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user131781
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Curves with the property that the curvature at a point is equal to the $y$-coordinate, more generally that it is proportional to some power thereof, have been studied in some detail in the arXiv paper arXiv:1102.1579 (the rather inelegant title is too long to reproduce here), where they are called Maclaurin catenaries. They have several other related properties, for example they are trajectories of particles moving under parallel force laws of the form $y^\alpha$ and are catenaries for such laws.

It is more convenient to use parametrised representations and the catenaries are of the form $(F_d(t),f_d(t))$ with $f_d(t)=(cos(d t))^{1/d}$ and $F_d$ its primitive. (There is a simple relationship between the exponents $\alpha$ and $d$). The crucial property of functions $f$ of this form is that they have the property that the differential expressions $f+f’’$, $f^2+f’^2$ and $(f f’’-f’^2)/(f^2+f’^2)^{3/2}$, which occur in the calculations, are all proportional to a power of $f$.

The justification for the use of the name of Maclaurin is that the distinguished scottish mathematician considered curves of the form $r f_d(\theta)=1$ with the above $f$ and showed that they have analogous properties for central forces proportional to $r^{\alpha}$. These are now known as Maclaurin spirals. The reason for this analogy between the parallel and the central case is that there are striking similarities between the geometrical and kinetic properties of curves with parametric representation $(F(t),f(t))$ where ($F$ a primitive of $f$) and those with polar representation $rf(\theta)=1$. These are elucidated in the above article.

These curves can initially be interpreted as the graphs of functions on an interval about the origin but can be extended in natural ways to the whole real line, either by translation or, as is more relevant to your question, by a glide reflection. These will have recurring singularities at the endpoints, the precise nature of which will depend on the index $d$. This is not discussed in the above article but the explicit solutions given there should make a detailed analysis possible.

Curves with the property that the curvature at a point is equal to the $y$-coordinate, more generally that it is proportional to some power thereof, have been studied in some detail in the arXiv paper arXiv:1102.1579 (the rather inelegant title is too long to reproduce here), where they are called Maclaurin catenaries. They have several other related properties, for example they are trajectories of particles moving under parallel force laws of the form $y^\alpha$ and are catenaries for such laws.

It is more convenient to use parametrised representations and the catenaries are of the form $(F_d(t),f_d(t))$ with $f_d(t)=(cos(d t))^{1/d}$ and $F_d$ its primitive. (There is a simple relationship between the exponents $\alpha$ and $d$). The crucial property of functions $f$ of this form is that they have the property that the differential expressions $f+f’’$, $f^2+f’^2$ and $(f f’’-f’^2)/(f^2+f’^2)^{3/2}$, which occur in the calculations, are all proportional to a power of $f$.

The justification for the use of the name of Maclaurin is that the distinguished scottish mathematician considered curves of the form $r f_d(\theta)=1$ with the above $f$ and showed that they have analogous properties for central forces proportional to $r^{\alpha}$. These are now known as Maclaurin spirals. The reason for this analogy between the parallel and the central case is that there are striking similarities between the geometrical and kinetic properties of curves with parametric representation $(F(t),f(t))$ ($F$ a primitive of $f$) and those with polar representation $rf(\theta)=1$. These are elucidated in the above article.

These curves can initially be interpreted as the graphs of functions on an interval about the origin but can be extended in natural ways to the whole real line, either by translation or, as is more relevant to your question, by a glide reflection. These will have recurring singularities at the endpoints, the precise nature of which will depend on the index $d$. This is not discussed in the above article but the explicit solutions given there should make a detailed analysis possible.

Curves with the property that the curvature at a point is equal to the $y$-coordinate, more generally that it is proportional to some power thereof, have been studied in some detail in the arXiv paper arXiv:1102.1579 (the rather inelegant title is too long to reproduce here), where they are called Maclaurin catenaries. They have several other related properties, for example they are trajectories of particles moving under parallel force laws of the form $y^\alpha$ and are catenaries for such laws.

It is more convenient to use parametrised representations and the catenaries are of the form $(F_d(t),f_d(t))$ with $f_d(t)=(cos(d t))^{1/d}$ and $F_d$ its primitive. (There is a simple relationship between the exponents $\alpha$ and $d$). The crucial property of functions $f$ of this form is that the differential expressions $f+f’’$, $f^2+f’^2$ and $(f f’’-f’^2)/(f^2+f’^2)^{3/2}$, which occur in the calculations, are all proportional to a power of $f$.

The justification for the use of the name of Maclaurin is that the distinguished scottish mathematician considered curves of the form $r f_d(\theta)=1$ with the above $f$ and showed that they have analogous properties for central forces proportional to $r^{\alpha}$. These are now known as Maclaurin spirals. The reason for this analogy between the parallel and the central case is that there are striking similarities between the geometrical and kinetic properties of curves with parametric representation $(F(t),f(t))$ where $F$ a primitive of $f$ and those with polar representation $rf(\theta)=1$. These are elucidated in the above article.

These curves can initially be interpreted as the graphs of functions on an interval about the origin but can be extended in natural ways to the whole real line, either by translation or, as is more relevant to your question, by a glide reflection. These will have recurring singularities at the endpoints, the precise nature of which will depend on the index $d$. This is not discussed in the above article but the explicit solutions given there should make a detailed analysis possible.

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user131781
user131781
  • 2.5k
  • 1
  • 8
  • 10

Curves with the property that the curvature at a point is equal to the $y$-coordinate, more generally that it is proportional to some power thereof, have been studied in some detail in the arXiv paper arXiv:1102.1579 (the rather inelegant title is too long to reproduce here), where they are called Maclaurin catenaries. They have several other related properties, for example they are trajectories of particles moving under parallel force laws of the form $y^\alpha$ and are catenaries for such laws.

It is more convenient to use parametrised representations and the catenaries are of the form $(F_d(t),f_d(t))$ with $f_d(t)=(cos(d t))^{1/d}$ and $F_d$ its primitive. (There is a simple relationship between the exponents $\alpha$ and $d$). The crucial property of functions $f$ of this form is that they have the property that the differential expressions $f+f’’$, $f^2+f’^2$ and $(f f’’-f’^2)/(f^2+f’^2)^{3/2}$, which occur in the calculations, are all proportional to a power of $f$.

The justification for the use of the name of Maclaurin is that the distinguished scottish mathematician considered curves of the form $r f_d(\theta)=1$ with the above $f$ and showed that they have analogous properties for central forces proportional to $r^{\alpha}$. These are now known as Maclaurin spirals. The reason for this analogy between the parallel and the central case is that there are striking similarities between the geometrical and kinetic properties of curves with parametric representation $(F(t),f(t))$ ($F$ a primitive of $f$) and those with polar representation $rf(\theta)=1$. These are elucidated in the above article.

These curves can initially be interpreted as the graphs of functions on an interval about the origin but can be extended in natural ways to the whole real line, either by translation or, as is more relevant to your question, by a glide reflection. These will have recurring singularities at the endpoints, the precise nature of which will depend on the index $d$. This is not discussed in the above article but the explicit solutions given there should make a detailed analysis possible.