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EPICYCLOID

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Originally appearing in Volume V09, Page 686 of the 1911 Encyclopedia Britannica.

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EPICYCLOID , the See also:

CURVE (Lat. curvus, bent)

curve traced out by a point on the circumference of a circle See also:

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 See also:

LOCUS (Lat. for " place "; in Gr. rinros)

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 See also:

cog-wheels having epicycloidal See also:

TEETH (O.E. Eel); plural of tooth, O.E. top)

teeth revolved with minimum See also:

FRICTION (from Lat. fricare, to rub)

friction (see See also:

MECHANICS

MECHANICS: Applied); this was also proved by See also:

Girard Desargues, Philippe de la Hire and See also:

Charles See also:

Stephen See also:

Louis See also:

Camus. Epicycloids also received See also:

ATTENTION (from Lat. ad-tendo, await, expect; the condition of being " stretched " or " tense ")

attention at the hands of See also:

Edmund See also:

HALLEY, EDMUND (1656–1742)

Halley, See also:

Sir See also:

ISAAC (Hebrew for " he laughs," on explanatory references to the name, see ABRAHAM)

Isaac See also:

Newton and others; spherical epicycloids, in which the moving circle is inclined at a See also:

CONSTANT, BENJAMIN (1845-1902)

constant See also:

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

angle to the See also:

PLANE

plane of the fixed circle, were studied by the Bernoullis, See also:

PIERRE

Pierre Louis M. de See also:

MAUPERTUIS, PIERRE LOUIS MOREAU DE (1698–1759)

Maupertuis, See also:

FRANCOIS (1558-1614)

Francois See also:

NICOLE, PIERRE (1625–1695)

Nicole, See also:

Alexis See also:

CLAUDE, JEAN (1619-1687)

Claude See also:

CLAIRAULT (or CLAIRAUT), ALEXIS CLAUDE (1713-1765)

Clairault and others. In the annexed figure, there are shown various examples of the curves named above, when the radii of the rolling and fixed circles are in the ratio of I to 3. Since the circumference of a circle is proportional to its See also:

RADIUS

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 See also:

case when the radii are in the ratio of I to 3 the epicycloid (curve a) will consist of three cusps See also:

EXTERNAL

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 See also:

FORM (Lat. forma)

form shown in the curve' c. It is interesting to compare the forms of these curves with the three forms of the See also:

CYCLOID (from Gr. iciidws, circle, and ennos, form)

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-See also:

FOIL

foil and is shown in curve e. Theepicycloid shown is termed the three-cusped epicycloid " or the epicycloid of See also:

Cremona." The cartesian See also:

EQUATION (from Lat. aequatio, aequare, to equalize)

equation to the epicycloid assumes the form x=(a+b) cosB–b See also:

cos (a+b/b)B, y=(a+b) See also:

sin B–b sin (a+b/b)B, when the centre of the fixed circle is the origin, and the See also:

AXIS (Lat. for " axle ")

axis of x passes through the initial. point of the curve (i.e. the See also:

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 through which the See also:

LINE

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 See also:

EULER, LEONHARD (1707-1783)

Euler (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 radii. These See also:

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 See also:

INTRINSIC (through Fr. intrinsique, from Lat. intrinsecus, inwardly; inter, within, secus, following, from root of sequi, to follow)

intrinsic equation is s=4(bla)(a+b) cos (a/aa--2b)p and the pedal equation is See also:

R2(rt-s2)

r2=See also:

A2(z)

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-See also:

HALF

half of the fixed circle, the hypocycloid becomes a See also:

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

diameter of this circle; this may be See also:

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 See also:

SIMPLE

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 See also:

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

condition a2+(32 =1/a, a constant. The epicycloid when the radii of the circles are equal is the See also:

CARDIOID

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.

Additional information and Comments

can you please explain why the Epicycloid of Cremona was called that in the first place?

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