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193. To describe a cycloid by continuous motion.—Let A B (fig. 162) be a straight line, as the edge of a ruler, and let a circle of given diameter C D roll along the line. A B; then a pencil fixed at any point on the rolling circle, start

Fig. 162.

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ing from A and returning to the straight line at B after having made one whole revolution, will mark out the curve AD B, called the cycloid.

194. To find points upon the perimeter of the cycloid.Divide the line A C (fig. 162) into eleven equal parts, and also the circumference of the half-circle CD into eleven equal parts. Let Cg be any number, as four, of those parts; draw g P parallel to A B, and take g P equal to Ag, then P is a point on the cycloid. In the same way as

many points can be found on the curve as there are divisions on the circle.

The part of the curve from D to B is exactly similar to the part from D to A.

195. To draw the tangent and normal at any given point on the cycloid.—Let P (fig. 162) be the given point on the curve. Draw Pg parallel to A B cutting the generating circle in g; draw Cg the chord of the circle, and at P draw PT parallel to Dg, PT will be the required tangent. In the same way the tangent at any other point can be found by drawing a line parallel to the corresponding chord of the circle. The tangent at D is parallel to A B, and the tangents at A and B are parallel to the diameter CD, or at right-angles to A B.

Since the chord Cg makes a right-angle with g D, the normal PN at the point P must be parallel to Cg. In this way the normal can be drawn at any point without drawing the tangent. The diameter C D is the normal at the vertex D, and the line A B is the normal to the curve at A and B.

The normal PN can also be drawn by taking Ad on AC equal to the arc Cy on the generating circle, and drawing PN through the point d on A C.

To approximate to the cycloid by means of arcs of circles, draw the normals at P and Q, meeting in N, and from N as a centre draw the arc P Q. Proceed in the same manner at the other points found on the curve, and the arcs of circles drawn from the intersections of consecutive normals as centres will give a curve nearly resembling the cycloid. The radius of curvature has its greatest value at the vertex

where it is equal to twice the diameter CD of the generating circle; the curvature being least at the rertex


D and increasing towards A and B, at which points the - radius of curvature is nothing. The length of the radius of curvature, PN, at any point P is equal to twice the chord Cg of the circle.

196. Properties and applications of the cycloid.The length of any arc as DP measured from the vertex D is equal to twice the length of the chord Dg of the generating circle, and the whole length of the arc AD or B D is equal to twice the diameter C D. The area contained between the curve and its base line A B is equal to three times the area of the generating circle.

If we invert the cycloid so as to make its vertex D the lowest point, 'A B being kept horizontal, and a ball is allowed to fall from any point on the curve and constrained to descend in a groove having the form of the cycloid, the time of its descent to the lowest point will be the same froin whatever point on the curve the descent commences; and if a pendulum is made to oscillate in a cycloid, its times of oscillation will be the same whatever the length of the arc it describes. The cycloid is also the curve of quickest descent of a body from one point to another not in the same vertical line.

The cycloid may be used as the contour of an arch, the height CD of which is to the span A B in the ratio of 7 to 22, or of 100 to 314. In this case the normals will represent the directions of the joints of the voussoirs of the arch.

The cycloid is the proper form for the teeth of a rack which drives a cogged or pinion wheel, the part from A towards P being used for that purpose.

197. To describe the curve called the companion to the cycloid.Let A B (fig. 163) represent the circumference of the circle whose diameter is CD, and its centre at 0. Let

C be the middle point of A B, and divide AC into eleven equal parts, and also the circumference of the semicircle into the same number of equal parts. Draw any ordinate as Ng P parallel to A C, and make N P equal to Dg or Ag, then P is a point on the required curve.

This curve has a point of contrary-flexure (155) at the ordinate OE equal to the quadrant of the circle. The tangent at E makes half a right-angle, or 45°, with the line

Fig. 163.

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EO. The tangent at D is parallel to A B, and at A and B the line A B is the tangent.

The tangent at any other point P can be drawn by taking OF equal to Ng, and the line FI is parallel to the tangent at P.

This curve can be applied to form the contour of architectural mouldings, such as the ogee or sima, or for an ogival arch, of which CD is the half-span, AC the height.

198. DEFINITION.—When the pencil is placed at a point within the circumference of the generating circle the curve marked out is called a trochoid.

199. To find points upon a trochoid.—Let A B be the line on which rolls the generating circle whose diameter is CD (fig. 164), Oq the distance from the centre of the

pencil which describes the curve. Divide, as before, the line AC into eleven equal parts, and also the semicircle CKD. Draw the radii O a, Ob, &c. Let the radius Od cut the inner circle in the point p, draw Np P parallel to AB, and take p P equal to the arc D d, or to the line Ad; then P is a point on the trochoid. Let mi represent the

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ratio of 0 E to 0 C, and make CM equal to the product of CO into the fraction (1 – m2); draw M9Q parallel to AC cutting the inner circle at q. Join Oq and produce it to meet the generating circle at l; take qQ equal to the are Dl, and the point Q will be one of contrary-flexure (155), and the tangent at Q is parallel to 09.

The curve has the line FG, parallel to AB, for its tangent at F and G; and the tangent at H is parallel to A B.

Draw O KL parallel to A B, meeting the curve at L, join H K, and the tangent at L will be parallel to H K.

200. To describe an epicycloid and hypocycloid.Let the generating circle move outside the circumference of another circle, instead of on a straight-line as in the cycloid ; then the curve which a pencil on the circum

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