Epitrochoid

# Epitrochoid

Overview
An epitrochoid is a roulette
Roulette (curve)
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
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...

r 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.
Discussion

Encyclopedia
An epitrochoid is a roulette
Roulette (curve)
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
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...

r 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
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

where is a parameter (not the polar angle).

Special cases include the limaçon
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
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
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
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....

curves.

The orbits of planets in the once popular geocentric Ptolemaic system are epitrochoids.

The combustion chamber
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
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.