We recall the parametric representation of a trochoid:
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 , so we fix ρ = 1:
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 an applet that displays the conformal map associated with a cycloid. The controls which determine the wheel ratio are familiar. Below these are two sliders which control the minimum and maximum values of t that are displayed. The applet draws 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.