Roulettes (curve)

right|frame|A cycloid generated by a rolling circle

thumb|A cardioidthumb|right|200px|The caustic (optics)|caustic appearing on the surface of this cup of coffee is a cardioid.

thumb|500px|The red curve is an epicycloid traced as the small circle (radius rolls around the outside of the large circle (radius . In geometry, an epicycloid (also called hypercycloid) is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle. It is a particular kind of roulette.

thumb|460px|The red path is a hypocycloid traced as the smaller black circle rolls around inside the larger black circle (parameters are R=4.0, r=1.0, and so k=4, giving an astroid). In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid created by rolling a circle on a line.

thumb|right|This chain, whose ends hang from two points, forms a catenary. thumb|right|The silk on this spider web forms multiple elastic catenaries.

thumb|Astroid thumb|The hypocycloid construction of the astroid. thumb|Astroid as the common Envelope (mathematics)|envelope of a family of [[ellipses of equation , where .]] [[File:sliding_ladder_in_astroid.svg|thumb|link=|The envelope of a ladder (coloured lines in the top-right quadrant) sliding down a vertical wall, and its reflections (other quadrants) is an astroid. The midpoints trace out a circle while other points trace out ellipses similar to the previous figure. [ ] hover over a ladder to highlight it.]] thumb|right|Astroid as an evolute of ellipse

thumb|Two involutes (red) of a parabola

thumb|A cycloid (a common trochoid) generated by a rolling circle

thumb|400px|The epitrochoid with , and

thumb|upright=1|The red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are ).

right|thumb|300px|Construction of the limaçon with polar coordinates' origin at

plane curve, 3-cusped hypocycloid

mathematical curve

mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle

mathematical curve generated by rolling other curves together

thumb|Nephroid: definition

In geometry, a cyclogon is the curve traced by a vertex of a regular polygon that rolls without slipping along a straight line.