# Questions tagged [frenet-frame]

Use this tag for questions on Frenet frames and the Frenet-Serret formulae. Related tags include (differential-geometry), (multivariable-calculus), and (curves).

103 questions
Filter by
Sorted by
Tagged with
50 views

### If acceleration is decomposed into the T and N directions, why can an object leave the plane?

I'm reading Thomas' Calculus, and had a question similar to this question, why-is-there-no-b-component-of-acceleration-in-my-multivariable-calculus-class I understand the math part, but cannot quite ...
33 views

### Can I find $b$ from $a = b \times c$?

So I just started studying the TNB (or Frenet-Serret) frame, where B = T × N. Then my book also goes on to say that T = N × B and N = B × T. Basically, we can find a new valid cross-product equation ...
31 views

### How to find rotation matrix of a plane curve of varying curvature?

I'm able to find a rotation matrix with respect to a fixed basis for a plane curve of constant curvature (example - circle) or a straight line (zero curvature). But in the case of a sinusoid, ...
11 views

58 views

### Torsion coefficient

We went over the Frenet-Serret formulas today in class and the professor wrote $$d\mathbf{\hat{B}}/dt=-\tau{}\mathbf{\hat{N}}.$$ He said that the coefficient is always negative (so that tau is always ...
29 views

48 views

34 views

59 views

### Proving that two curves that are symmetric about the origin have same curvature and same torsion (up to a sign)

This is an exercise in a differential geometry book I'm studying, and currently I can't fathom why it's there (and it doesn't feel at all intuitive why it would be true). A simple counter example ...
Let $\alpha(t)$ be a regular curve. Prove that if $\beta(s)$ and $\gamma(\bar{s})$ are two reparameterizations by arclength, then $s = \pm\bar{s} +a$, where $a \in \mathbb{R}$ is a constant. I know ...
Suppose $M$ is a smooth manifold of dimension $n$. Let $\gamma: [0,1] \mapsto M$ be a smooth curve on it. Assume $\dot{\gamma}(t)$ is never zero. Can we always find smooth vector fields \$e_1(t), e_2(t)...