Mathematics and the art of linearizing the circle












10












$begingroup$


One of the most prominent problems of ancient mathematics was the squaring of the circle: to construct the square with the same area as a given circle.



A related problem is linearizing the circle: to find a natural transition between a given line segment of length $L$ and the circle with circumference $U = 2pi R = L$ (which presupposes to find the radius $R = L/2pi$).



The main "problem" is: Along which paths are the points of the line segment to be moved to finally yield the circle such that the transition appears "natural".



By natural I mean this transition:



enter image description here



The points of the line segments follow these paths:



enter image description here



as can be seen here:



enter image description here



To be honest: Even though these paths look very much like circle segments, I'm not quite sure and I didn't define them by an explicit formula (which I didn't have at hand) but heuristically using some support points and splining.



My questions are:





  • Are these paths really circle segments?


  • If so: How to parametrize them?


  • If not so: What kind of curves are they otherwise?






Please allow me – freely associating – to compare the pictures above with this (artificially symmetrized) picture of The Great Wave off Kanagawa



enter image description here






modular-arithmetic euclidean-geometry projective-geometry visualization art







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    Do we have an actual definition of what it means for this transition to "appear natural"?
    $endgroup$
    – Morgan Rodgers
    13 hours ago






  • 4




    $begingroup$
    I think it would be "natural" to think of a family of curves $gamma_t$, $tin[0,1]$, of constant curvature such that $gamma_0$ is the flat line (curvature 0), $gamma_1$ is the circle (curvature 1/R) and each $gamma_t$ has curvature $tcdot1/R$.
    $endgroup$
    – Mars Plastic
    13 hours ago






  • 2




    $begingroup$
    I think you could probably make a map that "appears natural" using circles for the paths, and you could also probably make a different map without using circles that also "appears natural". So without a definition of what that term means, I don't know what answer you are looking for.
    $endgroup$
    – Morgan Rodgers
    13 hours ago








  • 2




    $begingroup$
    Maybe I'm missing something, but this 'transition' seems quite arbitrary to me.
    $endgroup$
    – rafa11111
    13 hours ago






  • 2




    $begingroup$
    You ask about making it "appear natural", but it's not clear what this means. You have things that look like circle segments, and you ask if they "really are circle segments", but we just have a picture, so how can we tell?
    $endgroup$
    – Morgan Rodgers
    13 hours ago
















10












$begingroup$


One of the most prominent problems of ancient mathematics was the squaring of the circle: to construct the square with the same area as a given circle.



A related problem is linearizing the circle: to find a natural transition between a given line segment of length $L$ and the circle with circumference $U = 2pi R = L$ (which presupposes to find the radius $R = L/2pi$).



The main "problem" is: Along which paths are the points of the line segment to be moved to finally yield the circle such that the transition appears "natural".



By natural I mean this transition:



enter image description here



The points of the line segments follow these paths:



enter image description here



as can be seen here:



enter image description here



To be honest: Even though these paths look very much like circle segments, I'm not quite sure and I didn't define them by an explicit formula (which I didn't have at hand) but heuristically using some support points and splining.



My questions are:





  • Are these paths really circle segments?


  • If so: How to parametrize them?


  • If not so: What kind of curves are they otherwise?






Please allow me – freely associating – to compare the pictures above with this (artificially symmetrized) picture of The Great Wave off Kanagawa



enter image description here






modular-arithmetic euclidean-geometry projective-geometry visualization art







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    Do we have an actual definition of what it means for this transition to "appear natural"?
    $endgroup$
    – Morgan Rodgers
    13 hours ago






  • 4




    $begingroup$
    I think it would be "natural" to think of a family of curves $gamma_t$, $tin[0,1]$, of constant curvature such that $gamma_0$ is the flat line (curvature 0), $gamma_1$ is the circle (curvature 1/R) and each $gamma_t$ has curvature $tcdot1/R$.
    $endgroup$
    – Mars Plastic
    13 hours ago






  • 2




    $begingroup$
    I think you could probably make a map that "appears natural" using circles for the paths, and you could also probably make a different map without using circles that also "appears natural". So without a definition of what that term means, I don't know what answer you are looking for.
    $endgroup$
    – Morgan Rodgers
    13 hours ago








  • 2




    $begingroup$
    Maybe I'm missing something, but this 'transition' seems quite arbitrary to me.
    $endgroup$
    – rafa11111
    13 hours ago






  • 2




    $begingroup$
    You ask about making it "appear natural", but it's not clear what this means. You have things that look like circle segments, and you ask if they "really are circle segments", but we just have a picture, so how can we tell?
    $endgroup$
    – Morgan Rodgers
    13 hours ago














10












10








10


4



$begingroup$


One of the most prominent problems of ancient mathematics was the squaring of the circle: to construct the square with the same area as a given circle.



A related problem is linearizing the circle: to find a natural transition between a given line segment of length $L$ and the circle with circumference $U = 2pi R = L$ (which presupposes to find the radius $R = L/2pi$).



The main "problem" is: Along which paths are the points of the line segment to be moved to finally yield the circle such that the transition appears "natural".



By natural I mean this transition:



enter image description here



The points of the line segments follow these paths:



enter image description here



as can be seen here:



enter image description here



To be honest: Even though these paths look very much like circle segments, I'm not quite sure and I didn't define them by an explicit formula (which I didn't have at hand) but heuristically using some support points and splining.



My questions are:





  • Are these paths really circle segments?


  • If so: How to parametrize them?


  • If not so: What kind of curves are they otherwise?






Please allow me – freely associating – to compare the pictures above with this (artificially symmetrized) picture of The Great Wave off Kanagawa



enter image description here






modular-arithmetic euclidean-geometry projective-geometry visualization art







share|cite|improve this question











$endgroup$




One of the most prominent problems of ancient mathematics was the squaring of the circle: to construct the square with the same area as a given circle.



A related problem is linearizing the circle: to find a natural transition between a given line segment of length $L$ and the circle with circumference $U = 2pi R = L$ (which presupposes to find the radius $R = L/2pi$).



The main "problem" is: Along which paths are the points of the line segment to be moved to finally yield the circle such that the transition appears "natural".



By natural I mean this transition:



enter image description here



The points of the line segments follow these paths:



enter image description here



as can be seen here:



enter image description here



To be honest: Even though these paths look very much like circle segments, I'm not quite sure and I didn't define them by an explicit formula (which I didn't have at hand) but heuristically using some support points and splining.



My questions are:





  • Are these paths really circle segments?


  • If so: How to parametrize them?


  • If not so: What kind of curves are they otherwise?






Please allow me – freely associating – to compare the pictures above with this (artificially symmetrized) picture of The Great Wave off Kanagawa



enter image description here






modular-arithmetic euclidean-geometry projective-geometry visualization art




modular-arithmetic euclidean-geometry projective-geometry visualization art






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 13 hours ago







Hans Stricker

















asked 14 hours ago









Hans StrickerHans Stricker

6,44743993




6,44743993








  • 4




    $begingroup$
    Do we have an actual definition of what it means for this transition to "appear natural"?
    $endgroup$
    – Morgan Rodgers
    13 hours ago






  • 4




    $begingroup$
    I think it would be "natural" to think of a family of curves $gamma_t$, $tin[0,1]$, of constant curvature such that $gamma_0$ is the flat line (curvature 0), $gamma_1$ is the circle (curvature 1/R) and each $gamma_t$ has curvature $tcdot1/R$.
    $endgroup$
    – Mars Plastic
    13 hours ago






  • 2




    $begingroup$
    I think you could probably make a map that "appears natural" using circles for the paths, and you could also probably make a different map without using circles that also "appears natural". So without a definition of what that term means, I don't know what answer you are looking for.
    $endgroup$
    – Morgan Rodgers
    13 hours ago








  • 2




    $begingroup$
    Maybe I'm missing something, but this 'transition' seems quite arbitrary to me.
    $endgroup$
    – rafa11111
    13 hours ago






  • 2




    $begingroup$
    You ask about making it "appear natural", but it's not clear what this means. You have things that look like circle segments, and you ask if they "really are circle segments", but we just have a picture, so how can we tell?
    $endgroup$
    – Morgan Rodgers
    13 hours ago














  • 4




    $begingroup$
    Do we have an actual definition of what it means for this transition to "appear natural"?
    $endgroup$
    – Morgan Rodgers
    13 hours ago






  • 4




    $begingroup$
    I think it would be "natural" to think of a family of curves $gamma_t$, $tin[0,1]$, of constant curvature such that $gamma_0$ is the flat line (curvature 0), $gamma_1$ is the circle (curvature 1/R) and each $gamma_t$ has curvature $tcdot1/R$.
    $endgroup$
    – Mars Plastic
    13 hours ago






  • 2




    $begingroup$
    I think you could probably make a map that "appears natural" using circles for the paths, and you could also probably make a different map without using circles that also "appears natural". So without a definition of what that term means, I don't know what answer you are looking for.
    $endgroup$
    – Morgan Rodgers
    13 hours ago








  • 2




    $begingroup$
    Maybe I'm missing something, but this 'transition' seems quite arbitrary to me.
    $endgroup$
    – rafa11111
    13 hours ago






  • 2




    $begingroup$
    You ask about making it "appear natural", but it's not clear what this means. You have things that look like circle segments, and you ask if they "really are circle segments", but we just have a picture, so how can we tell?
    $endgroup$
    – Morgan Rodgers
    13 hours ago








4




4




$begingroup$
Do we have an actual definition of what it means for this transition to "appear natural"?
$endgroup$
– Morgan Rodgers
13 hours ago




$begingroup$
Do we have an actual definition of what it means for this transition to "appear natural"?
$endgroup$
– Morgan Rodgers
13 hours ago




4




4




$begingroup$
I think it would be "natural" to think of a family of curves $gamma_t$, $tin[0,1]$, of constant curvature such that $gamma_0$ is the flat line (curvature 0), $gamma_1$ is the circle (curvature 1/R) and each $gamma_t$ has curvature $tcdot1/R$.
$endgroup$
– Mars Plastic
13 hours ago




$begingroup$
I think it would be "natural" to think of a family of curves $gamma_t$, $tin[0,1]$, of constant curvature such that $gamma_0$ is the flat line (curvature 0), $gamma_1$ is the circle (curvature 1/R) and each $gamma_t$ has curvature $tcdot1/R$.
$endgroup$
– Mars Plastic
13 hours ago




2




2




$begingroup$
I think you could probably make a map that "appears natural" using circles for the paths, and you could also probably make a different map without using circles that also "appears natural". So without a definition of what that term means, I don't know what answer you are looking for.
$endgroup$
– Morgan Rodgers
13 hours ago






$begingroup$
I think you could probably make a map that "appears natural" using circles for the paths, and you could also probably make a different map without using circles that also "appears natural". So without a definition of what that term means, I don't know what answer you are looking for.
$endgroup$
– Morgan Rodgers
13 hours ago






2




2




$begingroup$
Maybe I'm missing something, but this 'transition' seems quite arbitrary to me.
$endgroup$
– rafa11111
13 hours ago




$begingroup$
Maybe I'm missing something, but this 'transition' seems quite arbitrary to me.
$endgroup$
– rafa11111
13 hours ago




2




2




$begingroup$
You ask about making it "appear natural", but it's not clear what this means. You have things that look like circle segments, and you ask if they "really are circle segments", but we just have a picture, so how can we tell?
$endgroup$
– Morgan Rodgers
13 hours ago




$begingroup$
You ask about making it "appear natural", but it's not clear what this means. You have things that look like circle segments, and you ask if they "really are circle segments", but we just have a picture, so how can we tell?
$endgroup$
– Morgan Rodgers
13 hours ago










2 Answers
2






active

oldest

votes


















9












$begingroup$

What you want can be achieved using circle arcs, centered at $(0,r)$, of radius $r$ and central angle $2pi R/r$, with $r$ varying between $1$ and $+infty$. But I don't know if that is "natural" or not. Here's how it looks:



enter image description here






share|cite|improve this answer











$endgroup$













  • $begingroup$
    That's essentially my approach. Good to know, that an expert like you came to the same conclusion. Concerning "natural": which approach could be more "natural"?
    $endgroup$
    – Hans Stricker
    12 hours ago










  • $begingroup$
    How (and with which tools) did you create your elegant animated gif? (I had a hard time with mine.)
    $endgroup$
    – Hans Stricker
    11 hours ago












  • $begingroup$
    I used GeoGebra, which is free: www.geogebra.org
    $endgroup$
    – Aretino
    11 hours ago










  • $begingroup$
    Can you share this plot with me, i.e. save the plot and share the link?
    $endgroup$
    – Hans Stricker
    10 hours ago










  • $begingroup$
    Here it is: ggbm.at/kyx9q3he
    $endgroup$
    – Aretino
    10 hours ago



















3












$begingroup$

If we want this transition to have all the points on the boundary of a circle at all times, then it makes most sense to parameterize by the radius of this circle (and apply a transformation to get it in terms of finite time later). For simplicity, I will also have the transition be to a vertical line.



We shall have the radius of the circle $C_r$ be $r$ and centre be $(-r,0)$, such that $(0,0)$ is on $C_r$ for all $r$. The coordinates of the point at arclength $s$ from $(0,0)$ are then given by $(r (cos(s/r) - 1), r sin(s/r))$.



The most natural way to transition would likely be varying the curvature at a constant rate; thus we create a family of curves $f_t:[-pi, pi]tomathbb{R}^2$ where $tin[0,1]$ by
begin{align}
f_t(s) &= left(frac{cos(s(1-t))-1}{1-t}, frac{sin(s(1-t))}{1-t}right)&t<1\
f_1(s) &= (0, s)&
end{align}



Since the goals here seem to be rather subjective, I would attempt this and see how it looks to you (beyond making substitutions as needed to result in a horizontal line).






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Note that the paths of the endpoints are given by setting $s = pm pi$ in your parametric curve. The resulting paths are not circle segments (as the OP guessed they might be); this can be seen by calculating their curvature.
    $endgroup$
    – Michael Seifert
    9 hours ago













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


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3129072%2fmathematics-and-the-art-of-linearizing-the-circle%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









9












$begingroup$

What you want can be achieved using circle arcs, centered at $(0,r)$, of radius $r$ and central angle $2pi R/r$, with $r$ varying between $1$ and $+infty$. But I don't know if that is "natural" or not. Here's how it looks:



enter image description here






share|cite|improve this answer











$endgroup$













  • $begingroup$
    That's essentially my approach. Good to know, that an expert like you came to the same conclusion. Concerning "natural": which approach could be more "natural"?
    $endgroup$
    – Hans Stricker
    12 hours ago










  • $begingroup$
    How (and with which tools) did you create your elegant animated gif? (I had a hard time with mine.)
    $endgroup$
    – Hans Stricker
    11 hours ago












  • $begingroup$
    I used GeoGebra, which is free: www.geogebra.org
    $endgroup$
    – Aretino
    11 hours ago










  • $begingroup$
    Can you share this plot with me, i.e. save the plot and share the link?
    $endgroup$
    – Hans Stricker
    10 hours ago










  • $begingroup$
    Here it is: ggbm.at/kyx9q3he
    $endgroup$
    – Aretino
    10 hours ago
















9












$begingroup$

What you want can be achieved using circle arcs, centered at $(0,r)$, of radius $r$ and central angle $2pi R/r$, with $r$ varying between $1$ and $+infty$. But I don't know if that is "natural" or not. Here's how it looks:



enter image description here






share|cite|improve this answer











$endgroup$













  • $begingroup$
    That's essentially my approach. Good to know, that an expert like you came to the same conclusion. Concerning "natural": which approach could be more "natural"?
    $endgroup$
    – Hans Stricker
    12 hours ago










  • $begingroup$
    How (and with which tools) did you create your elegant animated gif? (I had a hard time with mine.)
    $endgroup$
    – Hans Stricker
    11 hours ago












  • $begingroup$
    I used GeoGebra, which is free: www.geogebra.org
    $endgroup$
    – Aretino
    11 hours ago










  • $begingroup$
    Can you share this plot with me, i.e. save the plot and share the link?
    $endgroup$
    – Hans Stricker
    10 hours ago










  • $begingroup$
    Here it is: ggbm.at/kyx9q3he
    $endgroup$
    – Aretino
    10 hours ago














9












9








9





$begingroup$

What you want can be achieved using circle arcs, centered at $(0,r)$, of radius $r$ and central angle $2pi R/r$, with $r$ varying between $1$ and $+infty$. But I don't know if that is "natural" or not. Here's how it looks:



enter image description here






share|cite|improve this answer











$endgroup$



What you want can be achieved using circle arcs, centered at $(0,r)$, of radius $r$ and central angle $2pi R/r$, with $r$ varying between $1$ and $+infty$. But I don't know if that is "natural" or not. Here's how it looks:



enter image description here







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 13 hours ago

























answered 13 hours ago









AretinoAretino

24.4k21443




24.4k21443












  • $begingroup$
    That's essentially my approach. Good to know, that an expert like you came to the same conclusion. Concerning "natural": which approach could be more "natural"?
    $endgroup$
    – Hans Stricker
    12 hours ago










  • $begingroup$
    How (and with which tools) did you create your elegant animated gif? (I had a hard time with mine.)
    $endgroup$
    – Hans Stricker
    11 hours ago












  • $begingroup$
    I used GeoGebra, which is free: www.geogebra.org
    $endgroup$
    – Aretino
    11 hours ago










  • $begingroup$
    Can you share this plot with me, i.e. save the plot and share the link?
    $endgroup$
    – Hans Stricker
    10 hours ago










  • $begingroup$
    Here it is: ggbm.at/kyx9q3he
    $endgroup$
    – Aretino
    10 hours ago


















  • $begingroup$
    That's essentially my approach. Good to know, that an expert like you came to the same conclusion. Concerning "natural": which approach could be more "natural"?
    $endgroup$
    – Hans Stricker
    12 hours ago










  • $begingroup$
    How (and with which tools) did you create your elegant animated gif? (I had a hard time with mine.)
    $endgroup$
    – Hans Stricker
    11 hours ago












  • $begingroup$
    I used GeoGebra, which is free: www.geogebra.org
    $endgroup$
    – Aretino
    11 hours ago










  • $begingroup$
    Can you share this plot with me, i.e. save the plot and share the link?
    $endgroup$
    – Hans Stricker
    10 hours ago










  • $begingroup$
    Here it is: ggbm.at/kyx9q3he
    $endgroup$
    – Aretino
    10 hours ago
















$begingroup$
That's essentially my approach. Good to know, that an expert like you came to the same conclusion. Concerning "natural": which approach could be more "natural"?
$endgroup$
– Hans Stricker
12 hours ago




$begingroup$
That's essentially my approach. Good to know, that an expert like you came to the same conclusion. Concerning "natural": which approach could be more "natural"?
$endgroup$
– Hans Stricker
12 hours ago












$begingroup$
How (and with which tools) did you create your elegant animated gif? (I had a hard time with mine.)
$endgroup$
– Hans Stricker
11 hours ago






$begingroup$
How (and with which tools) did you create your elegant animated gif? (I had a hard time with mine.)
$endgroup$
– Hans Stricker
11 hours ago














$begingroup$
I used GeoGebra, which is free: www.geogebra.org
$endgroup$
– Aretino
11 hours ago




$begingroup$
I used GeoGebra, which is free: www.geogebra.org
$endgroup$
– Aretino
11 hours ago












$begingroup$
Can you share this plot with me, i.e. save the plot and share the link?
$endgroup$
– Hans Stricker
10 hours ago




$begingroup$
Can you share this plot with me, i.e. save the plot and share the link?
$endgroup$
– Hans Stricker
10 hours ago












$begingroup$
Here it is: ggbm.at/kyx9q3he
$endgroup$
– Aretino
10 hours ago




$begingroup$
Here it is: ggbm.at/kyx9q3he
$endgroup$
– Aretino
10 hours ago











3












$begingroup$

If we want this transition to have all the points on the boundary of a circle at all times, then it makes most sense to parameterize by the radius of this circle (and apply a transformation to get it in terms of finite time later). For simplicity, I will also have the transition be to a vertical line.



We shall have the radius of the circle $C_r$ be $r$ and centre be $(-r,0)$, such that $(0,0)$ is on $C_r$ for all $r$. The coordinates of the point at arclength $s$ from $(0,0)$ are then given by $(r (cos(s/r) - 1), r sin(s/r))$.



The most natural way to transition would likely be varying the curvature at a constant rate; thus we create a family of curves $f_t:[-pi, pi]tomathbb{R}^2$ where $tin[0,1]$ by
begin{align}
f_t(s) &= left(frac{cos(s(1-t))-1}{1-t}, frac{sin(s(1-t))}{1-t}right)&t<1\
f_1(s) &= (0, s)&
end{align}



Since the goals here seem to be rather subjective, I would attempt this and see how it looks to you (beyond making substitutions as needed to result in a horizontal line).






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Note that the paths of the endpoints are given by setting $s = pm pi$ in your parametric curve. The resulting paths are not circle segments (as the OP guessed they might be); this can be seen by calculating their curvature.
    $endgroup$
    – Michael Seifert
    9 hours ago


















3












$begingroup$

If we want this transition to have all the points on the boundary of a circle at all times, then it makes most sense to parameterize by the radius of this circle (and apply a transformation to get it in terms of finite time later). For simplicity, I will also have the transition be to a vertical line.



We shall have the radius of the circle $C_r$ be $r$ and centre be $(-r,0)$, such that $(0,0)$ is on $C_r$ for all $r$. The coordinates of the point at arclength $s$ from $(0,0)$ are then given by $(r (cos(s/r) - 1), r sin(s/r))$.



The most natural way to transition would likely be varying the curvature at a constant rate; thus we create a family of curves $f_t:[-pi, pi]tomathbb{R}^2$ where $tin[0,1]$ by
begin{align}
f_t(s) &= left(frac{cos(s(1-t))-1}{1-t}, frac{sin(s(1-t))}{1-t}right)&t<1\
f_1(s) &= (0, s)&
end{align}



Since the goals here seem to be rather subjective, I would attempt this and see how it looks to you (beyond making substitutions as needed to result in a horizontal line).






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Note that the paths of the endpoints are given by setting $s = pm pi$ in your parametric curve. The resulting paths are not circle segments (as the OP guessed they might be); this can be seen by calculating their curvature.
    $endgroup$
    – Michael Seifert
    9 hours ago
















3












3








3





$begingroup$

If we want this transition to have all the points on the boundary of a circle at all times, then it makes most sense to parameterize by the radius of this circle (and apply a transformation to get it in terms of finite time later). For simplicity, I will also have the transition be to a vertical line.



We shall have the radius of the circle $C_r$ be $r$ and centre be $(-r,0)$, such that $(0,0)$ is on $C_r$ for all $r$. The coordinates of the point at arclength $s$ from $(0,0)$ are then given by $(r (cos(s/r) - 1), r sin(s/r))$.



The most natural way to transition would likely be varying the curvature at a constant rate; thus we create a family of curves $f_t:[-pi, pi]tomathbb{R}^2$ where $tin[0,1]$ by
begin{align}
f_t(s) &= left(frac{cos(s(1-t))-1}{1-t}, frac{sin(s(1-t))}{1-t}right)&t<1\
f_1(s) &= (0, s)&
end{align}



Since the goals here seem to be rather subjective, I would attempt this and see how it looks to you (beyond making substitutions as needed to result in a horizontal line).






share|cite|improve this answer









$endgroup$



If we want this transition to have all the points on the boundary of a circle at all times, then it makes most sense to parameterize by the radius of this circle (and apply a transformation to get it in terms of finite time later). For simplicity, I will also have the transition be to a vertical line.



We shall have the radius of the circle $C_r$ be $r$ and centre be $(-r,0)$, such that $(0,0)$ is on $C_r$ for all $r$. The coordinates of the point at arclength $s$ from $(0,0)$ are then given by $(r (cos(s/r) - 1), r sin(s/r))$.



The most natural way to transition would likely be varying the curvature at a constant rate; thus we create a family of curves $f_t:[-pi, pi]tomathbb{R}^2$ where $tin[0,1]$ by
begin{align}
f_t(s) &= left(frac{cos(s(1-t))-1}{1-t}, frac{sin(s(1-t))}{1-t}right)&t<1\
f_1(s) &= (0, s)&
end{align}



Since the goals here seem to be rather subjective, I would attempt this and see how it looks to you (beyond making substitutions as needed to result in a horizontal line).







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 13 hours ago









KhorossKhoross

811




811












  • $begingroup$
    Note that the paths of the endpoints are given by setting $s = pm pi$ in your parametric curve. The resulting paths are not circle segments (as the OP guessed they might be); this can be seen by calculating their curvature.
    $endgroup$
    – Michael Seifert
    9 hours ago




















  • $begingroup$
    Note that the paths of the endpoints are given by setting $s = pm pi$ in your parametric curve. The resulting paths are not circle segments (as the OP guessed they might be); this can be seen by calculating their curvature.
    $endgroup$
    – Michael Seifert
    9 hours ago


















$begingroup$
Note that the paths of the endpoints are given by setting $s = pm pi$ in your parametric curve. The resulting paths are not circle segments (as the OP guessed they might be); this can be seen by calculating their curvature.
$endgroup$
– Michael Seifert
9 hours ago






$begingroup$
Note that the paths of the endpoints are given by setting $s = pm pi$ in your parametric curve. The resulting paths are not circle segments (as the OP guessed they might be); this can be seen by calculating their curvature.
$endgroup$
– Michael Seifert
9 hours ago




















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














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3129072%2fmathematics-and-the-art-of-linearizing-the-circle%23new-answer', 'question_page');
}
);

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







-

Popular posts from this blog

If I really need a card on my start hand, how many mulligans make sense? [duplicate]

Image
5 This question already has an answer here: How do you calculate the likelihood of drawing certain cards in your opening hand? 3 answers There is a funny modern deck where you play Treasure Hunt, but you really need to have it on the starting hand. How many mulligans should I take, if my deck contains 60 cards and contains 4 copies of Treasure Hunt? magic-the-gathering probability share | improve this question asked Nov 23 '18 at 12:12 dan dan 152 4
Read more

Alcedinidae

Image
Alcedinidae Martin-pêcheur à dos bleu ( Alcedo azurea ) Classification (COI) Règne Animalia Embranchement Chordata Sous-embr. Vertebrata Classe Aves Ordre Coraciiformes Famille Alcedinidae Rafinesque, 1815 Les Alcédinidés ou Alcedinidae forment une famille d'oiseaux plus connus sous les noms de martins-pêcheurs et martins-chasseurs. Sommaire 1 Description 2 Répartition et habitat 3 Systématique 3.1 Liste alphabétique des genres 3.2 Liste des espèces 4 Chez les Anciens : l'alcyon 4.1 Étymologie (Littré) 4.2 Oiseau « fabuleux » ? 5 Notes 6 Articles connexes 7 Liens externes 7.1 Bibliographie Description | Ce sont des oiseaux compacts de taille petite à moyenne (10 à 46 cm), au bec droit et long, en forme de poignard. Leurs pattes sont courtes, et ils portent un plumage aux couleurs vives. Répartition et habitat | Cosmopolites, ils fréquentent surtout
Read more

Can an atomic nucleus contain both particles and antiparticles? [duplicate]

Image
9 $begingroup$ This question already has an answer here: What happens if we put together a proton and an antineutron? 2 answers Is it theoretically possible to make a "deuterium" atom containing a proton and an antineutron in its nucleus? Would the strong nuclear force cause attraction between a proton and an antineutron? Would such a nucleus be stable, or would the proton somehow annihilate the antineutron when close enough? nuclear-physics antimatter share | cite | improve this question asked yesterday
Read more