Derivative of surface normal || tangent unit vector

In summary, the conversation discusses the definition of a curvature line on a surface and the relationship between the surface normal, tangent unit vector, and principal curvature. The main point is that the unit normal along a curvature line is parallel to the tangent unit vector. This is proven by showing that the direction of a principal curvature is an eigen vector of the second fundamental form, which is defined as the derivative of the unit normal along a curve. Finally, the notation and formulas used in the conversation are explained.
  • #1
bennyzadir
18
0
Let be w=w(σ) a curve on a surface parametrized by the arc length (the natural parametrization). Consider the m surface normal along this curve as the function of the σ arc length of the curve. Prove that m'(σ) is parallel to the t(σ) tangent unit vector of the curve for all σ, IFF this curve is a curvature line.

I would appreciate if you could help me!
 
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  • #2
bennyzadir said:
Let be w=w(σ) a curve on a surface parametrized by the arc length (the natural parametrization). Consider the m surface normal along this curve as the function of the σ arc length of the curve. Prove that m'(σ) is parallel to the t(σ) tangent unit vector of the curve for all σ, IFF this curve is a curvature line.

I would appreciate if you could help me!

What definition of curvature line do you use?
 
  • #3
lavinia said:
What definition of curvature line do you use?

In my terminology curvature line is a curve on a surface whose tangents are always in the direction of principal curvature.

Thank you for your help in advance!
 
  • #4
bennyzadir said:
In my terminology curvature line is a curve on a surface whose tangents are always in the direction of principal curvature.

Thank you for your help in advance!

To me your equation is the definition of a line of curvature, that is a curve that points in the direction of an eigen vector of the second fundamental form.
 
  • #5
lavinia said:
To me your equation is the definition of a line of curvature, that is a curve that points in the direction of an eigen vector of the second fundamental form.

To tell the truth, I don't really see the point. Curvature line is a synonym of line of curvature.
The direction of an eigen vector of the second fundamental form is the same as the direction of the principal curvature.
How does the statement follow from the definition?
 
  • #6
bennyzadir said:
To tell the truth, I don't really see the point. Curvature line is a synonym of line of curvature.
The direction of an eigen vector of the second fundamental form is the same as the direction of the principal curvature.
How does the statement follow from the definition?

An eigen vector by definition means X.N = kX where N here is the unit normal.

On the other hand if you define a direction of principal curvature to be the direction of a curve of extremal curvature of all curves that lie on the intersection of the surface with a plane that contains the unit normal, then you need to show that such a direction is an eigen vector and visa versa. This must be what your question really is.

Since the unit normal,N, is orthogonal to X, <N,X> = 0 for any tangent direction X.

So 0 = X.<N,X> = <X.N,X> + <N,X.X> where X. means the derivative along a curve fitting X.

If X has length 1, then the second term is the curvature of a curve fitting X. So an extremal of the first term must be an extremal of the second.
 
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  • #7
Thank you very much for your detailed answer. I am afraid I don't really understand your notation. Does X. denote derivative, dot products or just a fullstop?
Thanks once more!
 
  • #8
bennyzadir said:
Thank you very much for your detailed answer. I am afraid I don't really understand your notation. Does X. denote derivative, dot products or just a fullstop?
Thanks once more!

On a surface, a function can be differentiated along curves. One just looks at the function as it varies with respect to the curve's parameter as you did using the parameter of arc length.

If X is a tangent vector at a point on the surface then X.f denotes the derivative of f
at that point along a curve whose velocity vector is X. Since vector fields, such as the unit normal to the surface, are just triples of functions, they can also be differentiated along a curve on the surface. So X.N means the derivative of the unit normal vector field with respect to a curve whose velocity is X. If X has length one, then this curve is parameterized by arc length as in your original formulation.

The derivative of the function that is the inner product (dot product) of two vector fields satisfies the Leibniz formula

X.<V,W> = <X.V,W> + <V,X.W>. ( <,> just means dot product.)

Note that the left hand side is the derivative of a function while the right hand side is the sum of inner products of derivatives of vector fields with vector fields.
 
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  • #9
Thanks for your quick help and for your patient.
All the best! Merry Cristmas!
 
  • #10
bennyzadir said:
Thanks for your quick help and for your patient.
All the best! Merry Cristmas!

Thanks. Merry Christmas to you
 

1. What is the derivative of surface normal?

The derivative of surface normal refers to the rate of change of the surface normal vector with respect to a parameter, typically a coordinate on the surface.

2. How is the derivative of surface normal calculated?

The derivative of surface normal can be calculated using the gradient operator applied to the surface normal vector. This involves taking partial derivatives of the components of the surface normal vector with respect to the chosen parameter.

3. What is the significance of the derivative of surface normal?

The derivative of surface normal is important in understanding the curvature and shape of a surface. It can also be used in various applications such as computer graphics and computer vision.

4. What is a tangent unit vector?

A tangent unit vector is a vector that lies in the tangent plane of a surface at a specific point and has a magnitude of 1. It is perpendicular to the surface normal vector at that point.

5. How is the tangent unit vector related to the derivative of surface normal?

The tangent unit vector can be obtained by normalizing the derivative of surface normal vector. This means dividing the derivative vector by its magnitude to get a unit vector in the same direction. The tangent unit vector and derivative of surface normal are closely related as they both provide information about the shape and curvature of a surface.

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