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A Theory of Surfaces and Theory of Curves Relationship

  1. Nov 14, 2016 #1
    Hello

    I am interested in the Frenet-Serret Formulas (theory of curves?) relationship to theory of surfaces.

    1) Can one arrive to the Frenet-Serret Formulas starting from the theory of surfaces? Any advice on where to begin?

    2) For a surface that contain a space curve: if the unit tangent vector to the curve is the very same unit tangent vector the surface then why/how is the unit normal vector to the curve not in the same direction as the surface's gradient vector?

    Thank you
     
  2. jcsd
  3. Nov 14, 2016 #2

    lavinia

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    Science Advisor

    A basic book on classical differential geometry is Lectures on Classical Differential Geometry by Struik. Without a specific question it is difficult to give a specific reply.
     
  4. Nov 16, 2016 #3
    Ok

    Why doesn't the normal to space curve contained on a surface not point in the same direction as that surfaces normal vector (gradient)?
     
  5. Nov 16, 2016 #4
    Think about latitude circles on a sphere.
     
  6. Nov 18, 2016 #5

    Oh ok!

    So the sphere's gradient vector would point in the same direction as the circle's negative normal vector?
     
  7. Nov 18, 2016 #6
    Hint: You can determine the circle's normal vector from the plane that it's in.
     
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