- #1

MxwllsPersuasns

- 101

- 0

## Homework Statement

Let γ: I → ℝ

^{2}be a smooth regular curve and let λ = γ ο φ with φ: I

_{λ}→ I be a reparameterisation of γ. Show, by using the general formula for curvature of a regular curve that κ

_{λ}= ±κ ο Φ where the ± depends on whether φ is orientation preserving (+) or reversing (-).

## Homework Equations

κ = Det{γ',γ''}/{absval(γ')}

^{3}

## The Attempt at a Solution

So, the way I would approach this problem I guess is by first taking the curvature of γ...

- Though the problem I immediately run into, and did on another problem I just posted, is whether this should be done in general or for a specific case of γ and λ

+ i.e., I would parameterize γ with t perhaps by <acos(t), bsin(t)> then take φ to be something orientation reversing (maybe like φ = -t) and again as something orientation preserving (like maybe φ = t

^{2}?) and then show that the curvature of each λ

_{±}equals the curvature of the original curve (up to a sign) reparameterized with Φ.

+ Another question is, why are we using k composed with Φ for the reparameterised curvature when the reparamterization is done by composing with the function φ? Wouldn't the curvature then be something like κ

_{λ}= ±κ ο φ?

Any help or guidance is extremely appreciated I'm having a bit of trouble grasping how to start and where to go with this.