Helix

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

$$x = r\cos t$$ (1)

$$y = r\sin t$$ (2)

$$z = ct,$$ (3)

where $c$ and $r$ are constants. The Curvature of the helix is given by

$$\kappa={r\over r^2+c^2},$$ (4)

and the Locus of the centers of Curvature of a helix is another helix. The Arc Length is given by

$$s=\int\sqrt{x'^2+y'^2+z'^2}\,dt = \sqrt{r^2+c^2}\,t.$$ (5)

The Torsion of a helix is given by

$$\tau = {1\over r^2(r^2+c^2)} \left\vert\begin{array}{ccc}-r\sin t & -r\cos t & r\sin t\\ r\cos t & -r\sin t & -r\cos t\\ c & 0 & 0\end{array}\right\vert$$

$$= {c\over r^2+c^2},$$ (6)

so

$${\kappa\over\tau}={{r\over r^2+c^2}\over {c\over r^2+c^2}}={r\over c},$$ (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

$$\left\vert\matrix{z_1-r\cos t & z_2-r\sin t & z_3-ct\cr -r\sin t & r\cos t & c\cr -r \cos t & -r\sin t & 0\cr}\right\vert=0$$ (8)

$$z_1c\sin t-z_2c\cos t+(z_3-ct)r=0.$$ (9)

The Minimal Surface of a helix is a Helicoid.

See also Generalized Helix, Helicoid, Spherical Helix

References

Geometry Center. ``The Helix.'' http://www.geom.umn.edu/zoo/diffgeom/surfspace/helicoid/helix.html.

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.