K may be marked on the straight edge of a strip of paper. This strip of paper may then be moved on the drawing paper until a position is found where H and K lie, one on AB and the other on AC. An ordinary drawing scale may be used in the same way.

T

B

E

/H

K

P

D

FIG. 123.

58. To draw the Path traced by one Angular Point of a Given Triangle while the other Angular Points move, one on each of two Given Lines. Let ABC (Fig. 123) be the given triangle. Let A be the tracing point, and let B move on the given line DE while C moves on the given line FH. Draw the triangle ABC on a piece of tracing paper TP. Make a small hole in the tracing paper at A with a needle. Place the tracing paper on the drawing paper so that B is on DE and C on FH. With a sharp round-pointed pencil make a mark on the drawing paper through the needle hole in the tracing paper at A. This will be one point in the path required. By moving the tracing paper into other positions other points may be obtained, and a fair curve KAL drawn through them will be the required path.

T

59. To draw an Arc of a Circle through Three Given Points without using the Centre of the Circle.-Let A, B, and C (Fig. 124) be the given points. Place a piece of tracing paper TP on the drawing paper, and draw on the former two straight lines AD and AE passing through B and C respectively. Make a small hole in the tracing paper

A

B

E

FIG. 124.

at A with a needle. Move the tracing paper round into different positions so that the lines AD and AE always pass through B and C respectively. For each position of the tracing paper make a mark on the drawing paper with a sharp round-pointed pencil through the hole at A. A fair curve drawn through the points obtained in this way will be the arc required. This construction is based on the fact that all angles in the same segment of a circle are of the same magnitude. Another form of this problem is-To describe on a given line BC a segment of a circle which shall contain an angle equal to a given angle BAC.

60. To draw any Roulette.-Let AHB (Fig. 125) be the directing line or base, and let CDE be the rolling curve. (In Fig. 125 the base is a straight line, but it may be any curved line; and the rolling curve is a circle, but it may also be any curved line.) The base is drawn on the drawing paper and the rolling curve is drawn on a piece of tracing paper. The tracing point P is also marked on the

tracing paper by two lines at right angles, and by a small needle hole. Place the tracing paper on the drawing paper so that the rolling curve CDE touches

FIG. 125.

the base, say at C. Make a mark on the drawing paper through the needle hole in the tracing paper at P. Place a needle through the tracing paper and into the drawing paper at C. Turn the tracing paper round about the needle at C until the rolling curve cuts the base at a near point F. Transfer the needle from C to F and turn the tracing paper until the rolling curve touches the base at F. The tracing point will now have moved from P to P1. Mark the drawing paper at P. Again turn the tracing paper until the rolling curve cuts the base at another near point H. Transfer the needle from F to H and turn the tracing paper until the rolling curve touches the base at H. The tracing point will now have moved to P2. Mark the drawing paper at P2, continue the process, obtaining the points Ps, P., etc. A fair curve drawn through the points P, P1, P2, P3, etc., will be the roulette required. In Fig. 125 the roulette is a trochoid.

and

61. To inscribe in a Given Figure a Figure similar to another Given Figure.-Let ABCD (Fig. 126) be the given figure in which it is required to inscribe a figure similar to a second given figure EFHK. Draw the figure EFHK on a piece of tracing paper. Take a point on the tracing paper as a pole (prefer

ably one angular point of the figure EFHK, say E) and join this point to each of the angular points of the figure on the tracing paper. Graduate these lines, the divisions being proportional to the distances of the pole from the angular points of the figure. In the example illustrated EF,

FIG. 126.

EH, and EK are each divided into two equal parts, and the graduations are extended on these lines produced. Place the tracing paper on the drawing paper and move the former about until the sides of the figure ABCD on the drawing paper cut the lines through the pole on the tracing paper at corresponding points L, M, and N; then prick the drawing paper through the tracing paper at these corresponding points and the positions of the angular points of the figure required are obtained.

It may happen that the corresponding points mentioned are not at points of graduation on the lines through the pole. In that case the graduations must be made finer in the neighbourhoods where they appear to be required. If the corresponding points required are still not at points of graduation, it may be possible, without further subdivision, to judge by the eye whether the points where the polar lines cut the sides of the first given figure are corresponding points.

62. Drawing Symmetrical Curves.-When a curve is symmetrical about an axis only the part on one side of that axis need be constructed, the part on the other side may be quickly and accurately drawn by means of a piece of tracing paper as follows.

Referring to Fig. 127, at (a) is shown one half of a curve which is symmetrical about the axis YY. A piece of tracing paper TP is

placed over this figure and the curve and the axis YY are traced in pencil. The tracing paper is then turned over and placed as shown at (b) and the curve on the tracing paper is traced with the pencil. It will then be found that the second half of the curve has appeared on the drawing paper in pencil, the lead for this curve having come from the lead which was put on the tracing paper when it was in the position (a). After removing the tracing paper the curve traced should be lined in to make it more distinct.

When a curve is symmetrical about two axes at right angles to one another only one quarter of the curve need be constructed, the other three quarters may be drawn, one quarter at a time, by means of a piece of tracing paper as just described.

Fig. 128 shows the method applied to an ellipse. At (1) is shown one quarter of the ellipse constructed, say, by the trammel method. A piece of tracing paper TP is placed over this and on it are traced the semi-axes and the curve lying between them. Turning the tracing paper over and placing it as shown at (2) the second quarter of the curve is transferred to the drawing paper, the lead coming from the first tracing. Turning the tracing paper over again and placing it as shown at (3) the third quarter of the curve is transferred to the drawing paper, the lead coming from the second tracing. Turning the

tracing paper over once more and placing it as shown at (4) the last quarter of the curve is obtained.

Exercises IV

The following exercises are intended to be worked out by the tracing
paper method

1. Find the circumference of a circle 3 inches in diameter, and compare the result with that got by calculation.

2. The sides of a figure ABC are arcs of circles. Radius of AB = 1 inches, radius of BC = 2 inches, radius of CA 2 inches. The arcs touch one another

in pairs at A, B, and C. Find the perimeter of the figure ABC.

3. Draw the involute of a circle 2 inches in diameter. Also draw a tangent to the circle to meet the involute, the length of this tangent between the involute and the point of contact with the circle to be 5 inches. Find also the length of the involute between the starting point and the point where the forementioned tangent meets it.

4. Draw an ellipse, major axis 2 inches, minor axis 1 inches, and draw that involute of the ellipse which starts from one extremity of the major axis. Find the circumference of the ellipse.

5. ABC is an equilateral triangle of 3 inches side. D is a point in AB 1 inch from A. Draw through D a straight line to cut the side AC at E, and the side CB produced at F, such that EF = 4 inches.

6. Draw an ellipse, major axis 3 inches, minor axis 2 inches. Take a point P, 2 inches from one extremity of the minor axis, and 14 inches from one extremity of the major axis. Through P draw a straight line to cut the ellipse in two points which shall be 2 inches apart.

7. X'OX and Y'OY are two straight lines intersecting at O. Angle XOY = 60°. A straight line AB, 3 inches long moves with the end A on X'OX, and the end B on Y'OY. Draw the complete curve traced by a point P in AB which is 1 inches from A. Draw also the curve traced by the middle point of AB.

8. ABC is an equilateral triangle of 3 inches side. The triangle moves with the point A on the circumference of a circle 24 inches in diameter, and the point B moves on a diameter of that circle produced. Draw the path traced by the point C. Draw also the path traced by the middle point of AB.

9. On a straight line 3 inches long describe a segment of a circle containing an angle of 120°, without using the centre of the circle.

10. O is the centre and OP a radius of a circle 2 inches in diameter. Q is a point in OP inch from O, and R is a point in OP produced, and 14 inches from O. Draw the cycloid described by P, also the trochoids described by Q and R, as the circle rolls on a straight line. Draw as much of each curve as is obtained by a little more than one revolution of the circle.

11. Taking the same rolling circle and the same points Q and R as in the preceding exercise, draw the epicycloid described by P, and the epitrochoids described by Q and R as the circle rolls on the outside of a base circle 3 inches in diameter; draw also the hypocycloid described by P, and the hypotrochoids described by Q and R as the circle rolls on the inside of the same base circle. Draw as much of each curve as is obtained by a little more than one revolution of the rolling circle.

12. Draw the roulette described by one extremity of the major axis of an ellipse (major axis, 24 inches, minor axis, 14 inches) which rolls on the outside of another ellipse (major axis, 3 inches, minor axis, 2 inches). The roulette to start from one extremity of the minor axis of the fixed ellipse. Draw also, on the same figure, the roulette described by one extremity of the minor axis of the rolling ellipse while the first roulette is being described.

13. AB is a straight line 3 inches long. BC is an arc of a circle whose centre

is in AB and 2 inches from B, and whose chord is 4 inches long.

AC is an arc of a circle of 3 inches radius whose centre is on that side of AC which is remote from B. Draw the largest possible equilateral triangle which has its angular points, one on each of the sides of the figure ABC.

14. ABCD is a quadrilateral. AB 24 inches, BC 1 inches, CD = 1} 2 inches, and AC 2 inches. In this quadrilateral inscribe a

inches, DA square.

=