# Show reparameterized curvature equals curvature up to a sign

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1. Feb 9, 2017

### MxwllsPersuasns

1. The problem statement, all variables and given/known data
Let γ: I → ℝ2 be a smooth regular curve and let λ = γ ο φ with φ: Iλ → I be a reparameterisation of γ. Show, by using the general formula for curvature of a regular curve that κλ = ±κ ο Φ where the ± depends on whether φ is orientation preserving (+) or reversing (-).

2. Relevant equations

κ = Det{γ',γ''}/{absval(γ')}3

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 = t2?) 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 κλ = ±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.

2. Feb 16, 2017