How can be set -theoretically defined the whole collection of possible circles in a given plane P ? ( not using analytic geometry) The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Minimal number of moves to construct the challenges (circle packings and regular polygons) in Ancient Greek Geometry?Credit Given - Geometricly Modeling Infinity with 3 planes and 9 circles - Ratio of CirclesGiven distances (shortest paths) between four cities, how to show that they cannot be in the same plane?Points closest to the center of the cubefinding the center of a circle (elementary geometry)Results of projective and Euclidean geometryRuler and Compasses without Ruler (Mohr-Mascheroni)Does anyone have information on using right circular cones to solve problems about circles in the plane?Geometric proof of equivalence between two constructs of ellipseProving Hilbert's Axioms as Theorems in $ℝ^n$
How to stretch delimiters to envolve matrices inside of a kbordermatrix?
What do you call a plan that's an alternative plan in case your initial plan fails?
What is this lever in Argentinian toilets?
Does Parliament hold absolute power in the UK?
Did the UK government pay "millions and millions of dollars" to try to snag Julian Assange?
When did F become S in typeography, and why?
Keeping a retro style to sci-fi spaceships?
How can I protect witches in combat who wear limited clothing?
Is above average number of years spent on PhD considered a red flag in future academia or industry positions?
How do you keep chess fun when your opponent constantly beats you?
How did the audience guess the pentatonic scale in Bobby McFerrin's presentation?
Why can't devices on different VLANs, but on the same subnet, communicate?
How to politely respond to generic emails requesting a PhD/job in my lab? Without wasting too much time
What can I do if neighbor is blocking my solar panels intentionally?
how can a perfect fourth interval be considered either consonant or dissonant?
Difference between "generating set" and free product?
Mortgage adviser recommends a longer term than necessary combined with overpayments
In horse breeding, what is the female equivalent of putting a horse out "to stud"?
How to split my screen on my Macbook Air?
What information about me do stores get via my credit card?
Relations between two reciprocal partial derivatives?
How is simplicity better than precision and clarity in prose?
He got a vote 80% that of Emmanuel Macron’s
How long does the line of fire that you can create as an action using the Investiture of Flame spell last?
How can be set -theoretically defined the whole collection of possible circles in a given plane P ? ( not using analytic geometry)
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Minimal number of moves to construct the challenges (circle packings and regular polygons) in Ancient Greek Geometry?Credit Given - Geometricly Modeling Infinity with 3 planes and 9 circles - Ratio of CirclesGiven distances (shortest paths) between four cities, how to show that they cannot be in the same plane?Points closest to the center of the cubefinding the center of a circle (elementary geometry)Results of projective and Euclidean geometryRuler and Compasses without Ruler (Mohr-Mascheroni)Does anyone have information on using right circular cones to solve problems about circles in the plane?Geometric proof of equivalence between two constructs of ellipseProving Hilbert's Axioms as Theorems in $ℝ^n$
$begingroup$
[ edited 9th april 2019]
I'd like to solve this question using only basic concepts of geometry ( without analytic geometry) and of elementary set theory.
If I am correct, the following process allows me to " build" the desired set. But, how could I express, in a single formula,the result of this process, using the proper logical and set theoretical symbolism?
(1) I choose an arbitrary point O (as first "center").
(2) I define an equivalence relation : the set (of points) A is equivalent to the set (of points) B iff all the elements of A are at the same distance from O as are all the elements of B ; I obtain an infinity of equivalence classes ( one for each possible " orbit")
(3) inside each equivalence class ( that is , for each " orbit") I choose the greatest set using the inclusion relation ( for the circle is the greatest set amongst sets whose points are all at a given distance from a given "center"); I obtain an infinity of circles for the first center O.
(4) I repeat (1)-(3) for each point in the plane, that is, for each possible "center".
(5) Using, maybe, the union operation, I gather all my ( infinite) collections of circles for a given center in a new set , which would be the set of circles in a plane P.
Remark - This is not homework; it is a question I ask myself, as a gratuitous exercise in logic/ elementary set theory
elementary-set-theory euclidean-geometry
asked Apr 8 at 13:48
Eleonore Saint JamesEleonore Saint James
1629
$endgroup$
|
show 1 more comment
$begingroup$
[ edited 9th april 2019]
I'd like to solve this question using only basic concepts of geometry ( without analytic geometry) and of elementary set theory.
If I am correct, the following process allows me to " build" the desired set. But, how could I express, in a single formula,the result of this process, using the proper logical and set theoretical symbolism?
(1) I choose an arbitrary point O (as first "center").
(2) I define an equivalence relation : the set (of points) A is equivalent to the set (of points) B iff all the elements of A are at the same distance from O as are all the elements of B ; I obtain an infinity of equivalence classes ( one for each possible " orbit")
(3) inside each equivalence class ( that is , for each " orbit") I choose the greatest set using the inclusion relation ( for the circle is the greatest set amongst sets whose points are all at a given distance from a given "center"); I obtain an infinity of circles for the first center O.
(4) I repeat (1)-(3) for each point in the plane, that is, for each possible "center".
(5) Using, maybe, the union operation, I gather all my ( infinite) collections of circles for a given center in a new set , which would be the set of circles in a plane P.
Remark - This is not homework; it is a question I ask myself, as a gratuitous exercise in logic/ elementary set theory
elementary-set-theory euclidean-geometry
asked Apr 8 at 13:48
Eleonore Saint JamesEleonore Saint James
1629
$endgroup$
1
$begingroup$
Pay attention to the difference between a set and its points. The set CIRCLE you defined is a set of subsets. The union of all subset is all the plane (every point is contained in some circle) but the set CIRCLE is not the plane
$endgroup$
– karmalu
Apr 8 at 13:56
1
$begingroup$
Why not $C=(x,y,r): rgt 0$?
$endgroup$
– John Douma
Apr 8 at 13:59
$begingroup$
@ John Douma - Is a triple sufficient? Would we need a quadruple in order to take the center into account ? Or not?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:05
1
$begingroup$
@Eleonore Saint James - I think (x,y) is the centre.
$endgroup$
– Paul
Apr 8 at 14:08
$begingroup$
I think I understand in which way a given circle can be identified by a triple < x, y, r> x and y being the coordinates of the center and r the radius. But I am wandering whether this triple is " able" to define the circle as a set of points. Is it the case?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:16
|
show 1 more comment
$begingroup$
[ edited 9th april 2019]
I'd like to solve this question using only basic concepts of geometry ( without analytic geometry) and of elementary set theory.
If I am correct, the following process allows me to " build" the desired set. But, how could I express, in a single formula,the result of this process, using the proper logical and set theoretical symbolism?
(1) I choose an arbitrary point O (as first "center").
(2) I define an equivalence relation : the set (of points) A is equivalent to the set (of points) B iff all the elements of A are at the same distance from O as are all the elements of B ; I obtain an infinity of equivalence classes ( one for each possible " orbit")
(3) inside each equivalence class ( that is , for each " orbit") I choose the greatest set using the inclusion relation ( for the circle is the greatest set amongst sets whose points are all at a given distance from a given "center"); I obtain an infinity of circles for the first center O.
(4) I repeat (1)-(3) for each point in the plane, that is, for each possible "center".
(5) Using, maybe, the union operation, I gather all my ( infinite) collections of circles for a given center in a new set , which would be the set of circles in a plane P.
Remark - This is not homework; it is a question I ask myself, as a gratuitous exercise in logic/ elementary set theory
elementary-set-theory euclidean-geometry
asked Apr 8 at 13:48
Eleonore Saint JamesEleonore Saint James
1629
$endgroup$
[ edited 9th april 2019]
I'd like to solve this question using only basic concepts of geometry ( without analytic geometry) and of elementary set theory.
If I am correct, the following process allows me to " build" the desired set. But, how could I express, in a single formula,the result of this process, using the proper logical and set theoretical symbolism?
(1) I choose an arbitrary point O (as first "center").
(2) I define an equivalence relation : the set (of points) A is equivalent to the set (of points) B iff all the elements of A are at the same distance from O as are all the elements of B ; I obtain an infinity of equivalence classes ( one for each possible " orbit")
(3) inside each equivalence class ( that is , for each " orbit") I choose the greatest set using the inclusion relation ( for the circle is the greatest set amongst sets whose points are all at a given distance from a given "center"); I obtain an infinity of circles for the first center O.
(4) I repeat (1)-(3) for each point in the plane, that is, for each possible "center".
(5) Using, maybe, the union operation, I gather all my ( infinite) collections of circles for a given center in a new set , which would be the set of circles in a plane P.
Remark - This is not homework; it is a question I ask myself, as a gratuitous exercise in logic/ elementary set theory
elementary-set-theory euclidean-geometry
elementary-set-theory euclidean-geometry
asked Apr 8 at 13:48
Eleonore Saint JamesEleonore Saint James
1629
asked Apr 8 at 13:48
Eleonore Saint JamesEleonore Saint James
1629
edited Apr 10 at 21:08
Eleonore Saint James
asked Apr 8 at 13:48
Eleonore Saint JamesEleonore Saint James
1629
asked Apr 8 at 13:48
Eleonore Saint JamesEleonore Saint James
1629
asked Apr 8 at 13:48
Eleonore Saint JamesEleonore Saint James
1629
1629
1
$begingroup$
Pay attention to the difference between a set and its points. The set CIRCLE you defined is a set of subsets. The union of all subset is all the plane (every point is contained in some circle) but the set CIRCLE is not the plane
$endgroup$
– karmalu
Apr 8 at 13:56
1
$begingroup$
Why not $C=(x,y,r): rgt 0$?
$endgroup$
– John Douma
Apr 8 at 13:59
$begingroup$
@ John Douma - Is a triple sufficient? Would we need a quadruple in order to take the center into account ? Or not?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:05
1
$begingroup$
@Eleonore Saint James - I think (x,y) is the centre.
$endgroup$
– Paul
Apr 8 at 14:08
$begingroup$
I think I understand in which way a given circle can be identified by a triple < x, y, r> x and y being the coordinates of the center and r the radius. But I am wandering whether this triple is " able" to define the circle as a set of points. Is it the case?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:16
|
show 1 more comment
1
$begingroup$
Pay attention to the difference between a set and its points. The set CIRCLE you defined is a set of subsets. The union of all subset is all the plane (every point is contained in some circle) but the set CIRCLE is not the plane
$endgroup$
– karmalu
Apr 8 at 13:56
1
$begingroup$
Why not $C=(x,y,r): rgt 0$?
$endgroup$
– John Douma
Apr 8 at 13:59
$begingroup$
@ John Douma - Is a triple sufficient? Would we need a quadruple in order to take the center into account ? Or not?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:05
1
$begingroup$
@Eleonore Saint James - I think (x,y) is the centre.
$endgroup$
– Paul
Apr 8 at 14:08
$begingroup$
I think I understand in which way a given circle can be identified by a triple < x, y, r> x and y being the coordinates of the center and r the radius. But I am wandering whether this triple is " able" to define the circle as a set of points. Is it the case?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:16
1
1
$begingroup$
Pay attention to the difference between a set and its points. The set CIRCLE you defined is a set of subsets. The union of all subset is all the plane (every point is contained in some circle) but the set CIRCLE is not the plane
$endgroup$
– karmalu
Apr 8 at 13:56
$begingroup$
Pay attention to the difference between a set and its points. The set CIRCLE you defined is a set of subsets. The union of all subset is all the plane (every point is contained in some circle) but the set CIRCLE is not the plane
$endgroup$
– karmalu
Apr 8 at 13:56
1
1
$begingroup$
Why not $C=(x,y,r): rgt 0$?
$endgroup$
– John Douma
Apr 8 at 13:59
$begingroup$
Why not $C=(x,y,r): rgt 0$?
$endgroup$
– John Douma
Apr 8 at 13:59
$begingroup$
@ John Douma - Is a triple sufficient? Would we need a quadruple in order to take the center into account ? Or not?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:05
$begingroup$
@ John Douma - Is a triple sufficient? Would we need a quadruple in order to take the center into account ? Or not?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:05
1
1
$begingroup$
@Eleonore Saint James - I think (x,y) is the centre.
$endgroup$
– Paul
Apr 8 at 14:08
$begingroup$
@Eleonore Saint James - I think (x,y) is the centre.
$endgroup$
– Paul
Apr 8 at 14:08
$begingroup$
I think I understand in which way a given circle can be identified by a triple < x, y, r> x and y being the coordinates of the center and r the radius. But I am wandering whether this triple is " able" to define the circle as a set of points. Is it the case?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:16
$begingroup$
I think I understand in which way a given circle can be identified by a triple < x, y, r> x and y being the coordinates of the center and r the radius. But I am wandering whether this triple is " able" to define the circle as a set of points. Is it the case?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:16
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
Mathematicians already have a symbol for a circle of radius $r$ centred at $(x_0 ,y_0 )$, it is $$B((x_0 ,y_0 ),r)= (x,y) $$ Then, the set of all circles is $$C=B((x_0 ,y_0 ),r) $$
answered Apr 8 at 14:27
NazimJNazimJ
880110
$endgroup$
$begingroup$
@ NazimJ - Thanks for your answer. ( I upvoted). I think this answer is correct. Do you think it would be possible to define the desired collection without using analytic geometry?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:58
1
$begingroup$
You could let $vecx$ denote a point on the plane, then as mentioned in previous comments, every open ball $B(vecx, r)$ can be "labelled" by an ordered pair $(vecx,r)$. And the set $(vecx,r) $ is isomorphic to the set of all circles, since every circle has a label, and every label corresponds to a circle (it is a bijection). Is that what you are looking for?
$endgroup$
– NazimJ
Apr 8 at 15:34
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%2f3179657%2fhow-can-be-set-theoretically-defined-the-whole-collection-of-possible-circles-i%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
$begingroup$
Mathematicians already have a symbol for a circle of radius $r$ centred at $(x_0 ,y_0 )$, it is $$B((x_0 ,y_0 ),r)= (x,y) $$ Then, the set of all circles is $$C=B((x_0 ,y_0 ),r) $$
answered Apr 8 at 14:27
NazimJNazimJ
880110
$endgroup$
$begingroup$
@ NazimJ - Thanks for your answer. ( I upvoted). I think this answer is correct. Do you think it would be possible to define the desired collection without using analytic geometry?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:58
1
$begingroup$
You could let $vecx$ denote a point on the plane, then as mentioned in previous comments, every open ball $B(vecx, r)$ can be "labelled" by an ordered pair $(vecx,r)$. And the set $(vecx,r) $ is isomorphic to the set of all circles, since every circle has a label, and every label corresponds to a circle (it is a bijection). Is that what you are looking for?
$endgroup$
– NazimJ
Apr 8 at 15:34
add a comment |
$begingroup$
Mathematicians already have a symbol for a circle of radius $r$ centred at $(x_0 ,y_0 )$, it is $$B((x_0 ,y_0 ),r)= (x,y) $$ Then, the set of all circles is $$C=B((x_0 ,y_0 ),r) $$
answered Apr 8 at 14:27
NazimJNazimJ
880110
$endgroup$
$begingroup$
@ NazimJ - Thanks for your answer. ( I upvoted). I think this answer is correct. Do you think it would be possible to define the desired collection without using analytic geometry?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:58
1
$begingroup$
You could let $vecx$ denote a point on the plane, then as mentioned in previous comments, every open ball $B(vecx, r)$ can be "labelled" by an ordered pair $(vecx,r)$. And the set $(vecx,r) $ is isomorphic to the set of all circles, since every circle has a label, and every label corresponds to a circle (it is a bijection). Is that what you are looking for?
$endgroup$
– NazimJ
Apr 8 at 15:34
add a comment |
$begingroup$
Mathematicians already have a symbol for a circle of radius $r$ centred at $(x_0 ,y_0 )$, it is $$B((x_0 ,y_0 ),r)= (x,y) $$ Then, the set of all circles is $$C=B((x_0 ,y_0 ),r) $$
answered Apr 8 at 14:27
NazimJNazimJ
880110
$endgroup$
Mathematicians already have a symbol for a circle of radius $r$ centred at $(x_0 ,y_0 )$, it is $$B((x_0 ,y_0 ),r)= (x,y) $$ Then, the set of all circles is $$C=B((x_0 ,y_0 ),r) $$
answered Apr 8 at 14:27
NazimJNazimJ
880110
answered Apr 8 at 14:27
NazimJNazimJ
880110
answered Apr 8 at 14:27
NazimJNazimJ
880110
answered Apr 8 at 14:27
NazimJNazimJ
880110
880110
$begingroup$
@ NazimJ - Thanks for your answer. ( I upvoted). I think this answer is correct. Do you think it would be possible to define the desired collection without using analytic geometry?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:58
1
$begingroup$
You could let $vecx$ denote a point on the plane, then as mentioned in previous comments, every open ball $B(vecx, r)$ can be "labelled" by an ordered pair $(vecx,r)$. And the set $(vecx,r) $ is isomorphic to the set of all circles, since every circle has a label, and every label corresponds to a circle (it is a bijection). Is that what you are looking for?
$endgroup$
– NazimJ
Apr 8 at 15:34
add a comment |
$begingroup$
@ NazimJ - Thanks for your answer. ( I upvoted). I think this answer is correct. Do you think it would be possible to define the desired collection without using analytic geometry?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:58
1
$begingroup$
You could let $vecx$ denote a point on the plane, then as mentioned in previous comments, every open ball $B(vecx, r)$ can be "labelled" by an ordered pair $(vecx,r)$. And the set $(vecx,r) $ is isomorphic to the set of all circles, since every circle has a label, and every label corresponds to a circle (it is a bijection). Is that what you are looking for?
$endgroup$
– NazimJ
Apr 8 at 15:34
$begingroup$
@ NazimJ - Thanks for your answer. ( I upvoted). I think this answer is correct. Do you think it would be possible to define the desired collection without using analytic geometry?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:58
$begingroup$
@ NazimJ - Thanks for your answer. ( I upvoted). I think this answer is correct. Do you think it would be possible to define the desired collection without using analytic geometry?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:58
1
1
$begingroup$
You could let $vecx$ denote a point on the plane, then as mentioned in previous comments, every open ball $B(vecx, r)$ can be "labelled" by an ordered pair $(vecx,r)$. And the set $(vecx,r) $ is isomorphic to the set of all circles, since every circle has a label, and every label corresponds to a circle (it is a bijection). Is that what you are looking for?
$endgroup$
– NazimJ
Apr 8 at 15:34
$begingroup$
You could let $vecx$ denote a point on the plane, then as mentioned in previous comments, every open ball $B(vecx, r)$ can be "labelled" by an ordered pair $(vecx,r)$. And the set $(vecx,r) $ is isomorphic to the set of all circles, since every circle has a label, and every label corresponds to a circle (it is a bijection). Is that what you are looking for?
$endgroup$
– NazimJ
Apr 8 at 15:34
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%2f3179657%2fhow-can-be-set-theoretically-defined-the-whole-collection-of-possible-circles-i%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
1
$begingroup$
Pay attention to the difference between a set and its points. The set CIRCLE you defined is a set of subsets. The union of all subset is all the plane (every point is contained in some circle) but the set CIRCLE is not the plane
$endgroup$
– karmalu
Apr 8 at 13:56
1
$begingroup$
Why not $C=(x,y,r): rgt 0$?
$endgroup$
– John Douma
Apr 8 at 13:59
$begingroup$
@ John Douma - Is a triple sufficient? Would we need a quadruple in order to take the center into account ? Or not?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:05
1
$begingroup$
@Eleonore Saint James - I think (x,y) is the centre.
$endgroup$
– Paul
Apr 8 at 14:08
$begingroup$
I think I understand in which way a given circle can be identified by a triple < x, y, r> x and y being the coordinates of the center and r the radius. But I am wandering whether this triple is " able" to define the circle as a set of points. Is it the case?
$endgroup$
– Eleonore Saint James
Apr 8 at 14:16