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EX. 27.-Draw a cam to give the following motions. It revolves uniformly at a rate of two revolutions per minute, a tappet is to be raised 4" at a uniform rate in 5 seconds, and allowed to remain in that position for 5 seconds; then allowed to drop 1" and remain there another 5 seconds, again raised to 4" for 10 seconds, and then allowed to drop suddenly to its original position, and remain there until again required to be raised. Diameter of shaft 3", of roller on end of tappet 11", least metal around shaft 2′′. Scale half full size.

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EX. 28.-Draw a line CA 3′′ long, and from end A draw a line AB so that angle CAB 150°, make AB 5" and AD=2". C is the centre of a shaft 1g" diameter, and A the centre of a roller 14" diameter in its lowest position. The roller is moved by a cam on the shaft along the line A B as follows:-One-third of a revolution raised from A to D uniformly, one-sixth of revolution remains at rest, one-fourth of revolution raised uniformly from D to B, one-fourth of revolution falls back from B to A. Scale half size. (S. & A. H., 1887.)

SECTION VII.

CONSTRUCTION OF ELLIPSE, PARABOLA, HYPERBOLA, CYCLOIDAL CURVES AND INVOLUTES.

It is very important for the draughtsman to understand the construction, and some of the more useful properties of certain well-known mathematical curves, such as are frequently made use of in practical work.

These curves include the ellipse, parabola, and hyperbola, known as the conic sections, in consequence of their being derived from three different plane sections of a cone (see Figs. 86a, b) and used for the curves of arches, bridges, and roofs; the cycloidal curves used in constructing the teeth of wheels; and the involute of a circle used for the same purpose, and for the blades and guides of turbines.

These curves, as in the case of spirals and paths of points (Section vi.) can only be geometrically constructed by finding a number of points through which it is known the curve must pass, and then drawing the curve through these points by freehand or with the aid of French curves. Arcs of circles

cannot be employed with any degree of accuracy; and as pointed out in previous examples, the greater the number of points found, the more accurate the curve, although for ordinary purposes it is usually sufficient to find the points not nearer than from" to 1" apart.

Ellipse, Parabola, and Hyperbola.*—Given a fixed straight line, and a fixed point, it is possible for another given point to move in three different ways with regard to its position from the fixed line and point. It can move, firstly, so that its distance from the line is always greater, in a constant ratio, than its distance from the point; secondly, so that its distance from the line shall be always equal to its distance from the point; and thirdly, so that its distance from the line is always less, in a constant ratio, than its distance from the point.

The three curves traced out under these different conditions are respectively, the ellipse, parabola, and hyperbola.

Hence we have for definitions of these curves

Ellipse. An ellipse is a curve traced out by a point moving in such a way that its distance from a fixed straight line is always greater than its distance from a fixed point, in a constant ratio.

Parabola.-A parabola is a curve traced out by a point moving in such a way that its distance from a fixed straight line is always equal to its distance from a fixed point.

Hyperbola.—An hyperbola is a curve traced out by a point moving in such a way that its distance from a fixed straight line is always less than its distance from a fixed point, in a constant ratio.

The fixed straight line is called the directrix, the fixed point the focus, and the line passing through the focus at right angles to the directrix, the axis. Lines at right angles to the axis terminated by the curve, are ordinates. The vertex of the curve is the point where the curve cuts the axis.

The ellipse is a closed curve, and has two directrices and two foci. The parabola is an open curve having one directrix and one focus. The hyperbola is an open double curve, having two directrices and foci.

These three curves can be constructed by an almost identical process, so that one example will suffice.

PROBLEM XXXVII. (Fig. 53a).-To construct an ellipse when a directrix and a focus are given, also the vertex and axis.

Let X Y be the directrix, F the focus, A the vertex, and the line through OAF the axis. What is required is to find a

*For the common geometrical constructions of an ellipse see Section iv., Figs. 19, 20, 21, 22.

PF AF

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where P F is the distance

number of points, P, so that of P from the focus, and PT its distance from the directrix. This is conveniently done by making A O the hypotenuse and A F the base of a right angled triangle, A BO, where O B = A F and angle ABO is a right angle. Then produce the line O B, mark off any points as 1.2.3. from A along the axis, and draw lines through each point parallel to A B meeting OB in the points 1' 2' 3' Draw lines through each of the points in the axis perpendicular to the axis, from F as centre, with distance Ol' cut the line through point 1 in the points 6 and 7, also from F with distance O 2' cut the line through point 2 in the points 8 and 9, and so on for each succeeding line. Then the

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points 6.7.8.9 are in the curve of the required ellipse, for 6 F 8 F во AF

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8T ΑΟ AO

where 6 T and 8T are the distances

of points 6 and 8 from the directrix.

Continue this method until the curve is completed.

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If the ratio of the distances from the focus and directrix is given (say, so that A F 2 and AO 3), it is unnecessary to draw the triangles. Any three equal distances can be marked along the axis from O, as to O 3 and two of these distances taken as radius from F to cut the line drawn through 3 parallel to the directrix, and so on for each point. Exactly the same

method is followed for the construction of the parabola and hyberbola.

The parabola is, however, more easily constructed by the method shown in Problem xxxi. (Fig. 38a) for drawing a curve equidistant from a point and a line, as we now know that curve to be a parabola. The curve which is equidistant from a straight line and the circumference of a circle (Fig. 38a) is also a parabola. The curve equidistant from the circumferences of two unequal circles is a hyperbola.

After completing the ellipse its second focus, F', and its second directrix, X'Y', can be found. In the right hand of Fig. 53a are shown a parabola, the curve P, and a hyperbola, the curve H, which are constructed together with the ellipse about the focus, F', and the directrix, X'Y'.

Tangents and Normals.-The usual methods of constructing tangents to an ellipse from points in the curve or outside it, have already been given in Section iv. (Figs. 19, 20). The rule which applies most conveniently to all three curves when constructed by the method just described, is the following:-"If the tangent to an ellipse, parabola, or hyperbola be produced to meet the directrix, and the meeting point be joined to the focus, the angle made by this line, with the line joining the focus to the point of contact, is a right angle." Thus, in Fig. 53a, to draw a tangent at the point N, join N to the focus F, draw a line from the focus towards the directrix at right angles to the line N F, meeting the directrix in M, then the line N M is a tangent to the curve.

Normals. The normal to a curve at any point is at right angles to the tangent at that point. Thus, in Fig. 53a, N R is a normal at N, being at right angles to the tangent N M.

EXAMPLES.

EX. 1.-A fixed point, F, is 2′′ from a fixed straight line X Y. Find eight points in the path of a point P moving as follows:— (a) distance of P from fixed point to its distance from the fixed line to be as 3 to 4; (b) point P to be equidistant from fixed point and fixed line; (c) distance of P from fixed point to its distance from the fixed line to be as 4 to 3. Draw the curves and name them.

EX. 2.-The focus F of an ellipse is 11" from the directrix X Y, and the vertex of the curve is " from the focus. Draw the ellipse, and draw a tangent and normal at any point in the

curve.

EX. 3.-Construct a parabola (finding, at least, twelve points in the curve) when the distance of the focus from the directrix is 1", and draw a tangent and normal at any point in the curve.

EX. 4.-Construct a hyperbola (finding at least twelve points in the curve), when the focus is 11" from the directrix, and the vertex 1" from the focus. Draw a tangent and normal at any point in the curve.

Construction of a Rectangular Hyperbola.-A special case of a hyperbola is one in which the point moves in such a way that the product of its distance from two fixed lines at right angles is a constant. Such a curve is a rectangular hyperbola and is exceedingly useful, because it represents graphically the relation between the pressure and volume of a gas which expands according to Boyle's law (pressure x volume = a constant), a condition often met with in steam diagrams. A simple construction of such a curve is as follows:

Fig. 536. Let O V and OP be two lines at right angles, and such that distances along O V represent volume, while distances along O P represent pressure, and let A be a point in the curve,

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which for ordinary practical problems will generally be the point from which the curve is required to start. Through A draw lines A B and A C parallel to O V and O P respectively, the line A C being produced as far as necessary since there is no limit to the curve. Mark any distances, equal or unequal, as 1, 2, 3 along B P, and draw ordinates through each point, parallel to A B, to meet the line A C in the points I', 2′, 3′ . Join each of the top points 1', 2', 3' to the point O, and mark the points where these lines cut the line A B, 12, 22, 32. Through each of these points draw lines parallel to OP to meet the ordinate through the corresponding top point, thus 1' meets ordinate 1 1', 32 meets ordinate 3 3', &c., these points are points in the required curve. If this curve satisfies the condition required, then O B × BA 02 × 2 d = 04 x 4 e, &c., and this we see is true, for VO 2 2' is a parallelogram, of which the figures

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