Epitrochoid

File:EpitrochoidIn3.gif

The epitrochoid with R = 3, r = 1 and d = 1/2

In geometry, an epitrochoid (Template:IPAc-en or Template:IPAc-en) is a roulette traced by a point attached to a circle of radius Template:Mvar rolling around the outside of a fixed circle of radius Template:Mvar, where the point is at a distance Template:Mvar from the center of the exterior circle.

The parametric equations for an epitrochoid are:

$\begin{array}{rl}& x(\theta )=(R+r)\cos \theta -d\cos \left(\frac{R+r}{r}\theta \right)\\ & y(\theta )=(R+r)\sin \theta -d\sin \left(\frac{R+r}{r}\theta \right)\end{array}$

The parameter Template:Mvar is geometrically the polar angle of the center of the exterior circle. (However, Template:Mvar is not the polar angle of the point $(x(\theta ),y(\theta ))$ on the epitrochoid.)

Special cases include the limaçon with R = r and the epicycloid with d = r.

The classic Spirograph toy traces out epitrochoid and hypotrochoid curves.

The paths of planets in the once popular geocentric system of deferents and epicycles are epitrochoids with $d>r,$ for both the outer planets and the inner planets.

The orbit of the Moon, when centered around the Sun, approximates an epitrochoid.

The combustion chamber of the Wankel engine is an epitrochoid with R = 2, r = 1 and d = 1.

See also

• Cycloid
• Cyclogon
• Epicycloid
• Hypocycloid
• Hypotrochoid
• Spirograph
• List of periodic functions
• Rosetta (orbit)
• Apsidal precession

External links

• Epitrochoid generator
• Visual Dictionary of Special Plane Curves on Xah Lee 李杀网
• Interactive simulation of the geocentric graphical representation of planet paths
• Plot Epitrochoid -- GeoFun

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