- #1

mXCSNT

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## Homework Statement

Assume that [tex]\tau(s) \neq 0[/tex] and [tex]k'(x) \neq 0[/tex] for all [tex]s \in I[/tex]. Show that a necessary and sufficient condition for [tex]\alpha(I)[/tex] to lie on a sphere is that [tex]R^2 + (R')^2T^2 = const[/tex] where [tex]R = 1/k[/tex], [tex]T = 1/\tau[/tex], and [tex]R' = \frac{dr}{ds}[/tex]

## Homework Equations

[tex]\alpha(s)[/tex] is a curve in R3 parametrized by arc length

[tex]k = curvature = |\alpha''|[/tex]

[tex]\tau = torsion = -\frac{\alpha' \times \alpha'' \cdot \alpha'''}{k^2}[/tex] (note sign; this is opposite of some conventions)

## The Attempt at a Solution

I've approached this from 2 directions, but I haven't gotten them to meet. First, a necessary and sufficient condition is that [tex]|\alpha - P|[/tex] is constant, where P is the center of the circle. Alternatively, [tex](\alpha - P) \cdot \alpha' = 0[/tex].

And I've expanded out [tex]R^2 + (R')^2T^2 = const[/tex] to get

[tex]\frac{(\alpha' \times \alpha'' \cdot \alpha''')^2 + (\alpha'' \cdot \alpha''')^2}{(\alpha'' \cdot \alpha'')(\alpha' \times \alpha'' \cdot \alpha''')^2} = const[/tex]

Also, I'm going to guess that the const on the right hand side is some function of the radius of the sphere, maybe the square of it (which would be [tex](\alpha - P) \cdot (\alpha - P)[/tex]), because what else is constant in a sphere?

But I don't know where to go from here. I'm just looking for a hint at an avenue of approach, please nothing specific.