# Epicycloid Sentence Examples

- It may be regarded as an
**epicycloid**in which the rolling and fixed circles are equal in diameter, as the inverse of a parabola for its focus, or as the caustic produced by the reflection at a spherical surface of rays emanating from a point on the circumference. - The path of contact which it traces is identical with itself; and the flanks of the teeth c are internal and their faces ex ternal
for wheels, and both flanks and faces are cycloids For a pitch-circle of twice the P, - / radius of the rolling or describing /, -~- circle (as it is called) the internal B ~,**epicycloids****epicycloid**is a straight line, being, / E in fact, a diameter of the pitch- circle, so that the flanks of the teeth for such a pitch-circle are planes radiating from the axis. - Nearly
**Epicycloidal**Teeth: Williss Method.To facilitate the drawing of**epicycloidal**teeth in practice, Willis showed how to approximate to their figure by means of two circular arcsone concave, for the flank, and the other convex, for the faceand each having for its radius the mean radius of curvature of the**epicycloidal**arc. \Villiss formulae are founded on the following properties ofLet R be the radius of the pitch-circle; r that of the describing circle; 8 the angle made by the normal TI to the**epicycloids****epicycloid**at a given point T, with a tangent-to the circle at Ithat is, the obliquity of the action at T. - Then the radius of curvature of the
**epicycloid**at T is For an internal**epicycloid**, p =4r sin o~1 - For an external
**epicycloid**, p = 4r sin O~iJ **EPICYCLOID**, the curve traced out by a point on the circumference of a circle rolling externally on another circle.- 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. - The
**epicycloid**shown is termed the "three-cusped**epicycloid**" or the "**epicycloid**of Cremona." - The cartesian equation to the
**epicycloid**assumes the form x = (a +b) cos 0 - b cos (a -Fb/b)8, y = (a +b) sin 0 - b sin (a -1--b/b)6, when the centre of the fixed circle is the origin, and the axis of x passes through the initial point of the curve (i.e. - Leonhard Euler (Acta Petrop. 1784) showed that the same hypocycloid can be generated by circles having radii of; (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. - The tangential polar equation to the
**epicycloid**, as given above, is p= (a+2b) sin (a a+2b),I', while the intrinsic equation is s=4(bla)(a+b) cos (ala+2b)>G and the pedal equation is r2=a2+ (4b.a+b)p 2 l(a+2b) . - Therefore any
**epicycloid**or hypocycloid may be represented by the equations p = A sin B+,' or p---A cos B,,G, s = A sin B11. - An
**epicycloid**in which the radii of the fixed and rolling circles are equal. - The polar form is {(u+p) cos 26} a+{(u-p) sin 20) a = (2k)t, where p and k are the reciprocals of c and a, and u the reciprocal of the radius vector of any point on the caustic. When c =a or = oo the curve reduces to the cardioid or the two cusped
**epicycloid**previously discussed.