CHAPTER VI

ROULETTES AND GLISSETTES

70. Roulettes.-When one curve rolls without sliding on another curve, any point connected with the first curve describes on the plane of the second a curve called a roulette. The curve which rolls is called the rolling curve or generating curve, and the curve on which it rolls is called the directing curve or base. The directing curve is generally assumed to be fixed, and is sometimes called the fixed curve.

When the rolling and directing curves are circles the roulette becomes a cycloidal curve.

71. General Construction for Drawing a Roulette.-The best practical method of drawing any roulette is the tracing paper or transparent templet method which is described in Art. 60, p. 56.

The following is a general construction which may be used for drawing any roulette. ABCD (Fig. 137) is the base, Abcd is the rolling curve, and P is the tracing point. Take

a number of points b, c, d, etc. on the rolling curve and determine points B, C, D, etc. on the base such that the arcs AB, BC, CD, etc. are equal to the arcs Ab, bc, cd, etc. respectively. If the points are sufficiently near to one another the arcs may be assumed equal to their chords. Draw tangents to the base at B, C, D, etc. and tangents to the rolling curve at b, c, d, etc. From P draw Pm perpendicular to the tangent at b. On the tangent at B make BM = bm, and draw MP, perpendicular to BM and equal to Pm. P, will be the position of P when the rolling curve touches the base at B,

FIG. 137.

and will therefore be a point in the roulette described by P. In like manner other points may be determined.

Centre of Curvature

72. General Construction for the of a Roulette.-Three cases are illustrated in Figs. 138, 139, and 140. The description which follows applies to each.

O, is the centre of curvature of the base line AQB at Q. (If AQB

is an arc of a circle then O, is the centre of the circle, and if AQB is a straight line (Fig. 140), then O, is at an infinite distance from Q in a straight line through Q at right angles to AQB.)

O is the centre of curvature of the rolling curve CQD at Q. (If CQD is an arc of a circle, then O, is the centre of the circle.)

P is the position of the tracing point when the rolling curve is in the position shown.

Join PQ. Draw QR at right angles to PQ to meet PO, or PO2 produced at R. Join RO, and produce it if necessary to meet PQ or PQ produced at S. S is the centre of curvature of the roulette

at P.

When P, O2, and Q are in a straight line the above construction

If AB is a straight line then QO, is infinite and

73. The Cycloid. The cycloid is the curve described by a point on the circumference of a circle which rolls on a straight line, the circle and straight line remaining in the same plane. The ordinary geometrical construction for drawing the cycloid is shown in Fig. 141. mo'e' is the straight line upon which the circle Pbo'q rolls and P is the point which describes the cycloid. Pbo'q is the rolling circle in its middle position, the diameter POo' being at right angles to mo'e'. Make o'e' equal to half the circumference of the rolling circle. Divide o'e' into a number of equal parts at a', b', c', etc. (preferably six, but for

the sake of a clearer figure o'e' in Fig. 141 has been divided into five equal parts). Divide the semicircle Pbo' into the same number of equal parts at a, b, c, etc. Through O draw OE parallel to o'e'. From a', b', c', etc. draw perpendiculars to o'e' to meet OE at A', B', C', etc. With centres A', B', C', etc. describe arcs of circles touching o'e', and through a, b, c, etc. draw parallels to o'e' to cut these arcs at A, B, C, etc. as shown. The points A, B, C, etc. are points on the half of the cycloid traced by the point P as the circle Pbo' makes half a revolution to the right. Points on the other half of the cycloid may be obtained in a similar manner, or, since the curve is symmetrical about Po' the part to the left of Po' may be copied from the part to the right.

If from a point q on the circle Pbo'q a line be drawn parallel to the base line mo' to meet the cycloid at Q then QR parallel to qo' is the normal and QT parallel to qP is the tangent to the cycloid at Q.

Again, if PT be drawn parallel to mo' to meet at T the tangent to the cycloid at Q, the length of the arc PQ of the cycloid will be twice the length of the tangent QT or twice the length of the chord qP. Hence the total length of the cycloid is four times the diameter of the rolling circle.

If QR be produced to S and RS is made equal to QR then S is the centre of curvature of the cycloid at Q. The locus of S is the evolute of the cycloid. If the rolling circle be drawn below the base line mo' and touching that line at R it is obvious that this circle will pass through S. Also if the rolling circle be drawn above the base line and touching it at R this circle will pass through Q and since the chord RS is equal to the chord QR the arc RS must be equal to the arc QR. But the arc QR is equal to mR, and mo' is equal to half the circumference of the rolling circle, therefore if RU is a diameter of the circle RSU, the arc SU is equal to Ro'. Hence if a line UL be drawn parallel to mo' and the circle RSU be made to roll on this line the

point S will describe a cycloid equal to the original cycloid. The evolute of a cycloid is therefore an equal cycloid.

74. The Trochoid. When a circle rolls on a straight line and remains in the same plane, a point in the plane of the circle, connected to the circle but not on its circumference, describes a trochoid. If the describing point is outside the rolling circle the trochoid is called a superior trochoid. If the describing point is inside the rolling circle the trochoid is called an inferior trochoid. A superior trochoid is also called a curtate cycloid and an inferior trochoid is also called a prolate cycloid.

The geometrical construction for finding points on a trochoid is similar to that already given for the cycloid and is shown in Fig. 142. mo'm' is the base line. P is the position of the describing point when it is furthest from the base line and O is the corresponding position of

the centre of the rolling circle. o'm' is made equal to half the circumference of the rolling circle. The points a', b', c', etc. and A', B', C', etc. are determined as for the cycloid.

With centre O and radius OP describe a circle, and divide the half of this circle which is to the right of OP into as many equal parts as o'm' is divided into. This determines the points a, b, c, etc. Through a, b, c, etc. draw parallels to the base line and with centres A', B', C', etc. and radius equal to OP describe arcs of circles to cut these parallels at A, B, C, etc. as shown. A, B, C, etc. are points on the half of the trochoid described by the point P as the rolling circle makes half a revolution to the right. Points on the other half of the trochoid may be obtained in a similar manner, or, since the curve is symmetrical about OP, the part to the left of OP may be copied from the part to the right.

If from a point q on the circle qPb a line be drawn parallel to the base line to meet the trochoid at Q, then QR parallel to qo' is the

normal and QT perpendicular to QR is the tangent to the trochoid at Q.

Draw RV perpendicular to the base line to meet the parallel through O to the base line at V. V is the position of the centre of the rolling circle when the describing point is at Q.

To find S the centre of curvature of the trochoid at Q, draw RN perpendicular to QR to meet QV produced at N and draw NS perpendicular to the base line to meet QR produced at S.

The evolute of the trochoid is the locus of the centre of curvature S. The evolute of the superior trochoid is shown in Fig. 142. The evolute of the inferior trochoid is partly on one side and partly on the other side of the curve, and the normals which have the least inclination to the base line are asymptotes of the evolute.

75. The Epicycloid. When a circle rolls on the outside of a fixed circle, the two circles being in the same plane, a point on the circumference of the rolling circle describes an epicycloid.

The ordinary geometrical construction for finding points on an epicycloid is shown in Fig. 143. P is the position of the tracing point

when it is furthest from the fixed or base circle, O is the corresponding position of the centre of the rolling circle and o' is the corresponding point of contact of the rolling and base circles. O, is the centre of the base circle. Make the arc o'm equal to half the circumference of the rolling circle. Join O,m and produce it to meet at M the circle through O concentric with the base circle. Divide the arc OM into a number of equal parts (preferably six or eight) at A', B', C', etc. and divide the semicircle Pco' into the same number of equal parts at a, b, c, etc.