Theory of Surfaces and Theory of Curves Relationship

  • #1
37
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
 

Answers and Replies

  • #2
lavinia
Science Advisor
Gold Member
3,258
642
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
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.
 
  • #3
37
1
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.

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)?
 
  • #4
399
118
Think about latitude circles on a sphere.
 
  • #5
37
1
Think about latitude circles on a sphere.


Oh ok!

So the sphere's gradient vector would point in the same direction as the circle's negative normal vector?
 
  • #6
399
118
Hint: You can determine the circle's normal vector from the plane that it's in.
 

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