',C''$), you can always cover the whole triangle with the blue circle. But the blue circle cannot be any smaller because in that case it could not cover the triangle $ABC$. And in that case:
$$M=Rsqrt3$$
answered Mar 27 at 16:53
OldboyOldboy
9,53411138
$endgroup$
$begingroup$
Thanks! Just realized my flaw for the $n=3$ case. Theoretically, since points can be arbitrarily close together, I think $M$ cannot be any larger than $Rsqrt3$, and since adding more points only puts more restrictions on where the next one can go, I don't think $M$ gets any smaller, either. For my practical application, though, it would be interesting to see if $M$ could be made larger with a relatively high probability for points that are reasonably "spread out."
$endgroup$
– Dennis
Mar 27 at 21:07
$begingroup$
@Dennis If you find the answer useful, consider upvoting it, you don’t have to accept it.
$endgroup$
– Oldboy
Mar 27 at 22:19
$begingroup$
I did. Just didn't register because apparently I don't have enough "reputation points" or whatever.
$endgroup$
– Dennis
Mar 27 at 22:31
add a comment |
$begingroup$
I don't have a complete solution but I can prove that your conclusion for $n=3$ points is wrong. Suppose that the maximum pairwise distance between 3 points is $M$. WLOG, suppose that the maximum distance $M$ is reached between points $A$ and $B$.
Where is the third point $C$? It has to be somewhere in the pink region. The borders of that region are segment $AB$ and two arcs, each encompassing $60^circ$ with centers at points $A$ and $B$. Wherever you put the third point ($C',C''$), you can always cover the whole triangle with the blue circle. But the blue circle cannot be any smaller because in that case it could not cover the triangle $ABC$. And in that case:
$$M=Rsqrt3$$
answered Mar 27 at 16:53
OldboyOldboy
9,53411138
$endgroup$
I don't have a complete solution but I can prove that your conclusion for $n=3$ points is wrong. Suppose that the maximum pairwise distance between 3 points is $M$. WLOG, suppose that the maximum distance $M$ is reached between points $A$ and $B$.
Where is the third point $C$? It has to be somewhere in the pink region. The borders of that region are segment $AB$ and two arcs, each encompassing $60^circ$ with centers at points $A$ and $B$. Wherever you put the third point ($C',C''$), you can always cover the whole triangle with the blue circle. But the blue circle cannot be any smaller because in that case it could not cover the triangle $ABC$. And in that case:
$$M=Rsqrt3$$
answered Mar 27 at 16:53
OldboyOldboy
9,53411138
answered Mar 27 at 16:53
OldboyOldboy
9,53411138
answered Mar 27 at 16:53
OldboyOldboy
9,53411138
answered Mar 27 at 16:53
OldboyOldboy
9,53411138
9,53411138
$begingroup$
Thanks! Just realized my flaw for the $n=3$ case. Theoretically, since points can be arbitrarily close together, I think $M$ cannot be any larger than $Rsqrt3$, and since adding more points only puts more restrictions on where the next one can go, I don't think $M$ gets any smaller, either. For my practical application, though, it would be interesting to see if $M$ could be made larger with a relatively high probability for points that are reasonably "spread out."
$endgroup$
– Dennis
Mar 27 at 21:07
$begingroup$
@Dennis If you find the answer useful, consider upvoting it, you don’t have to accept it.
$endgroup$
– Oldboy
Mar 27 at 22:19
$begingroup$
I did. Just didn't register because apparently I don't have enough "reputation points" or whatever.
$endgroup$
– Dennis
Mar 27 at 22:31
add a comment |
$begingroup$
Thanks! Just realized my flaw for the $n=3$ case. Theoretically, since points can be arbitrarily close together, I think $M$ cannot be any larger than $Rsqrt3$, and since adding more points only puts more restrictions on where the next one can go, I don't think $M$ gets any smaller, either. For my practical application, though, it would be interesting to see if $M$ could be made larger with a relatively high probability for points that are reasonably "spread out."
$endgroup$
– Dennis
Mar 27 at 21:07
$begingroup$
@Dennis If you find the answer useful, consider upvoting it, you don’t have to accept it.
$endgroup$
– Oldboy
Mar 27 at 22:19
$begingroup$
I did. Just didn't register because apparently I don't have enough "reputation points" or whatever.
$endgroup$
– Dennis
Mar 27 at 22:31
$begingroup$
Thanks! Just realized my flaw for the $n=3$ case. Theoretically, since points can be arbitrarily close together, I think $M$ cannot be any larger than $Rsqrt3$, and since adding more points only puts more restrictions on where the next one can go, I don't think $M$ gets any smaller, either. For my practical application, though, it would be interesting to see if $M$ could be made larger with a relatively high probability for points that are reasonably "spread out."
$endgroup$
– Dennis
Mar 27 at 21:07
$begingroup$
Thanks! Just realized my flaw for the $n=3$ case. Theoretically, since points can be arbitrarily close together, I think $M$ cannot be any larger than $Rsqrt3$, and since adding more points only puts more restrictions on where the next one can go, I don't think $M$ gets any smaller, either. For my practical application, though, it would be interesting to see if $M$ could be made larger with a relatively high probability for points that are reasonably "spread out."
$endgroup$
– Dennis
Mar 27 at 21:07
$begingroup$
@Dennis If you find the answer useful, consider upvoting it, you don’t have to accept it.
$endgroup$
– Oldboy
Mar 27 at 22:19
$begingroup$
@Dennis If you find the answer useful, consider upvoting it, you don’t have to accept it.
$endgroup$
– Oldboy
Mar 27 at 22:19
$begingroup$
I did. Just didn't register because apparently I don't have enough "reputation points" or whatever.
$endgroup$
– Dennis
Mar 27 at 22:31
$begingroup$
I did. Just didn't register because apparently I don't have enough "reputation points" or whatever.
$endgroup$
– Dennis
Mar 27 at 22:31
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3163510%2fshortest-maximum-pairwise-distance-of-points-in-a-circle-of-radius-r%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
