Relationship between two centers of circles in a Venn diagram
9
$begingroup$
Let $S$ be a circle of 1 unit area. No part of circles $A$ and $B$ are outside the circle $S$.
Let $n(S)=1$, $n(A)$, and $n(B)$ be the area of circle $S$, $A$, and $B$, respectively.
For the given values $n(A)=a$, $n(B)=b$, and $n(A cap B)=c$, find the relationship of their centers in terms of $a$, $b$, and $c$.
The objective is to draw both inner circles.
geometry circle
asked Jul 12 '12 at 8:16
kiss my armpitkiss my armpit
2,16541834
$endgroup$
|
show 1 more comment
9
$begingroup$
Let $S$ be a circle of 1 unit area. No part of circles $A$ and $B$ are outside the circle $S$.
Let $n(S)=1$, $n(A)$, and $n(B)$ be the area of circle $S$, $A$, and $B$, respectively.
For the given values $n(A)=a$, $n(B)=b$, and $n(A cap B)=c$, find the relationship of their centers in terms of $a$, $b$, and $c$.
The objective is to draw both inner circles.
geometry circle
asked Jul 12 '12 at 8:16
kiss my armpitkiss my armpit
2,16541834
$endgroup$
2
$begingroup$
When you say 'relationship', do you mean distance? Also, apart from giving bounds for $a$, $b$ and $c$, $S$ doesn't really do anything, does it?
$endgroup$
– Karolis Juodelė
Jul 12 '12 at 8:26
$begingroup$
You've tried making a drawing?
$endgroup$
– J. M. is not a mathematician
Jul 12 '12 at 8:26
$begingroup$
@J.M.: I haven't drawn it yet because there might be many solutions.
$endgroup$
– kiss my armpit
Jul 12 '12 at 8:36
$begingroup$
If you're trying to draw an area-proportional Venn diagram, you could have $a=b=frac12$ but $c=0$, in which case there is no solution.
$endgroup$
– Rahul
Jan 22 '13 at 20:03
$begingroup$
there might be many solutions... Are you sure? For any given values of $a$ and $b$, any $0lt cltmin(a,b)$ uniquely determines the distance between their centers.
$endgroup$
– Did
Jan 26 '13 at 14:08
|
show 1 more comment
9
9
9
1
$begingroup$
Let $S$ be a circle of 1 unit area. No part of circles $A$ and $B$ are outside the circle $S$.
Let $n(S)=1$, $n(A)$, and $n(B)$ be the area of circle $S$, $A$, and $B$, respectively.
For the given values $n(A)=a$, $n(B)=b$, and $n(A cap B)=c$, find the relationship of their centers in terms of $a$, $b$, and $c$.
The objective is to draw both inner circles.
geometry circle
asked Jul 12 '12 at 8:16
kiss my armpitkiss my armpit
2,16541834
$endgroup$
Let $S$ be a circle of 1 unit area. No part of circles $A$ and $B$ are outside the circle $S$.
Let $n(S)=1$, $n(A)$, and $n(B)$ be the area of circle $S$, $A$, and $B$, respectively.
For the given values $n(A)=a$, $n(B)=b$, and $n(A cap B)=c$, find the relationship of their centers in terms of $a$, $b$, and $c$.
The objective is to draw both inner circles.
geometry circle
geometry circle
asked Jul 12 '12 at 8:16
kiss my armpitkiss my armpit
2,16541834
asked Jul 12 '12 at 8:16
kiss my armpitkiss my armpit
2,16541834
edited Jul 12 '12 at 8:29
kiss my armpit
asked Jul 12 '12 at 8:16
kiss my armpitkiss my armpit
2,16541834
asked Jul 12 '12 at 8:16
kiss my armpitkiss my armpit
2,16541834
asked Jul 12 '12 at 8:16
kiss my armpitkiss my armpit
2,16541834
2,16541834
2
$begingroup$
When you say 'relationship', do you mean distance? Also, apart from giving bounds for $a$, $b$ and $c$, $S$ doesn't really do anything, does it?
$endgroup$
– Karolis Juodelė
Jul 12 '12 at 8:26
$begingroup$
You've tried making a drawing?
$endgroup$
– J. M. is not a mathematician
Jul 12 '12 at 8:26
$begingroup$
@J.M.: I haven't drawn it yet because there might be many solutions.
$endgroup$
– kiss my armpit
Jul 12 '12 at 8:36
$begingroup$
If you're trying to draw an area-proportional Venn diagram, you could have $a=b=frac12$ but $c=0$, in which case there is no solution.
$endgroup$
– Rahul
Jan 22 '13 at 20:03
$begingroup$
there might be many solutions... Are you sure? For any given values of $a$ and $b$, any $0lt cltmin(a,b)$ uniquely determines the distance between their centers.
$endgroup$
– Did
Jan 26 '13 at 14:08
|
show 1 more comment
2
$begingroup$
When you say 'relationship', do you mean distance? Also, apart from giving bounds for $a$, $b$ and $c$, $S$ doesn't really do anything, does it?
$endgroup$
– Karolis Juodelė
Jul 12 '12 at 8:26
$begingroup$
You've tried making a drawing?
$endgroup$
– J. M. is not a mathematician
Jul 12 '12 at 8:26
$begingroup$
@J.M.: I haven't drawn it yet because there might be many solutions.
$endgroup$
– kiss my armpit
Jul 12 '12 at 8:36
$begingroup$
If you're trying to draw an area-proportional Venn diagram, you could have $a=b=frac12$ but $c=0$, in which case there is no solution.
$endgroup$
– Rahul
Jan 22 '13 at 20:03
$begingroup$
there might be many solutions... Are you sure? For any given values of $a$ and $b$, any $0lt cltmin(a,b)$ uniquely determines the distance between their centers.
$endgroup$
– Did
Jan 26 '13 at 14:08
2
2
$begingroup$
When you say 'relationship', do you mean distance? Also, apart from giving bounds for $a$, $b$ and $c$, $S$ doesn't really do anything, does it?
$endgroup$
– Karolis Juodelė
Jul 12 '12 at 8:26
$begingroup$
When you say 'relationship', do you mean distance? Also, apart from giving bounds for $a$, $b$ and $c$, $S$ doesn't really do anything, does it?
$endgroup$
– Karolis Juodelė
Jul 12 '12 at 8:26
$begingroup$
You've tried making a drawing?
$endgroup$
– J. M. is not a mathematician
Jul 12 '12 at 8:26
$begingroup$
You've tried making a drawing?
$endgroup$
– J. M. is not a mathematician
Jul 12 '12 at 8:26
$begingroup$
@J.M.: I haven't drawn it yet because there might be many solutions.
$endgroup$
– kiss my armpit
Jul 12 '12 at 8:36
$begingroup$
@J.M.: I haven't drawn it yet because there might be many solutions.
$endgroup$
– kiss my armpit
Jul 12 '12 at 8:36
$begingroup$
If you're trying to draw an area-proportional Venn diagram, you could have $a=b=frac12$ but $c=0$, in which case there is no solution.
$endgroup$
– Rahul
Jan 22 '13 at 20:03
$begingroup$
If you're trying to draw an area-proportional Venn diagram, you could have $a=b=frac12$ but $c=0$, in which case there is no solution.
$endgroup$
– Rahul
Jan 22 '13 at 20:03
$begingroup$
there might be many solutions... Are you sure? For any given values of $a$ and $b$, any $0lt cltmin(a,b)$ uniquely determines the distance between their centers.
$endgroup$
– Did
Jan 26 '13 at 14:08
$begingroup$
there might be many solutions... Are you sure? For any given values of $a$ and $b$, any $0lt cltmin(a,b)$ uniquely determines the distance between their centers.
$endgroup$
– Did
Jan 26 '13 at 14:08
|
show 1 more comment
1 Answer
1
active
oldest
votes
0
$begingroup$
I haven't gone into calculations but tried to device an algorithm for the prob
- Fix an origin at one of the inner circle's center say of $A$ and let us call it point $O$
- From the area of the $2$ inner circle find their respective radi
- Now assign variable center to the second inner circle $B$ let us call it point $X$
- From the radius and center find the equation of each of the inner circles respectively
- From the equation of the $2$ inner circles find the equation of 'chord of contact'(also apply the intersection constraint for the same)
- Find the points of intersection of the the chord of the contact and any of the $2$ circles and call it points $Y$ and $Z$
- Now find the angle $YOZ$ and $YXZ$ and called it $alpha$ and $beta$
- With the help of the angles above find area of sector $YOZ$ and $YXZ$ and call it $S_a$ and $S_b$
- now the center of the circle $B$ will depend on the equation
$S_a + S_b - c = S_{triangle{YOZ}} + S_{triangle{YXZ}}$
Now for the relation with outer circle:
- distance between the two centers of the inner circles and call it as $m$. Let their radii are $R_a$, $R_b$.
- $l = R_a+R_b+m$
- $b = max{R_a,R_b}$
- Now calculate the area $l cdot b$
- put the condition that area less than area of outer circle S.
this algorithm for relation wiht outer circle can be improved!
edited Jul 16 '12 at 15:04
Norbert
45.7k774161
answered Jul 12 '12 at 9:55
nimeshkiranvermanimeshkiranverma
248412
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
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
});
}
});
draft saved
draft discarded
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%2f169838%2frelationship-between-two-centers-of-circles-in-a-venn-diagram%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
0
$begingroup$
I haven't gone into calculations but tried to device an algorithm for the prob
- Fix an origin at one of the inner circle's center say of $A$ and let us call it point $O$
- From the area of the $2$ inner circle find their respective radi
- Now assign variable center to the second inner circle $B$ let us call it point $X$
- From the radius and center find the equation of each of the inner circles respectively
- From the equation of the $2$ inner circles find the equation of 'chord of contact'(also apply the intersection constraint for the same)
- Find the points of intersection of the the chord of the contact and any of the $2$ circles and call it points $Y$ and $Z$
- Now find the angle $YOZ$ and $YXZ$ and called it $alpha$ and $beta$
- With the help of the angles above find area of sector $YOZ$ and $YXZ$ and call it $S_a$ and $S_b$
- now the center of the circle $B$ will depend on the equation
$S_a + S_b - c = S_{triangle{YOZ}} + S_{triangle{YXZ}}$
Now for the relation with outer circle:
- distance between the two centers of the inner circles and call it as $m$. Let their radii are $R_a$, $R_b$.
- $l = R_a+R_b+m$
- $b = max{R_a,R_b}$
- Now calculate the area $l cdot b$
- put the condition that area less than area of outer circle S.
this algorithm for relation wiht outer circle can be improved!
edited Jul 16 '12 at 15:04
Norbert
45.7k774161
answered Jul 12 '12 at 9:55
nimeshkiranvermanimeshkiranverma
248412
$endgroup$
add a comment |
0
$begingroup$
I haven't gone into calculations but tried to device an algorithm for the prob
- Fix an origin at one of the inner circle's center say of $A$ and let us call it point $O$
- From the area of the $2$ inner circle find their respective radi
- Now assign variable center to the second inner circle $B$ let us call it point $X$
- From the radius and center find the equation of each of the inner circles respectively
- From the equation of the $2$ inner circles find the equation of 'chord of contact'(also apply the intersection constraint for the same)
- Find the points of intersection of the the chord of the contact and any of the $2$ circles and call it points $Y$ and $Z$
- Now find the angle $YOZ$ and $YXZ$ and called it $alpha$ and $beta$
- With the help of the angles above find area of sector $YOZ$ and $YXZ$ and call it $S_a$ and $S_b$
- now the center of the circle $B$ will depend on the equation
$S_a + S_b - c = S_{triangle{YOZ}} + S_{triangle{YXZ}}$
Now for the relation with outer circle:
- distance between the two centers of the inner circles and call it as $m$. Let their radii are $R_a$, $R_b$.
- $l = R_a+R_b+m$
- $b = max{R_a,R_b}$
- Now calculate the area $l cdot b$
- put the condition that area less than area of outer circle S.
this algorithm for relation wiht outer circle can be improved!
edited Jul 16 '12 at 15:04
Norbert
45.7k774161
answered Jul 12 '12 at 9:55
nimeshkiranvermanimeshkiranverma
248412
$endgroup$
add a comment |
0
0
0
$begingroup$
I haven't gone into calculations but tried to device an algorithm for the prob
- Fix an origin at one of the inner circle's center say of $A$ and let us call it point $O$
- From the area of the $2$ inner circle find their respective radi
- Now assign variable center to the second inner circle $B$ let us call it point $X$
- From the radius and center find the equation of each of the inner circles respectively
- From the equation of the $2$ inner circles find the equation of 'chord of contact'(also apply the intersection constraint for the same)
- Find the points of intersection of the the chord of the contact and any of the $2$ circles and call it points $Y$ and $Z$
- Now find the angle $YOZ$ and $YXZ$ and called it $alpha$ and $beta$
- With the help of the angles above find area of sector $YOZ$ and $YXZ$ and call it $S_a$ and $S_b$
- now the center of the circle $B$ will depend on the equation
$S_a + S_b - c = S_{triangle{YOZ}} + S_{triangle{YXZ}}$
Now for the relation with outer circle:
- distance between the two centers of the inner circles and call it as $m$. Let their radii are $R_a$, $R_b$.
- $l = R_a+R_b+m$
- $b = max{R_a,R_b}$
- Now calculate the area $l cdot b$
- put the condition that area less than area of outer circle S.
this algorithm for relation wiht outer circle can be improved!
edited Jul 16 '12 at 15:04
Norbert
45.7k774161
answered Jul 12 '12 at 9:55
nimeshkiranvermanimeshkiranverma
248412
$endgroup$
I haven't gone into calculations but tried to device an algorithm for the prob
- Fix an origin at one of the inner circle's center say of $A$ and let us call it point $O$
- From the area of the $2$ inner circle find their respective radi
- Now assign variable center to the second inner circle $B$ let us call it point $X$
- From the radius and center find the equation of each of the inner circles respectively
- From the equation of the $2$ inner circles find the equation of 'chord of contact'(also apply the intersection constraint for the same)
- Find the points of intersection of the the chord of the contact and any of the $2$ circles and call it points $Y$ and $Z$
- Now find the angle $YOZ$ and $YXZ$ and called it $alpha$ and $beta$
- With the help of the angles above find area of sector $YOZ$ and $YXZ$ and call it $S_a$ and $S_b$
- now the center of the circle $B$ will depend on the equation
$S_a + S_b - c = S_{triangle{YOZ}} + S_{triangle{YXZ}}$
Now for the relation with outer circle:
- distance between the two centers of the inner circles and call it as $m$. Let their radii are $R_a$, $R_b$.
- $l = R_a+R_b+m$
- $b = max{R_a,R_b}$
- Now calculate the area $l cdot b$
- put the condition that area less than area of outer circle S.
this algorithm for relation wiht outer circle can be improved!
edited Jul 16 '12 at 15:04
Norbert
45.7k774161
answered Jul 12 '12 at 9:55
nimeshkiranvermanimeshkiranverma
248412
edited Jul 16 '12 at 15:04
Norbert
45.7k774161
edited Jul 16 '12 at 15:04
Norbert
45.7k774161
edited Jul 16 '12 at 15:04
Norbert
45.7k774161
45.7k774161
answered Jul 12 '12 at 9:55
nimeshkiranvermanimeshkiranverma
248412
answered Jul 12 '12 at 9:55
nimeshkiranvermanimeshkiranverma
248412
answered Jul 12 '12 at 9:55
nimeshkiranvermanimeshkiranverma
248412
248412
add a comment |
add a comment |
draft saved
draft discarded
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.
draft saved
draft discarded
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%2f169838%2frelationship-between-two-centers-of-circles-in-a-venn-diagram%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
2
$begingroup$
When you say 'relationship', do you mean distance? Also, apart from giving bounds for $a$, $b$ and $c$, $S$ doesn't really do anything, does it?
$endgroup$
– Karolis Juodelė
Jul 12 '12 at 8:26
$begingroup$
You've tried making a drawing?
$endgroup$
– J. M. is not a mathematician
Jul 12 '12 at 8:26
$begingroup$
@J.M.: I haven't drawn it yet because there might be many solutions.
$endgroup$
– kiss my armpit
Jul 12 '12 at 8:36
$begingroup$
If you're trying to draw an area-proportional Venn diagram, you could have $a=b=frac12$ but $c=0$, in which case there is no solution.
$endgroup$
– Rahul
Jan 22 '13 at 20:03
$begingroup$
there might be many solutions... Are you sure? For any given values of $a$ and $b$, any $0lt cltmin(a,b)$ uniquely determines the distance between their centers.
$endgroup$
– Did
Jan 26 '13 at 14:08