We recall the parametric representation of a trochoid:

T(w; ρ)(θ) =
(1-w)e^{iθ} + ρwe^{i(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^{iθ} + we^{t/w}e^{i(1-1/w)θ}) .

As θ varies, this traces out

e^{-t}T(w; e^{t/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.