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I have a question about the definition of Frenet frame here (pg 6) https://www.math.cuhk.edu.hk/~martinli/teaching/4030lectures.pdf .

The definition above is summarized below:

We are given a regular curve, $c$, in $\mathbb{R}^n$ parametrized by arc length, such that $c'(s),c''(s),\dots,c^{n-1}(s)$ are linearly independent for each $s$. A Frenet frame for $c$ is defined to be a positively oriented orthonormal basis $\{ e_1(s), \dots, e_n(s) \}$ such that

  1. $\text{Span}\{e_1(s),\dots,e_k(s)\} = \text{Span}\{c'(s),\dots,c^{k}(s)\}$ for $k= 1,\dots, \color{red} n$ and
  2. $\langle c^{k}(s) , e_k(s) \rangle > 0$.

Condition 1. above does not look right to me because no assumptions about $c^{n}(s)$ are made. Should 1. be replaced by

  1. $\text{Span}\{e_1(s),\dots,e_k(s)\} = \text{Span}\{c'(s),\dots,c^{k}(s)\}$ for $k= 1,\dots, \color{red} {n - 1}?$

A similar assumption consistent with the document above is also made here http://www.cs.elte.hu/geometry/csikos/dif/dif2.pdf (pg 7. first definition on top of the page) so I wonder if I am missing something.

Arin Chaudhuri
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2 Answers2

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Note that the author specifically comments that for a Frenet curve we might have $c^{(n)}(s)=0$ for some $s$, so both the conditions you listed should, as you said, hold for $k=1,2,\dots,n-1$. You construct $e_n(s)$ by orthonormality and orientation.

Ted Shifrin
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I found this question by coincidence and I have a different answer, so I decide to post it here. Since we might have $c^{(n)}(s)=0$ for some $s$, in that case $$\text{Span}\{c'(s),\dots,c^{(n)}(s)\}=\text{Span}\{c'(s),\dots,c^{(n-1)}(s)\}$$ But $\text{Span}\{e_1(s),\dots,e_n(s))\neq \text{Span}(e_1(s),\dots,e_{n-1}(s)\}$ by orthonormality, so the identity $$\text{Span}\{e_1(s),\dots,e_k(s)\}=\text{Span}\{c'(s),\dots,c^{(k)}(s)\}$$ can't be true for both $n-1$ and $n$. In my opinion, OP was right; then $e^n(s)$ can be uniquely determined from $e_1(s),\dots,e_n(s)$ by orthonormality and orientation, just like the last line of the previous answer. In the book Differential Geometry: Curves - Surfaces - Manifolds pg. 13, Wolfgang Kühnel use the same definition.

withoutfeather
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