UniThread Chromatics: When each thread circle is color coded with hue, the interunion turns out to be the color blending operation at the eye (incidence between the threads). Here I demonstrate the quadratic form of [i]a[/i] ⋇ [i]b[/i], where the result is an [url=https://en.wikipedia.org/wiki/Epicycloid]epicycloid[/url] for temporal interunion and a [url=https://en.wikipedia.org/wiki/Hypocycloid]hypocycloid[/url] for spatial interunion. Note that when [i]a < b[/i], the temporal interunion produces a special kind of epicycloid called "[url=https://mathcurve.com/courbes2d.gb/peritrochoid/peritrochoid.shtml]peritrochoid[/url]". For higher order forms, see the applet [url=https://www.geogebra.org/m/etenmkub]Interunion-Euclidean[/url].

Spirograph: [url=https://en.wikipedia.org/wiki/Spirograph]The toy spirograph[/url] has similar structures, [url=https://en.wikipedia.org/wiki/Epitrochoid]epitrochoid[/url] and [url=https://en.wikipedia.org/wiki/Hypotrochoid]hypotrochoid[/url], corresponding to the [url=https://en.wikipedia.org/wiki/Epicycloid]epicycloid[/url] and [url=https://en.wikipedia.org/wiki/Hypocycloid]hypocycloid[/url] in this applet.[br]See [url=https://nathanfriend.io/inspiral-web/]Nathan Friend's applet[/url] for interactive spirographs.[br][br]Cyclocycloid: epicycloid, hypocycloid, epitrochoid and hypotrochoid are in general called [url=https://en.wikipedia.org/wiki/Cyclocycloid]cyclocycloid[/url].[br][br]Epicycles & Fourier series: The higher order of cyclocycloid is a curve traced by the [url=https://en.wikipedia.org/wiki/Deferent_and_epicycle]epicycles[/url]. When the number of epicycles is unlimited, it becomes a [url=https://en.wikipedia.org/wiki/Fourier_series]Fourier series[/url].