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A helix is also called a Curve of Constant Slope. It can be defined as a curve for which the Tangent makes a constant Angle with a fixed line. The helix is a Space Curve with parametric equations

$\displaystyle x$ $\textstyle =$ $\displaystyle r\cos t$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle r\sin t$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle ct,$ (3)

where $c$ and $r$ are constants. The Curvature of the helix is given by
\kappa={r\over r^2+c^2},
\end{displaymath} (4)

and the Locus of the centers of Curvature of a helix is another helix. The Arc Length is given by
s=\int\sqrt{x'^2+y'^2+z'^2}\,dt = \sqrt{r^2+c^2}\,t.
\end{displaymath} (5)

The Torsion of a helix is given by
$\displaystyle \tau$ $\textstyle =$ $\displaystyle {1\over r^2(r^2+c^2)} \left\vert\begin{array}{ccc}-r\sin t & -r\cos t & r\sin t\\  r\cos t & -r\sin t & -r\cos t\\  c & 0 & 0\end{array}\right\vert$  
  $\textstyle =$ $\displaystyle {c\over r^2+c^2},$ (6)

{\kappa\over\tau}={{r\over r^2+c^2}\over {c\over r^2+c^2}}={r\over c},
\end{displaymath} (7)

which is a constant. In fact, Lancret's Theorem states that a Necessary and Sufficient condition for a curve to be a helix is that the ratio of Curvature to Torsion be constant. The Osculating Plane of the helix is given by
\left\vert\matrix{z_1-r\cos t & z_2-r\sin t & z_3-ct\cr
-r\sin t & r\cos t & c\cr
-r \cos t & -r\sin t & 0\cr}\right\vert=0
\end{displaymath} (8)

z_1c\sin t-z_2c\cos t+(z_3-ct)r=0.
\end{displaymath} (9)

The Minimal Surface of a helix is a Helicoid.

See also Generalized Helix, Helicoid, Spherical Helix


Geometry Center. ``The Helix.''

Gray, A. ``The Helix and Its Generalizations.'' §7.5 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 138-140, 1993.

Isenberg, C. Plate 4.11 in The Science of Soap Films and Soap Bubbles. New York: Dover, 1992.

Pappas, T. ``The Helix--Mathematics & Genetics.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 166-168, 1989.

Wolfram, S. The Mathematica Book, 3rd ed. Champaign, IL: Wolfram Media, p. 163, 1996.

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© 1996-9 Eric W. Weisstein