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Differential geoemtry tangent lines parallel proof

  • Thread starter hlin818
  • Start date
  • #1
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Homework Statement



Prove that a(s) is a straight line if and only if its tangent lines are all parallel.

Homework Equations



Frenet serret theorem

The Attempt at a Solution



I'm confused on the direction "if the tangent lines are parallel then a(s) is a straight line".

Assume all the tangent lines of a(s) are parallel. So the tangent vector T is the same for all points xo on the curve a(s) and the values of T(s) of any two points on the curve are parallel. Thus T(s) is constant, and T'(s)=0 which implies that the curvature is zero, and thus a(s) must be a straight line.
 
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Answers and Replies

  • #2
lanedance
Homework Helper
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so you know the direction fo the tanget vector at all times, eg.

T(t) = a.g(t)

where a is a constant vector and g is a scalar function. Think about integrating this to get the original curve and/or the effect on other parameters, curvature etc.
 
  • #3
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Is the way I did it incorrect?
 
  • #4
Dick
Science Advisor
Homework Helper
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Is the way I did it incorrect?
It seems fine to me.
 
  • #5
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Thanks!
 

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