How exactly to obtain Frenet Frame via Gram-Schmidt process?

In summary: Yeah, sorry about that)In summary, there are two ways to obtain an orthonormal frame for a curve in ℝN: using Gram-Schmidt on a given basis or using the motion along the curve. The first method is simpler but does not always produce a valid frame.
  • #1
weetabixharry
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I have a regular curve, [itex]\underline{a}(s)[/itex], in ℝN (parameterised by its arc length, [itex]s[/itex]).

To a running point on the curve, I want to attach the (Frenet) frame of orthonormal vectors [itex]\underline{u}_1(s),\underline{u}_2(s),\dots, \underline{u}_N(s)[/itex]. However, looking in different books, I find different claims as to how these should be obtained. Specifically, some books suggest that Gram-Schmidt should be applied to:[tex]\underline{a}^{\prime}(s), \underline{a}^{\prime \prime}(s), \dots , \underline{a}^{(N-1)}(s)[/tex]while another book suggests that [itex]\underline{u}_{k+1}(s)[/itex] is obtained by applying Gram-Schmidt to [itex]\underline{u}_k^{\prime}(s)[/itex].

Which should I use?
 
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  • #2
To add a little more detail...

Since [itex]s[/itex] is an invariant parameter, I start with:[tex]\underline{u}_1(s) = \underline{a}^{\prime}(s)[/tex]
Then, using [itex]\underline{a}^{\prime \prime}(s) = \underline{u}_1^{\prime}(s)[/itex] as the next linearly independent vector for Gram-Schmidt gives:[tex]\underline{u}_2 = \frac{ \underline{u}_1^{\prime} - (\underline{u}_1^T\underline{u}_1^{\prime}) \underline{u}_1}{\Vert numerator \Vert}[/tex]

However, for [itex]\underline{u}_3, \underline{u}_4, \dots [/itex] the two approaches appear to become different.
 
  • #3
In three dimension a curve parameterized by arc length has acceleration perpendicular to the tangent. The cross product of the unit tangent with the normalized acceleration is perpendicular to both and this gives you the third vector in the frame. I do not believe that the Frenet frame can include other vectors so starting with an arbitrary basis and Gram-Schmiditfying will not work.

In higher dimensions there are many frames that extend the unit tangent and normalized acceleration. Gram Schmitt would work on a given basis but I am not sure what its geometric meaning would be.
 
  • #4
Thanks for your response.
lavinia said:
In higher dimensions there are many frames that extend the unit tangent and normalized acceleration.
Could you expand on what you mean? Is it possible/likely that both methods I mentioned are valid?

lavinia said:
Gram Schmitt would work on a given basis but I am not sure what its geometric meaning would be.
What I'd like to do is define the Cartan matrix (containing curvature, tortion, ..., general curvatures) and so develop expressions relating the moving frame, [itex]\underline{u}_1(s),\underline{u}_2(s),\dots, \underline{u}_N(s)[/itex], and its derivative, [itex]\underline{u}_1^{ \prime}(s),\underline{u}_2^{ \prime}(s),\dots, \underline{u}_N^{ \prime}(s)[/itex]. I find that this is straightforward using the second method I mentioned in my original post ("while another book suggests that..."). However, before I had checked in books, I had thought the first method would be the way to go...
 
  • #5
weetabixharry said:
Thanks for your response.
Could you expand on what you mean? Is it possible/likely that both methods I mentioned are valid?

What I'd like to do is define the Cartan matrix (containing curvature, tortion, ..., general curvatures) and so develop expressions relating the moving frame, [itex]\underline{u}_1(s),\underline{u}_2(s),\dots, \underline{u}_N(s)[/itex], and its derivative, [itex]\underline{u}_1^{ \prime}(s),\underline{u}_2^{ \prime}(s),\dots, \underline{u}_N^{ \prime}(s)[/itex]. I find that this is straightforward using the second method I mentioned in my original post ("while another book suggests that..."). However, before I had checked in books, I had thought the first method would be the way to go...

The second method seems right because it is defined by the motion along the curve. Each successive vector in the frame points in the direction that the hyperplane spanned by the previously defined vectors is moving. It is possible though that there will be points on the curve where these derivatives are zero.

Rather than defining a frame this way in terms of the motion along the curve. one could just pick some basis at every points and Gram Schmitify it to get an orthonormal frame. This would not be the Frenet frame in general.
 
  • #6
lavinia said:
Each successive vector in the frame points in the direction that the hyperplane spanned by the previously defined vectors is moving.
Ah, this is very insightful. In fact, I hadn't realized that this is probably the whole point of what I'm trying to do.

So, going back a few steps, is the following roughly true?
If I zoom in super close to the curve, it looks like a straight line pointing in the direction of [itex]\underline{u}_1(s)[/itex]. I zoom out a bit, and actually the curve looks like a little circular arc lying in the plane of [itex]\underline{u}_1(s),\underline{u}_2(s)[/itex] and with radius equal to the inverse of the first curvature. Then I zoom out a bit more, and see that the curve actually lifts out from the plane of [itex]\underline{u}_1(s),\underline{u}_2(s)[/itex] in the direction of [itex]\underline{u}_3(s)[/itex]... so (locally) the curve looks like a piece of helix (?)

(... and so on in N dimensions...)
 
  • #7
(I may have abused my notation there... I mean in the vicinity of some specific [itex]s[/itex], rather than the general definition of [itex]s[/itex] as arc length)
 
  • #8
Hmm, it seems like this thread has dried up. Perhaps I can rephrase the original question.

Do the following two approaches yield the same result?[tex]\underline{u}_k(s)=\frac{\underline{a}^{(k)}(s) - \sum\limits_{m=1}^{k-1}\left(\underline{u}_m^T(s)\underline{a}^{(k)}(s)\right) \underline{u}_m(s)}{\Vert numerator \Vert}[/tex]... suggested in, for example, [1, p. 13] (link) and [2] (link).
[tex]\underline{u}_k(s)=\frac{\underline{u}_{k-1}^{\prime}(s) - \sum\limits_{m=1}^{k-1}\left(\underline{u}_m^T(s)\underline{u}_{k-1}^{\prime}(s) \right) \underline{u}_m(s)}{\Vert numerator \Vert}[/tex]... suggested in, for example, [3, p. 159].

In other words, is the subspace spanned by [itex]\left\{\underline{a}^{\prime}, \underline{a}^{ \prime \prime}, \dots, \underline{a}^{(k)}\right\}[/itex] the same as the subspace spanned by [itex]\left\{\underline{u}_1, \underline{u}_2, \dots, \underline{u}_{k-1}, \underline{u}_{k-1}^{\prime} \right\}[/itex]?

References:
[1] W. Kühnel, "Differential Geometry: Curves - Surfaces - Manifolds".
[2] Wikipedia, "Frenet–Serret formulas".
[3] H. W. Guggenheimer, "Differential Geometry", McGraw Hill (or Dover Edition), 1963 (1977).
 

What is the Frenet Frame?

The Frenet Frame is a mathematical concept used in differential geometry to describe the local geometry of a curve in three-dimensional space. It consists of three orthonormal vectors, known as the tangent, normal, and binormal vectors, that together form a basis for the tangent plane at each point along the curve.

What is the Gram-Schmidt process?

The Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space. It involves taking a set of linearly independent vectors and transforming them into a set of orthogonal vectors, where each vector is perpendicular to all the others. This process is commonly used in mathematics and physics, including in the calculation of the Frenet Frame.

How does the Gram-Schmidt process help obtain the Frenet Frame?

The Gram-Schmidt process is used to transform the tangent vector of a curve into the normal and binormal vectors, which are perpendicular to each other and the tangent vector. This process involves taking the cross product of the tangent vector with a new, orthogonal vector, and then normalizing the resulting vector to create a unit vector. Repeating this process with each new vector creates the orthonormal Frenet Frame.

What is the significance of the Frenet Frame?

The Frenet Frame is important because it provides a way to describe the local geometry of a curve without relying on a specific coordinate system. This means that the Frenet Frame can be used to analyze curves in any orientation or location in space. It is also used in various applications, including computer graphics, motion planning, and robotics.

Are there any other methods for obtaining the Frenet Frame besides the Gram-Schmidt process?

Yes, there are other methods for obtaining the Frenet Frame, including the parallel transport method and the rotation matrix method. However, the Gram-Schmidt process is one of the most commonly used and efficient methods for calculating the Frenet Frame.

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