EPICYCLOID

This article appears in Volume V09, Page 686 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: EMS-EUD

EPICYCLOID , the curve traced out by a point on the circumference of a circle

rolling

ROLLING

externally on another circle. If the moving circle rolls internally on the fixed circle, a point on the circumference describes a " hypocycloid " (from inro, under). The locus of any other carried point is an " epitrochoid " when the circle rolls externally, and a "hypotrochoid " when the circle rolls internally. The epicycloid was so named by Ole Romer in 1674, who also demonstrated that cog-wheels having epicycloidal teeth revolved with minimum friction (see MECHANICS: Applied); this was also proved by Girard Desargues, Philippe de la Hire and Charles Stephen Louis

Camus

CAMUS, ARMAND GASTON (1740-1804)
CAMUS, CHARLES
ETIENNE
- ETIENNE, CHARLES GUILLAUME (1778-1845)
LOUIS (1699-1768)
CAMUS, FRANCOIS JOSEPH DES (1672-1732)

. Epicycloids also received attention at the hands of Edmund Halley, Sir Isaac Newton and others; spherical epicycloids, in which the moving circle is inclined at a constant

angle

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

to the plane of the fixed circle, were studied by the Bernoullis, Pierre Louis M. de Maupertuis, Francois

Nicole

NICOLE, PIERRE (1625–1695)

, Alexis Claude Clairault and others.
In the annexed figure, there are shown various examples of the curves named above, when the radii of the

rolling

ROLLING

and fixed circles are in the ratio of I to 3. Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself. In the particular case when the radii are in the ratio of I to 3 the epicycloid (curve a) will consist of three cusps external to the circle and placed at equal distances along its circumference. Similarly, the corresponding epitrochoids will exhibit three loops or nodes (curve b), or assume the form shown in the curve' c. It is interesting to compare the forms of these curves with the three forms of the cycloid (q.v.). The hypocycloid derived from the same circles is shown as curve d, and is seen to consist of three cusps arranged internally to the fixed circle; the corresponding hypotrochoid consists of a three-foil and is shown in curve e. Theepicycloid shown is termed the three-cusped epicycloid " or the epicycloid of Cremona."
The cartesian equation to the epicycloid assumes the form
x=(a+b) cosBb cos (a+b/b)B, y=(a+b) sin Bb sin (a+b/b)B,
when the centre of the fixed circle is the origin, and the

axis

AXIS (Lat. for " axle ")

of x
passes through the initial. point of the curve (i.e. the

original

ORIGINAL

position
of the moving point on the fixed circle), a and b being the radii of the fixed and rolling circles, and B the

angle

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

through which the line joining the centres of the two circles has passed. It may be shown that if the distance of the carried point from the centre of the rolling circle be mb, the equation to the epitrochoid is
x = (a+b) cos B mb cos (a+b/b)B, y = (a+b) sin o mb sin (a+b/b)9. The equations to the hypocycloid and its corresponding trochoidal curves are derived from the two preceding equations by changing the sign of b. Leonhard

Euler

EULER, LEONHARD (1707-1783)

(Acta Petrop. 1784) showed that the same hypocycloid can be generated by circles having radii of a (a+b) rolling on a circle of radius a; and also that the hypocycloid formed when the radius of the rolling circle is greater than that of the fixed circle is the same as the epicycloid formed by the rolling of a circle whose radius is the difference of the

original

ORIGINAL

radii. These pro-positions may be derived from the formulae given above, or proved directly by purely geometrical methods.
The tangential polar equation to the epicycloid, as given above, is p=(a+2b) sin (a a+2b)+,G, while the intrinsic equation is s=4(bla)(a+b) cos (a/aa--2b)p and the pedal equation is r2=a2+ (4b.a+b)p1/(a+2b)2, Therefore any epicycloid or hypocycloid may be represented by the equations p =A sin or p =A cos B,', s=A sin B or s =A cos BC or r2=A+Bpi, the constants A and B being readily determined by the above considerations.
If the radius of the rolling circle be one-half of the fixed circle, the hypocycloid becomes a

diameter

DIAMETER (from the Cr. &a, through, µErpov, measure)

of this circle; this may be con-firmed from the equation to the hypocycloid. If the ratio of the radii be as 1 to 4, we obtain the four-cusped hypocycloid, which has the simple cartesian equation x213+yE13=a213. This curve is the envelope of a line of constant length, which moves so that its extremities are always on two fixed lines at right angles to each other, i.e.. of the line x/a+y/a=1, with the

condition

CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

a2+(32 =1/a, a constant. The epicycloid when the radii of the circles are equal is the cardioid (q.v ), and the corresponding trochoidal curves are limagons (q.v.). Epicycloids are also examples of certain caustics (q.v.).
For the methods of determining the formulae and results stated above see J. Edwards, Differential Calculus, and for geometrical constructions see T. H. Eagles, Plane Curves.

End of Article: EPICYCLOID

If you wish, you can link directly to this article.

<a href="http://jcsm.org/StudyCenter/Encyclopedia/EMS_EUD/EPICYCLOID.html">
EPICYCLOID
</a>

(Previous)
EPICYCLE (Gr. upon, and KuKXos, circle)

(Next)
EPIDAURUS

JCSM was founded in 1997 and exists to help the community and bring people into a life-changing and productive relationship with Jesus Christ. JCSM offers over 200,000 free web pages, including its weekly inspirational emails that were sent continuously for over a decade.

Jesus Christ Saves Ministries
P.O. Box 9297
San Diego, CA 92169
1-888-887-0417 or Email