Derivative of surface normal || tangent unit vector

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Discussion Overview

The discussion revolves around the mathematical properties of curves on surfaces, specifically focusing on the relationship between the derivative of the surface normal and the tangent unit vector of a curve parameterized by arc length. Participants explore definitions and implications of curvature lines and lines of curvature.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the derivative of the surface normal, m'(σ), is parallel to the tangent unit vector, t(σ), if and only if the curve is a curvature line.
  • There is a request for clarification on the definition of curvature line, with one participant defining it as a curve whose tangents align with principal curvature directions.
  • Another participant suggests that the equation presented is the definition of a line of curvature, which points in the direction of an eigenvector of the second fundamental form.
  • Some participants express skepticism about the distinction between curvature lines and lines of curvature, suggesting they may be synonymous.
  • One participant discusses the mathematical implications of eigenvectors and principal curvature directions, questioning how these concepts relate to the definitions provided.
  • There is a clarification regarding notation used in the discussion, particularly concerning the meaning of X. in mathematical expressions.
  • Participants engage in a detailed explanation of differentiating functions along curves and the application of the Leibniz formula to vector fields.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of curvature lines versus lines of curvature, with no consensus reached on whether they are synonymous. The discussion remains unresolved regarding the relationship between the definitions and the mathematical properties being explored.

Contextual Notes

Participants highlight the need for clarity in definitions and notation, indicating that assumptions about terminology may affect the understanding of the mathematical relationships discussed.

bennyzadir
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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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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?
 
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!
 
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.
 
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?
 
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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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!
 
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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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
 

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