Can Parametrized Plane Curves Have Constant Curvature?

In summary, if the norm of a smooth plane curve \sigma(t) depends entirely on the difference between two parameters t_1 and t_0, then the image of the curve is a subset of either a circle or a line. This is because circles and lines are the only plane curves with constant curvature, and the given condition implies that the unit vector \hat n in the curve's equation is constant.
  • #1
Mandelbroth
611
24

Homework Statement


Suppose ##\sigma:I\subseteq\mathbb{R}\to\mathbb{R}^2## is a smooth plane curve parametrized by a parameter ##t\in I##. Prove that if ##\|\sigma(t_1)-\sigma(t_0)\|## depends entirely on ##|t_1-t_0|##, then the image of ##I## under ##\sigma## is a subset of either ##S^1## or a line.

The Attempt at a Solution


Embarrassingly enough, I'm having trouble setting up a proof here. I understand intuitively why this is true, but I can't see where to start. Can someone just nudge me in the right direction?

Thank you.
 
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  • #2
Mandelbroth said:

Homework Statement


Suppose ##\sigma:I\subseteq\mathbb{R}\to\mathbb{R}^2## is a smooth plane curve parametrized by a parameter ##t\in I##. Prove that if ##\|\sigma(t_1)-\sigma(t_0)\|## depends entirely on ##|t_1-t_0|##, then the image of ##I## under ##\sigma## is a subset of either ##S^1## or a line.

The Attempt at a Solution


Embarrassingly enough, I'm having trouble setting up a proof here. I understand intuitively why this is true, but I can't see where to start. Can someone just nudge me in the right direction?

Thank you.

Suppose [itex]\|\sigma(t_1) - \sigma(t_0)\| = f(|t_1 - t_0|)[/itex] for some real-valued [itex]f[/itex] such that [itex]f(t) \geq 0[/itex] for all [itex]t[/itex] with [itex]f(0) = 0[/itex]. Then
[tex]
\sigma(t_1) = \sigma(t_0) + \hat n(t_0,t_1) f(|t_1 - t_0|)\qquad(1)
[/tex]
where [itex]\hat n[/itex] is a unit vector. This can be rearranged to give
[tex]
\sigma(t_0) = \sigma(t_1) - \hat n(t_0,t_1) f(|t_1 - t_0|)
[/tex]
and swapping [itex]t_0[/itex] and [itex]t_1[/itex] and comparing with (1) yields
[tex]\hat n(t_0, t_1) = -\hat n(t_1,t_0)[/tex]

I don't know whether this will lead anywhere. However I do see that if [itex]\hat n[/itex] is constant then [itex]\sigma(I)[/itex] will be a line segment.

EDIT: It also occurs to me that circles and lines are the only plane curves of constant curvature.
 
Last edited:
  • #3
pasmith said:
Suppose [itex]\|\sigma(t_1) - \sigma(t_0)\| = f(|t_1 - t_0|)[/itex] for some real-valued [itex]f[/itex] such that [itex]f(t) \geq 0[/itex] for all [itex]t[/itex] with [itex]f(0) = 0[/itex]. Then
[tex]
\sigma(t_1) = \sigma(t_0) + \hat n(t_0,t_1) f(|t_1 - t_0|)\qquad(1)
[/tex]
where [itex]\hat n[/itex] is a unit vector. This can be rearranged to give
[tex]
\sigma(t_0) = \sigma(t_1) - \hat n(t_0,t_1) f(|t_1 - t_0|)
[/tex]
and swapping [itex]t_0[/itex] and [itex]t_1[/itex] and comparing with (1) yields
[tex]\hat n(t_0, t_1) = -\hat n(t_1,t_0)[/tex]

I don't know whether this will lead anywhere. However I do see that if [itex]\hat n[/itex] is constant then [itex]\sigma(I)[/itex] will be a line segment.

EDIT: It also occurs to me that circles and lines are the only plane curves of constant curvature.
I realized that they are both the only plane curves with constant curvature, then I proceeded to cook up a proof.

Thank you.
 

1. What is a parametrized plane curve?

A parametrized plane curve is a mathematical representation of a curve in the plane using a set of equations, called parametric equations, that describe the coordinates of points on the curve as functions of one or more parameters.

2. How are parametrized plane curves different from regular curves?

Parametrized plane curves are different from regular curves in that they are defined by parametric equations, which introduce a parameter that varies along the curve and allows for a more flexible and precise representation of the curve's shape and properties.

3. What are the advantages of using parametrized plane curves?

Parametrized plane curves offer several advantages over regular curves, including the ability to represent complex curves and surfaces, easily calculate derivatives and integrals, and analyze the behavior of the curve as the parameter varies.

4. How are parametrized plane curves used in real-world applications?

Parametrized plane curves have many real-world applications, such as in computer graphics, robotics, and engineering. They can be used to model and design curves and surfaces in 3D computer graphics, plan and control the motion of robotic arms, and analyze and optimize the shape of structures in engineering.

5. Are there any limitations to using parametrized plane curves?

While parametrized plane curves offer many advantages, they also have some limitations. For example, they may not always accurately represent certain types of curves, such as sharp corners or cusps. Additionally, the choice of parameterization can affect the shape of the curve and may require some experimentation to find the most appropriate representation.

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