geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, an epitrochoid ( or ) is a

roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...

traced by a point attached to a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

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 defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...

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 In geometry, an epicycloid 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. Equati ...

with . The classic

Spirograph Spirograph is a geometric drawing device that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. The well-known toy version was developed by British engineer Denys Fisher and first sold ...

toy traces out epitrochoid and

hypotrochoid In geometry, a hypotrochoid is a 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 parametric equations f ...

curves. The orbits of planets in the once popular geocentric

Ptolemaic system In astronomy, the geocentric model (also known as geocentrism, often exemplified specifically by the Ptolemaic system) is a superseded description of the Universe with Earth at the center. Under most geocentric models, the Sun, Moon, stars, and ...

are epitrochoids (see

deferent and epicycle In the Hipparchian, Ptolemaic, and Copernican systems of astronomy, the epicycle (, meaning "circle moving on another circle") was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun ...

). 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 fuel/air mix is burned. For steam engines, the term has also been used for an extension of the firebox which is used to allow a more complete combustion process. Interna ...

of the

Wankel engine The Wankel engine (, ) is a type of internal combustion engine using an Eccentric (mechanism), eccentric rotary combustion engine, rotary design to convert pressure into rotating motion. It was invented by German engineer Felix Wankel, and desi ...

is an epitrochoid.

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