A Theory of Surfaces and Theory of Curves Relationship

[ltkach2015]

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

 

[lavinia]

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.

 

[ltkach2015]

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)?

 

[zinq]

Think about latitude circles on a sphere.

 

[ltkach2015]

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?

 

[zinq]

Hint: You can determine the circle's normal vector from the plane that it's in.