Trochoids are Parts of Conformal Maps

We recall the parametric representation of a trochoid:

T(w; ρ)(θ) = (1-w)e + ρwei(1-1/w)θ .

The trochoid is the image of a line segment under this mapping. The mapping can be extended from the real line to the complex plane, if we simply replace θ with θ + i t. Then trochoids with varying arm ratios are all images of lines with different values of the imaginary part t, so we fix ρ = 1:

T(w)(θ + it) = (1-w)e-t(e + wet/wei(1-1/w)θ) .

As θ varies, this traces out

e-tT(w; et/w).

The appealing property of a conformal map is that, while it sends straight lines into curves, it preserves the angles between the lines. Another definition of this property is that the image of a small square, under a conformal map, is (approximately) another small square.

Here is a display of the conformal map associated with a cycloid. The controls which determine the wheel ratio are familiar. Below these are sliders which control the minimum and maximum values of t that are displayed. The display shows one or more trochoids corresponding to equally spaced values of t. There are three sliders to control the values of t; the first controls the spacing between successive values, and the others control the minimum and maximum values. There is a check box which gives you the option of drawing only a single arc (1/D of the whole curve). Another check box gives you the option to draw curves, perpendicular to the trochoids, corresponding to distinct values of θ.

Last modified on \$Date: 2015-05-02 12:51:16 -0400 (Sat, 02 May 2015) \$

Christopher J. Henrich