Epitrochoid

EPITROCHOID

From the Greek epi "over" and trokhos "wheel".
Applet for tracing epitrochoids: www.wordsmith.org/~anu/java/spirograph.html

Complex parametrization:

, i.e.

where a is the radius of the base circle, b = a / q that of the rolling circle and d = k b the distance between the point and the centre of the moving circle.
Cartesian parametrization:

.
Polar parametrization:

.
Area encompassed by the curve in the case

, integer, and

(obtained by the Holditch theorem).

The epitrochoids are the curves described by a point linked to a circle (C) rolling without slipping on a base circle (C₀), the open disks with boundaries (C) and (C₀) being disjoint; they are the curves obtained with a Spirograph with an external disk.

In other words, they are also the roulettes of the movement of a plane over a fixed plane the base of which is a circle and the rolling curve of which is a circle outside the first one.

For d = b, i.e. k = 1, we get the epicycloids.
For d = a + b, we get the roses.

For k < 1, the curve is also called shortened epicycloid.
For k > 1, the curve is also called elongated epicycloid.

Every epitrochoid is also a peritrochoid (property called "of double generation"): fixed circle with radius , moving circle with radius and distance between the point and the centre of the moving circle equal to d' = a + b.

Here are the different shapes depending on the values of k.

It is remarkable that for , the shortened epicycloid constantly turns its concavity towards the center (opposite, case q = 3, k = 1/8).
From this paper, this is the case of the trajectory of the moon relative to the sun, for which .
Note that for the right trochoide, there are always changes in concavity.

The limit case is ; there is then a flat point.

For , the curve undulates, with points of inflection.

For k = 1, we obtain the epicycloid, with cusps.

For , the curve makes loops, with concave and convex portions, (positive angular speed, then negative
alternately).

For k = q + 1 (so d = a + b)), We get a rose with index n < 1 , of polar equation .

Case k > q – 1.

Epitrochoids can also be defined by the trajectories of a movement composed of two circular motions in the same direction and different angular speeds, with complex parametrization: (); they are epicycloids if , elongated epicycloids if , and shortened epicycloid if (we can then take , , d = r₂, thus ).
The angular speed (with respect to the fixed plane) of the first arm is the quadruple of that of the second arm: we get an epitrochoid with parameter q = 4 - 1 = 3.

Writing provides the following interpretation of epitrochoids: if two bodies are in uniform rotation in the same direction in a fixed plane, the visible trajectory of one of them in the plane linked to the second one and in translation with respect to the fixed plane is an epitrochoid.

Approximating the movements of the planets around the Sun by coplanar uniform circular movements, we can consider that the visible trajectories of the planets with respect to one another are epitrochoids. These trajectories were called epicycles (see also this animation).
Opposite, a sketch drawn by Kepler in Astronomia nova, representing the evolution of the laces of the orbit of Mars from the Earth.

Shape of the curves in the different cases:

Value of q Value of

1 2 limaçons of Pascal

2 3

3 4

4 5

5 6

3/2 5/2

4/3 7/3

The epitrochoids are also the plane projections of the Capareda curve, or the satellite curves, as well as the curves of constant precession.

The epitrochoids along with the hypotrochoids are the centred trochoids (see also a generalisation on this page).

Given a hypocycloid with parameter , there exists a circumscribed hypotrochoid, with parameter , with and .

Here are various epitrochoids represented in 3D and then tied:

q = 3, link 9.2.24 q = 3, knot 9.1.40 q = 3/2, link with 12 crossings q = 4, see this page

q = 4, knot with 12 crossings q = 5, link with 15 crossings q = 5/2, knot 5.1.1 q = 3/2 and q =3 intertwined, link with 15 crossings

Gear belt, in the form of an epitrochoid by Lévi Capareda during a class of industrial sciences... p = 4, k = 4
p = 4, k = 4

Ditto with q = 2 (Dürer folium)

This pattern of an Arab tapestry has the same structure than the epitrochoid with p = 8, k = 5 (rotate by 1/16th of a turn).

next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

It is remarkable that for , the shortened epicycloid constantly turns its concavity towards the center (opposite, case q = 3, k = 1/8). From this paper, this is the case of the trajectory of the moon relative to the sun, for which . Note that for the right trochoide, there are always changes in concavity.
The limit case is ; there is then a flat point.
For , the curve undulates, with points of inflection.
For k = 1, we obtain the epicycloid, with cusps.
For , the curve makes loops, with concave and convex portions, (positive angular speed, then negative alternately).
For k = q + 1 (so d = a + b)), We get a rose with index n < 1 , of polar equation .
Case k > q – 1.

Epitrochoids can also be defined by the trajectories of a movement composed of two circular motions in the same direction and different angular speeds, with complex parametrization: (); they are epicycloids if , elongated epicycloids if , and shortened epicycloid if (we can then take , , d = r₂, thus ).	The angular speed (with respect to the fixed plane) of the first arm is the quadruple of that of the second arm: we get an epitrochoid with parameter q = 4 - 1 = 3.
Writing provides the following interpretation of epitrochoids: if two bodies are in uniform rotation in the same direction in a fixed plane, the visible trajectory of one of them in the plane linked to the second one and in translation with respect to the fixed plane is an epitrochoid.
Approximating the movements of the planets around the Sun by coplanar uniform circular movements, we can consider that the visible trajectories of the planets with respect to one another are epitrochoids. These trajectories were called epicycles (see also this animation). Opposite, a sketch drawn by Kepler in Astronomia nova, representing the evolution of the laces of the orbit of Mars from the Earth.


q = 3, link 9.2.24	q = 3, knot 9.1.40	q = 3/2, link with 12 crossings	q = 4, see this page

q = 4, knot with 12 crossings	q = 5, link with 15 crossings	q = 5/2, knot 5.1.1	q = 3/2 and q =3 intertwined, link with 15 crossings

Gear belt, in the form of an epitrochoid by Lévi Capareda during a class of industrial sciences...	p = 4, k = 4		p = 4, k = 4
	Ditto with q = 2 (Dürer folium)