geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an epitrochoid ( or ) is a

roulette Roulette (named after the French language, French word meaning "little wheel") is a casino game which was likely developed from the Italy, Italian game Biribi. In the game, a player may choose to place a bet on a single number, various grouping ...

traced by a point attached to a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

rolling around the outside of a fixed circle of radius , where the point is at a distance from the center of the exterior circle. The

parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case ...

s for an epitrochoid are: :

\begin
& x (\theta) = (R + r)\cos\theta - d\cos\left(\theta\right) \\
& y (\theta) = (R + r)\sin\theta - d\sin\left(\theta\right)
\end

The parameter is geometrically the polar angle of the center of the exterior circle. (However, is not the polar angle of the point

(x(\theta),y(\theta))

on the epitrochoid.) Special cases include the

limaçon In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. I ...

with and the epicycloid with . The classic Spirograph toy traces out epitrochoid and

hypotrochoid In geometry, a hypotrochoid is a roulette (curve), roulette traced by a point attached to a circle of radius rolling around the inside of a fixed circle of radius , where the point is a distance from the center of the interior circle. The par ...

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 A combustion chamber is part of an internal combustion engine in which the air–fuel ratio, fuel/air mix is burned. For steam engines, the term has also been used for an extension of the Firebox (steam engine), firebox which is used to allow a mo ...

of the

Wankel engine The Wankel engine (, ) is a type of internal combustion engine using an eccentric (mechanism), eccentric Pistonless rotary engine, rotary design to convert pressure into rotating motion. The concept was proven by German engineer Felix Wankel, f ...

is an epitrochoid with , and .

References

External links

Epitrochoid generator
*

* ttp://gerdbreitenbach.de/planet/planet.html Interactive simulation of the geocentric graphical representation of planet paths *{{MacTutor, class=Curves, id=Epitrochoid, title=Epitrochoid
Plot Epitrochoid -- GeoFun
Roulettes (curve) ja:トロコイド#外トロコイド

See also

References

External links