Prove a unit speed curve is a helix...

dammahom59 · **Joined:** Thu, 26 Jan 2012 19:40:01 UTC **Posts:** 3

Prove a unit speed curve α(s) with k <> 0 is a helix if and only if there is a constant c such that τ=ck.

Shadow · **Posted:** Thu, 26 Jan 2012 20:43:33 UTC

dammahom59 wrote:

Prove a unit speed curve α(s) with k <> 0 is a helix if and only if there is a constant c such that τ=ck.

What is your definition of a helix? I've never heard that as a technical term before.

dammahom59 · **Joined:** Thu, 26 Jan 2012 19:40:01 UTC **Posts:** 3

Shadow wrote:

dammahom59 wrote:

Prove a unit speed curve α(s) with k <> 0 is a helix if and only if there is a constant c such that τ=ck.

What is your definition of a helix? I've never heard that as a technical term before.

A helix is a curve in 3-dimensional space. The following parametrization in Cartesian coordinates defines a helix:

x(t) = cos(t),
y(t) = sin(t),
z(t) = t.

Shadow · **Posted:** Thu, 26 Jan 2012 21:38:27 UTC

dammahom59 wrote:

Shadow wrote:

dammahom59 wrote:

Prove a unit speed curve α(s) with k <> 0 is a helix if and only if there is a constant c such that τ=ck.

What is your definition of a helix? I've never heard that as a technical term before.

A helix is a curve in 3-dimensional space. The following parametrization in Cartesian coordinates defines a helix:

x(t) = cos(t),
y(t) = sin(t),
z(t) = t.

OK, then what is $\tau$ ? and what do you mean by k<>0

?

mathematic · **Joined:** Fri, 1 Jul 2011 01:17:26 UTC **Posts:** 321

dammahom59 wrote:

Prove a unit speed curve α(s) with k <> 0 is a helix if and only if there is a constant c such that τ=ck.

Your question is too cryptic. You should define your symbols and how they relate to each other. Also "unit speed curve".

outermeasure · **Posted:** Fri, 27 Jan 2012 08:45:47 UTC

Shadow wrote:

dammahom59 wrote:

Shadow wrote:

dammahom59 wrote:

Prove a unit speed curve α(s) with k <> 0 is a helix if and only if there is a constant c such that τ=ck.

What is your definition of a helix? I've never heard that as a technical term before.

A helix is a curve in 3-dimensional space. The following parametrization in Cartesian coordinates defines a helix:

x(t) = cos(t),
y(t) = sin(t),
z(t) = t.

OK, then what is $\tau$ ? and what do you mean by k<>0

?

I suspect dammahom59 meant $\kappa\neq 0$ and $\tau$ , the curvature and torsion. In which case, it follows immediately from the Frenet-Serret formula and uniqueness of solutions, remembering a characterisation of helix is a space curve with a constant direction orthogonal to the principal normal vector (exercise: prove this characterisation from the usual definition that a helix is a space curve where the tangent vector makes a constant angle to a fixed direction).

dammahom59 · **Joined:** Thu, 26 Jan 2012 19:40:01 UTC **Posts:** 3

outermeasure wrote:

Shadow wrote:

dammahom59 wrote:

Shadow wrote:

dammahom59 wrote:

Prove a unit speed curve α(s) with k <> 0 is a helix if and only if there is a constant c such that τ=ck.

What is your definition of a helix? I've never heard that as a technical term before.

A helix is a curve in 3-dimensional space. The following parametrization in Cartesian coordinates defines a helix:

x(t) = cos(t),
y(t) = sin(t),
z(t) = t.

OK, then what is $\tau$ ? and what do you mean by k<>0

?

I suspect dammahom59 meant $\kappa\neq 0$ and $\tau$ , the curvature and torsion. In which case, it follows immediately from the Frenet-Serret formula and uniqueness of solutions, remembering a characterisation of helix is a space curve with a constant direction
orthogonal to the principal normal vector (exercise: prove this characterisation from the usual definition that a helix is a space curve where the tangent vector makes a constant angle to a fixed direction).

Yes, that is exactly what I meant, thank you.

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Prove a unit speed curve is a helix...

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