is a roulette
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes....
traced by a point attached to a circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
rolling around the outside of a fixed circle of radius R
, where the point is a distance d
from the center of the exterior circle.
The parametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
s for an epitrochoid are
is a parameter (not the polar angle).
Special cases include the limaçon
In geometry, a limaçon or limacon , also known as a limaçon of Pascal, is defined as a roulette formed when a circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller...
with R = r
and the epicycloid
In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called an epicycle — which rolls without slipping around a fixed circle...
with d = r
The classic Spirograph
Spirograph is a geometric drawing toy that produces mathematical curves of the variety technically known as hypotrochoids and epitrochoids. The term has also been used to describe a variety of software applications that display similar curves, and applied to the class of curves that can be produced...
toy traces out epitrochoid and hypotrochoid
A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle....
The orbits of planets in the once popular geocentric Ptolemaic system are epitrochoids.
The combustion chamber
A combustion chamber is the part of an engine in which fuel is burned.-Internal combustion engine:The hot gases produced by the combustion occupy a far greater volume than the original fuel, thus creating an increase in pressure within the limited volume of the chamber...
of the Wankel engine
The Wankel engine is a type of internal combustion engine using an eccentric rotary design to convert pressure into a rotating motion instead of using reciprocating pistons. Its four-stroke cycle takes place in a space between the inside of an oval-like epitrochoid-shaped housing and a rotor that...
is an epitrochoid.