4. Construct Figs. 76, 77, and 78, p. 33, as explained in Art. 35, to the following dimensions:

For Fig. 76, vertical angle of cone tvu inches.

=

=

50°, va = 2.5 inches, va, = 1.25 For Fig. 77, vertical angle of cone tva = 60°, va = 1 inch, xx, parallel to vt. For Fig. 78, vertical angle of cone tvu = 80°, va 1 inch, va 0.75 inch. The eccentricity of the conic being the ratio of AF to AX, construct each conic separately on another part of the paper by the method of Art. 34 and Fig. 75. Make a tracing of each conic obtained in this way and test whether it agrees with the conic determined as a section of the cone.

-

5. PTQ is a triangle, PT = 2.75 inches, TQ = 1.75 inches, and PQ 2 inches. R is a point in PT 1 inch from P. RF is perpendicular to PT, and RF = 1.25 inches. F and Q are on the same side of PT.

PT is a tangent to a conic, P being the point of contact. F is the focus, and Q is another point on the curve. Find the directrix and draw the conic. [Art. 37, (1) and (2).]

6. PFNQ is a quadrilateral. The angles at F and N are right angles. PF = 1.3 inches, FN = NQ = 1.6 inches. P and Q are points on a conic of which F is the focus and FN the direction of the axis. Construct the conic.

7. F is the focus and S is any fixed point on the axis of a conic. From S a perpendicular is drawn to the tangent at a point P on the curve meeting FP at Q. Show by actual drawing that the locus of Q is a circle.

8. A focal chord of a parabola is 2.3 inches long and it is inclined at 30° to the directrix. The middle point of the chord is at a perpendicular distance of 1-2 inches from the directrix. Draw the parabola.

9. AP, a chord of a parabola, is 1.8 inches long and is inclined at 50° to the axis. The point A being the vertex of the parabola, draw the curve.

10. TP, a tangent to a parabola from a point T on the axis, is inclined at 30° to the axis. P is the point of contact, and TP is 3 inches long. Draw as much of the parabola as lies between the vertex and a double ordinate whose distance from the vertex is 2.2 inches.

11. Construct a triangle FPQ. FP = 1.5 inches, PQ 3.4 inches, and QF 2-6 inches. Draw a parabola whose focus is F and which passes through P and Q.

12. Make a careful tracing of the parabola of the preceding exercise without any lines other than the curve; then determine the axis, focus, and directrix of the curve by constructions on the tracing.

13. Draw a line FS 1.3 inches long and a line SP making the angle FSP 45°. SP is a tangent to a parabola of which F is the focus and S a point on the directrix. Find the point of contact of the tangent and draw the parabola.

14. PNP,, a double ordinate of a parabola, is 3-6 inches long. A being the vertex of the parabola, the area bounded by the curve PAP, and the double ordinate PNP, is 4-8 square inches. Draw the curve PAP1.

15. Construct a triangle RST; RS

1.7 inches, ST = TR = 2.6 inches. ST and SR, both produced, are tangents to a parabola and TR is parallel to their chord of contact. TR contains the focus. Draw the parabola.

=

1.1 inches.

16. ABC is a triangle. AB = 2.2 inches, BC = 2·3 inches, and CA D is a point in BC 0-9 inch from B. Draw the parabola which touches BC at D and AB and AC produced.

17. The normal PG to a parabola at a point P on the curve is 1.9 inches long, the point G being on the axis. The parameter of the diameter through P is 4.8 inches long. Construct the parabola.

18. The major and minor axes of an ellipse are 4 inches and 3 inches long respectively. Construct the curve by the trammel method and find the foci. 19. The major axis of an ellipse is 3-5 inches long and the distance between the foci is 2.5 inches. Draw the ellipse.

2.2

20. ABC is a triangle. AB 2.3 inches, BC 11 inches, and CA = inches. Draw an ellipse whose foci are A and B and which passes through C. 21. The minor axis of an ellipse is 2·2 inches long and the distance between

the foci is 2 inches. Draw the ellipse, and construct the locus of the middle points of all the chords through one focus.

22. The major and minor axes of an ellipse are 3 inches and 2 inches long respectively. Draw a half of the ellipse which lies on one side of the major axis. Divide the curve into twelve parts whose chords are equal, and from the points of division draw normals to the ellipse, each normal to project 0.5 inch outside the ellipse. Lastly, draw a fair curve through the outer extremities of the normals. 23. The distance between the foci of an ellipse is 2 inches. A tangent to the ellipse is inclined to the major axis at an angle of 30° and cuts that axis produced at a point 2.5 inches from the centre of the ellipse. Draw the ellipse.

24. SABT is a quadrilateral. The angles at A and B are right angles. SA = 0.75 inch, AB = 3 inches, and BT = 2.25 inches. The points S and T are on the same side of AB. ST is a tangent to an ellipse of which AB is the major axis. Construct the ellipse.

25. Two conjugate diameters of an and the angle between them is 60°. methods described in Art. 47, p. 42.

ellipse are 3 inches and 3.5 inches long, Draw the ellipse by each of the three

26. CO is a straight link, 2 inches long, which revolves about a fixed axis at C. PON is another straight link, 4 inches long, jointed at its middle point O to the outer end of CO. N is constrained to move in a straight line which passes through C. Draw the loci of the middle points of OP and ON and also the locus of P.

27. Draw a quadrilateral FPOQ. FP

3.2 inches, angle PFQ = 70°, angle FPO 56, FQ = 1.2 inches, and PO 16 inches. Draw an ellipse touching OP at P and OQ at Q, and having F for one focus.

=

28. The transverse axis of an hyperbola is 2 inches long and the distance between the foci is 2.7 inches. Draw the hyperbola.

29. The transverse and conjugate axes of an hyperbola are 2.2 inches and 1·7 inches long respectively. Draw the hyperbola and the conjugate hyperbola.

30. The foci of an hyperbola are 2.3 inches apart. One point on the curve is 2.9 inches from one focus and 1.1 inches from the other. Draw the hyperbola.

31. The distance between the foci of an hyperbola is 2.9 inches. A tangent to the hyperbola is inclined to the transverse axis at 48° and cuts that axis at a point 06 inch from its centre. Draw the hyperbola.

32. The transverse axis of an hyperbola is 2 inches long. A tangent to the hyperbola is inclined at 58° to the transverse axis and cuts that axis at a point 0.8 inch from its centre. Draw the hyperbola.

=

33. PMM, P, is a quadrilateral. MM, 2.5 inches, the angles at M and M, are right angles, PM 0.9 inch, P,M, 1-2 inches. P and P, are points on an hyperbola of which MM, (produced both ways) is a directrix, and whose eccentricity is. Find the foci and draw the hyperbola.

34. The asymptotes of an hyperbola are at right angles to one another and one point on the curve is 1 inch from each asymptote. Construct the two branches of the curve by the method illustrated by Fig. 108, p. 47.

35. One cubic foot of air at a pressure of 100 lbs. per square inch expands until its volume is 10 cubic feet. The relation between the pressure p and volume v is given by the formula pv = 100. Construct the expansion curve. Pressure scale, 1 inch to 20 lbs. per square inch; volume scale, 1 inch to 2 cubic feet. Draw the tangent and normal to the curve at the point where the volume is 3 cubic feet.

A

XI

C

PB

30°

D

FIG. 119.

36. AB and CD (Fig. 119) are two straight lines of unlimited length. AB revolves with uniform angular velocity about the centre P, and CD revolves with the same angular velocity, but in the opposite direction, about the centre Q. PQ = 2.5 inches. O is the middle point of PQ. X,OX and YOY, are two lines at right angles to one another, the angle POX being 30. The initial positions

of AB and CD are parallel to X,OX. Show by actual drawing that the locus of the point of intersection of AB and CD is a rectangular hyperbola of which X,OX and YOY, are the asymptotes and P and Q points on the curve.

37. Draw the complete evolute of an ellipse whose major and minor axes are 4 inches and 2.75 inches long respectively.

38. The focal distance of the vertex of a parabola is 0.3 inch. Draw that part of the evolute of the parabola which lies between the vertex and a double ordinate whose distance from the vertex is 5 inches.

39. The angle between the asymptotes of an hyperbola is 60° and the vertex is at a distance of 1 inch from their intersection. Draw that part of the evolute of one branch of the hyperbola which lies between the vertex and a double ordinate whose distance from the vertex is 5 inches.

40. Draw an ellipse, major axis 3 inches, minor axis 2 inches. Take a point P within the ellipse 0.8 inch from the centre C and lying on a line through C inclined at 30° to the major axis. Through P draw a number of chords of the ellipse and at their extremities draw tangents to the ellipse. Find the point of intersection of each pair of tangents and see whether the points thus obtained are in one straight line. [By a pair of tangents is meant the tangents at the ends of a chord.] Repeat the construction for a point Q lying on CP produced. CQ 2 inches.

CHAPTER IV

TRACING PAPER PROBLEMS

54. Use of Tracing Paper in Practical Geometry.-Frequently draughtsmen have to make geometrical constructions on complicated drawings in order to determine some point, line, or figure, and in such cases the fewer the construction lines the better. By using a piece of tracing paper in the manner explained in his chapter the desired result may be obtained very accurately in many cases without making any construction lines whatever on the drawing paper, and some problems can be easily solved by this method which would be impossible by ordinary geometrical methods, or which would otherwise involve very complicated constructions.

55. To find the Length of a Given Curved Line. -Let ABCD (Fig. 120) be the given curved line. On a piece of tracing paper TP draw a straight line 1 1. Mark a point A on this line and place the tracing paper on the drawing paper so that this point coincides with one end A of the curved line to be measured. Put a needle point through the tracing paper and into the drawing paper at A. Now turn the tracing paper round until the line 11 cuts the curve at a point B not far from A. Remove the needle point from A

FIG. 120.

to B, taking care that the tracing paper does not change its position during the operation. Next turn the tracing paper round until the line on it takes up the position 2 2, cutting the curve at a point C not far from B. The needle point must then be moved to C and the operations continued until a point on the straight line coincides with the last point on the curve. The last point obtained on the straight line must be marked distinctly. The distance between the first and last points marked on the straight line will be approximately equal to the length of the curved line. The approximation will be closer the shorter the steps AB, BC, etc. When the curve has a larger radius of curvature the steps such as AB and BC may be longer than when the radius of curvature is smaller. In Fig. 120 the steps, for the

sake of clearness, are of greater length than would be adopted in practice.

The foregoing method is equivalent to stepping off the length of the curve with the dividers, but the tracing paper method has the advantage that the lengths of the different steps may be made to suit the variations of curvature when the curve is not an arc of a circle.

It is obvious that this method may also be used to mark off a portion of a given curved line which shall be of a given length.

This

56. To draw an Involute of a Given Curved Line.-Let ABCD (Fig. 121) be the given curved line. On a piece of tracing paper TP draw a straight line 11. Mark a point A on this line. point will be called the tracing point. Place the tracing paper on the drawing paper so that the tracing point coincides with the point A on the curve from which the involute is to start. Put a needle point through the tracing paper and into the drawing paper at A. 4 Now turn the tracing paper round until the straight line 1 1 cuts the curve at a point B not far from A. Remove the needle point from A to B, taking care that the tracing paper does not change its position during the operation. Next turn the tracing paper round until the straight line on it takes up the position 22, touching the curve at B, and with a sharp round-pointed pencil make a mark on the drawing paper through the needle hole at the tracing point. If these operations be continued, a number of points are obtained and a fair curve drawn through them will be an approximation to the involute required. The approximation will be closer the shorter the steps AB, BC, etc.

FIG. 121.

57. To draw a Straight Line to pass through a Given Point and cut two Given Lines so that the Portion intercepted between them shall have a Given Length.-Let AB and AC (Fig. 122) be the given lines and D the given point. On a piece of tracing paper TP draw a straight line EF, and mark two points H and K on this line such that HK is equal to the given length. Move the tracing paper into a number of different positions on the drawing paper, the point K being on the line AC and the A line EF passing through D. A position will quickly be reached in which the point

FIG. 122.

H is also on the line AB. Now make a mark on the drawing paper at F; a line joining this mark with D will be the line required. Instead of using tracing paper for this problem, the points H and