Show existence of arc-length parameterized period p'....

In summary, we have shown that for any regular closed curve γ with period p, there exists an orientation preserving diffeomorphism ϕ and a number p' such that ϕ(s + p') = ϕ(s) + p and γ' = γ ◦ ϕ is an arclength parametrized closed curve with period p'.
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Homework Statement



Let γ : R → Rn be a regular (smooth) closed curve with period p. Show that there exist an orientation preserving diffeomorphism ϕ: R → R, a number p' ∈ R such that ϕ(s + p') = ϕ(s) + p and γ' = γ ◦ ϕ is an arclength parametrized closed curve with period p'

Homework Equations


Arc-length Parameterized Curve: Length = ||dγ/dt|| = 1
Orientation preserving: dΦ/dt > 0 for all t

The Attempt at a Solution


So essentially this question is asking for the existence of a number p' such that when we reparameterize by arc-length we have a new (arclength parameterized) period. Is this correct? And if so, how should I go about proving the existence of such a number? Thanks so much in advance folks, I'm looking forward to the lively discussion!
 
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  • #2


Yes, your understanding of the problem is correct. To prove the existence of p', we can use the periodicity of γ and the fact that it is a smooth curve.

First, let's consider the case where p is the length of the curve γ. In this case, we can define ϕ(s) = s/p, which is an orientation preserving diffeomorphism from R to R. We can easily verify that ϕ(s + p) = ϕ(s) + p, and that dϕ/ds = 1/p, so ϕ is indeed an arclength parametrization.

Now, if p is not equal to the length of γ, we can use the fact that γ is a smooth curve to construct a new parametrization that has the desired period. Let p' = p/||dγ/dt||, which is a positive number since ||dγ/dt|| = 1 for an arclength parametrized curve. Then, we can define ϕ(s) = ∫_0^s ||dγ/dt|| dt, which is an orientation preserving diffeomorphism from R to R. We can verify that ϕ(s + p') = ϕ(s) + p, and that dϕ/ds = ||dγ/dt||, so ϕ is also an arclength parametrization.

Finally, we can compose ϕ with γ to get the desired arclength parametrized curve with period p'. That is, γ' = γ ◦ ϕ is an arclength parametrized curve with period p'. This completes the proof of the existence of p'.
 

FAQ: Show existence of arc-length parameterized period p'....

1. What is an arc-length parameterized period p'?

An arc-length parameterized period p' is a mathematical concept used in the study of curves and surfaces. It represents the length of the curve or surface over a certain period, usually denoted by p, that is measured in units of arc length.

2. Why is it important to show the existence of an arc-length parameterized period p'?

The existence of an arc-length parameterized period p' is important because it allows us to measure the length of curves and surfaces in a standardized way, which is necessary for many mathematical and scientific applications. It also helps us to understand the properties and behavior of curves and surfaces more accurately.

3. How is the existence of an arc-length parameterized period p' shown?

The existence of an arc-length parameterized period p' can be shown using mathematical techniques such as differential calculus and integral calculus. These methods involve finding the derivative of the arc length function and proving that it is continuous and differentiable, which is necessary for the existence of p'.

4. What are the benefits of using an arc-length parameterized period p'?

There are several benefits of using an arc-length parameterized period p'. Firstly, it allows us to measure the length of curves and surfaces in a consistent and precise manner. Secondly, it helps us to analyze and compare different curves and surfaces more easily. Finally, it is useful for solving problems in various fields such as physics, engineering, and computer graphics.

5. Can an arc-length parameterized period p' be negative?

No, an arc-length parameterized period p' cannot be negative. This is because it represents a physical length, which cannot be negative. However, it is possible for p' to be zero if the curve or surface has no length, such as a single point or a straight line.

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