Editing Frenet–Serret formulas (section)

=== Frenet–Serret formulas in calculus ===
The Frenet–Serret formulas are frequently introduced in courses on [[multivariable calculus]] as a companion to the study of space curves such as the [[helix]]. A helix can be characterized by the height {{math|2π''h''}} and radius {{mvar|r}} of a single turn. The curvature and torsion of a helix (with constant radius) are given by the formulas
<math display=block>\begin{align}
  \kappa &= \frac{r}{r^2+h^2} \\[4pt]
  \tau &= \pm\frac{h}{r^2+h^2}.
\end{align}</math>
[[Image:Frenet-Serret helices.png|right|thumb|Two helices (slinkies) in space. (a) A more compact helix with higher curvature and lower torsion. (b) A stretched out helix with slightly higher torsion but lower curvature.]]
The sign of the torsion is determined by the right-handed or left-handed [[right-hand rule|sense]] in which the helix twists around its central axis. Explicitly, the parametrization of a single turn of a right-handed helix with height {{math|2π''h''}} and radius {{mvar|r}} is
<math display=block>\begin{align}
  x &= r \cos t \\
  y &= r \sin t \\
  z &= ht \\
  (0 &\leq t \leq 2 \pi)
\end{align}</math>
and, for a left-handed helix,
<math display=block>\begin{align}
  x &= r \cos t \\
  y &= -r \sin t \\
  z &= ht \\
  (0 &\leq t \leq 2 \pi).
\end{align}</math>
Note that these are not the arc length parametrizations (in which case, each of {{math|''x'', ''y'', ''z''}} would need to be divided by <math>\sqrt{h^2+r^2}</math>.)

In his expository writings on the geometry of curves, [[Rudy Rucker]]<ref name = rucker>{{cite web|last = Rucker |first = Rudy|date = 1999|title = Watching Flies Fly: Kappatau Space Curves |publisher = San Jose State University|url = http://www.mathcs.sjsu.edu/faculty/rucker/kaptaudoc/ktpaper.htm|url-status = dead |archive-url =https://web.archive.org/web/20041015020304/http://www.mathcs.sjsu.edu/faculty/rucker/kaptaudoc/ktpaper.htm|archive-date = 15 October 2004}}</ref> employs the model of a [[slinky]] to explain the meaning of the torsion and curvature. The slinky, he says, is characterized by the property that the quantity
<math display=block> A^2 = h^2+r^2</math>
remains constant if the slinky is vertically stretched out along its central axis. (Here {{math|2π''h''}} is the height of a single twist of the slinky, and {{mvar|r}} the radius.) In particular, curvature and torsion are complementary in the sense that the torsion can be increased at the expense of curvature by stretching out the slinky.