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epicycloid

an epicycloid in which the radii of the fixed and rolling circles are equal.

10Epicycloid >>

00It 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.

00The path of contact which it traces is identical with itself; and the flanks of the teeth c are internal and their faces ex ternal epicycloids 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 ~, 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.

00Nearly 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 of epicycloids Let R be the radius of the pitch-circle; r that of the describing circle; 8 the angle made by the normal TI to the epicycloid at a given point T, with a tangent-to the circle at Ithat is, the obliquity of the action at T.

00Then the radius of curvature of the epicycloid at T is For an internal epicycloid, p =4r sin o~1

00For an external epicycloid, p = 4r sin O~iJ

00EPICYCLOID, the curve traced out by a point on the circumference of a circle rolling externally on another circle.

00In 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.

00The epicycloid shown is termed the "three-cusped epicycloid" or the "epicycloid of Cremona."

00The 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.

00Leonhard 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.

00Therefore any epicycloid or hypocycloid may be represented by the equations p = A sin B+,' or p---A cos B,,G, s = A sin B11.

00For epiand hypo-cycloids and epiand hypo-trochoids see Epicycloid.

00The 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.

00It 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.

00The curve may be regarded as an epitrochoid (see Epicycloid) in which the rolling and fixed circles have equal radii.

00The path of contact which it traces is identical with itself; and the flanks of the teeth c are internal and their faces ex ternal epicycloids 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 ~, 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.

00Nearly 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 of epicycloids Let R be the radius of the pitch-circle; r that of the describing circle; 8 the angle made by the normal TI to the epicycloid at a given point T, with a tangent-to the circle at Ithat is, the obliquity of the action at T.

00Then the radius of curvature of the epicycloid at T is For an internal epicycloid, p =4r sin o~1

00For an external epicycloid, p = 4r sin O~iJ

00EPICYCLOID, the curve traced out by a point on the circumference of a circle rolling externally on another circle.

00The 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.

00In 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.

00The epicycloid shown is termed the "three-cusped epicycloid" or the "epicycloid of Cremona."

00The 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.

00Leonhard 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.

00The 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) .

00Therefore any epicycloid or hypocycloid may be represented by the equations p = A sin B+,' or p---A cos B,,G, s = A sin B11.

00For epiand hypo-cycloids and epiand hypo-trochoids see Epicycloid.

00an epicycloid in which the radii of the fixed and rolling circles are equal.

00remaining circular, the question can be similarly treated, and it is found that the caustic is an epicycloid in which the radius of the fixed circle is twice that of the rolling circle (fig.

00The 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.

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