# 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).

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### Proving that two curves are orthogonal

Let $α(s)=(x(s)),y(s))$ be a regular plane curve that is parameterized by arc length, and let $n(s)$ be its normal vector. Consider the family of curves: $β(s,r)=α(s)+rn(s),−ϵ≤r≤ϵ$ I need to prove ...
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### Finding the curvature & torsion of the derivative of a smooth regular curve in $\mathbb{R}^3$

Let $\beta(s)$ be a smooth regular curve in $\mathbb{R}^3$ parameterized by arclength with nowhere vanishing curvature. Let $\gamma(s) = \beta'(s)$. Find $\kappa_\gamma(s)$ and $\tau_\gamma(s)$ in ...
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### What is $\left( \frac{t^2}{t^2+2}i+\frac{2}{t^2+2}j+\frac{2t}{t^2+2}k \right) \times \frac{2ti-2tj-(t^2+2)k}{t^2+2}$?

I did \begin{align} & \left( \frac{t^2}{t^2+2}i+\frac{2}{t^2+2}j+\frac{2t}{t^2+2}k \right) \times \frac{2ti-2tj-(t^2+2)k}{t^2+2} \\[8pt] = {} & \frac{2t^2i-4tj-2t(t^2+2)k}{(t^2+2)^2} \\[8pt] = ...
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### Jacobian Determinant of frenet transformation

If anybody can help with this. Given a point $\boldsymbol x = (x,y)$, it can be represented as $\boldsymbol x=\boldsymbol p(s)+r \boldsymbol u(s)$. p is a curve parametrized with arc length s. $r$ is ...
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### Finding curavature and torsion of a curve in terms of its Bishop frame

Suppose we are given a regular, naturally parametrized curve $\gamma :I \rightarrow \mathbb{R}^3$, how does one then compute the curvature and torsion of $\gamma$ in terms of the entries of the Bishop ...
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