PROBLEM 69.

Given any two conjugate diameters AB and CD, inclined to each other at 60°, to construct the ellipse.

Draw AB, CD, two lines bisecting each other in o, and including an angle of 60°, as conjugate diameters. If the

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3

E

R

D

60°

S

parallelogram AOCE be first drawn,

the method of construction is then similar to that of the preceding 8 problem. The text for the one will suffice for the other, the letters used being identical in both cases. The advantage of this method of construction lies in the fact that it admits of general application,-whether the diameters given are at right angles, or inclined to each other. It is, however, desirable to learn other simple methods as well, as given in the following figures.

Fig. 87.

PROBLEM 70.

Having given the major axis, AB, of an ellipse, and any point, P, on the curve, to construct the ellipse.

Note that the minor axis is not yet known. Bisect AB in o.

Fig. 88.

Having the axes, the ellipse may now be completed by Prob. 68,

or by the method shown in dotted lines in fig. 88, as

follows:-From o draw two or three more radial lines corresponding to oq. These lines intersect the circles (described on the respective axes as diameters) in points corresponding to Q and R. If lines perpendicular to the major and minor axes be now drawn through them, so as to correspond to QP, RP, further points in the ellipse can readily be determined. Remaining points in the curve can then be copied from those already known, in a manner already described in preceding problems. A neat curve drawn freehand through them completes the ellipse.

The student should now construct ellipses mechanically, 1st, By means of a strip of drawing paper having a straight edge, or a piece of tracing paper, provided the major and minor axes be known. This method leaves the diagram clear of lines, but it is not recommended when the difference between the lengths of the axes is inconsiderable. Taking fig. 89, draw the major and minor axes. On the straight edge of a strip of paper, or thin copper-sheeting, mark off three points o, c, and A, making OA = the semi-major axis, and oc = the semi-minor axis. Move the strip about in such a manner that A is always on the minor axis, while c is on the other, and tick off the path of o, which is an ellipse. If tracing paper be preferred, draw a straight line oA on it, and mark off c as before; when the tracingpaper is being placed in the successive positions referred to, the locus of o may be pricked through.

Fig. 89.

AC is the difference between the lengths of the two axes. The principle involved is the basis of the construction of the elliptic trammel. It is interesting to note that any other point in the edge of the paper will trace out an ellipse; if the point o be taken midway between A and C, it will trace a circle-a special case of the ellipse.

2nd, By using a piece of thread and pencil only, in the following manner :- Make knots at the ends of a piece of flexible string, such that its virtual length is, say, 4"; pass pins through the knots, and insert the former through the

paper into the board, about 24" apart, as shown at F, and F2 in fig. 90. If the pencil be now taken, and the point gently pressed against the bight of the string on the paper as indicated in the photograph, an ellipse can be plotted with the point by moving the pencil along, always keeping the string uniformly taut.

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the major

F, and F, are the foci, and the length of the string axis AB. If the minor axis CD be drawn, from a property already

=

=

AB
2

Also, if any point, as P, on the curve, be

=

For

joined to the foci, we have the sum of its focal distances, viz., PF, + PF2 the major axis AB = a constant quantity. Hence, if a series of triangles, such as FFP, be drawn, having a common base FF, and the sum of whose other two sides the major axis, a curve traced through the apices gives an ellipse. Though this method of plotting out points in the curve on paper is not recommended, the principle involved is very useful in practice. example, in the case of pairs of geared elliptical wheels in machines, a focus in each wheel is made the centre of revolution, the object being to convert the uniform angular velocity of one wheel into an angular velocity of another wheel that is not required to be uniform, to give a slow cutting and a quick return stroke. With the same length of string, and the same distance apart of the foci, we get equal ellipses; but alter the length of the string, or the relative position of the foci, or both, the resultant curves would on trial be different, showing that there is an infinite number of possible ellipses.

Exercises.

1. In a rectangle (size 3′′ × 2′′) inscribe the principal ellipse. Hint.-Obtain the major and minor axes by joining the middle points of opposite sides, and proceed as in fig. 86 or 88.

2. Draw a parallelogram (3" x 2", included angle 45°) and circumscribe it with an ellipse.

Hint. The diagonals are conjugate diameters.

Proceed as in fig. 87.

3. Assuming that only the outline of the ellipse of the answer to Ex. 2 is known, draw its major and minor axes, and find the foci. Hint.-Refer to figs. 84-5.

4. A point, P, on an ellipse, is 1" from each of the two axes. The major axis 21". Construct the ellipse.

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Hint.-Refer to fig. SS.

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5. Assuming the ellipse of Ex. 4 is drawn, draw a parallel ellipse," "outside it all the way round.

Hint.-Draw a number of normals, 3" long, outwards from points on the ellipse. Join their extremities with a neat curve. It is not a true ellipse, but is required in construction of arches.

6. Construct an ellipse (axes 3.25", 2.375" respectively) by the method shown in fig. 89. Draw a diameter making 40° with the major axis, and find the corresponding conjugate diameter of the ellipse. 29",

7. Draw the locus of the vertex of a triangle whose base and perimeter = 61".

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Hint.—The locus is an ellipse whose foci are 25′′ apart, and major axis = 61′′ – 25" 35". Refer to fig. 90.

8. Place your 45° and 60° set-squares together one on top of the other on your paper, so that the two right angles coincide. Now move the 45° square in such a manner, that each of the corners where the acute angles are, always lies on one of the sides (not the hypotenuse) of the other square. Draw the squares half size, and plot the locus of the 90° corner of the moving square, and of the centre point of its hypotenuse.

Hint. The loci are a straight line and a quadrant of a circle respectively. See fig. 89.

9. Using fig. 96 (three times the size), plot the locus of any point, C, in the connecting-rod centre line, assuming that the crank revolves once, and the centre of the gudgeon, J, moves to and fro on the centre line of the stroke.

Hint. The path of the crank-pin is a circle. Rule the length of the connectingrod on a piece of tracing paper, and mark c on it. Superpose this tracing on the drawing, so that one end is on the line of stroke, and the other end is on the circle, and prick off the point c. Repeat this operation for different positions, when the locus of c (an ellipse) can be sketched.

10. Construct a rhombus, side 24", diagonal 44". In this rhombus inscribe an ellipse. (1889)

11. Draw an ellipse, the distances between the foci being 21′′, and the major axis 3" long.

(1891)

12. The axes of an ellipse are 4" and 3" long respectively. Draw the curve, and determine the foci.

(1892)

13. The minor axis of an ellipse is 21" long, and the distance between the foci is 2". Find the major axis and draw the curve.

(1894)