3. By finding the centre and axes of the ellipse. The centre of the ellipse is the projection of the centre of the circle, and is found by making ob= 23 inches, measured from scale (2), and be, parallel to om, = 23 inches, from scale (3). The major axis is the projection of that diameter of the circle which is parallel to the plane of projection, that is a line through e parallel to the horizontal trace of the plane of the axes ol and om. But the trace of that plane is at right angles to on. Therefore drawing through e the line fg at right angles to on, and making ef, eg each equal to 23 inches, on scale (1), fg is the major axis of the ellipse. The minor axis is found from fig. 108. It is the projection of a line 42 inches long in the plane lom, at right angles to the trace of that plane. Hence finding the inclination of AF, fig. 108, and the projection of a line of the required length, measured on it from scale (1), gives the minor axis, hk. The ellipse can now be drawn from the axes by any of the well-known methods.

The ellipse sqt has the same centre e and its axes are in the same ratio to one another; so that making eq and er each 14 inches, measured from scale (1), and drawing through q a line qt parallel to ƒk, the axes of the ellipse are found.

All the other ellipses of the Axometric Projection may be drawn in a similar manner. Their planes are all parallel to the plane of A,B,C,, and their axes are consequently parallel to fg and hk respectively, and are in the same ratio to each

other.

1

ISOMETRIC PROJECTION.

If two of the angles in fig. 108 were made equal to each other the two lines which contain the third angle would have the same inclination, and the same scale would in consequence serve for both. Thus if BaC and BaD were equal, aC and aD would be equal. For in the two right-angled triangles BaF, Ba G, the angles BaF and BaG being equal, and Ba common to the two triangles, aF would be equal to a G; and in the two right-angled triangles CaF, DaG, the sides aF, aG being equal, and also the angles CaF and DaG, the side a C would be equal to aD. As can be readily seen from the figure, aC and aD being equal must be equally inclined to the horizontal plane.

Fig.14.

m

Again, if the three angles were equal to one another the three axes would have the same inclination. When the same scale serves for the three axes, the projection is said to be Isometric.

Fig. 114 is the isometric projection of a cube of 1 inch side. The three edges of the cube are taken for axes. These lines being drawn making angles of 120° with one another,

the projection of 1 inch measured on each may be found either as in fig. 108 or as follows:

The line mn being at right angles to ol is parallel to the plane of projection, and is therefore equal to the real length of the diagonal of the square face of the cube. If then eA, be drawn perpendicular to en and made equal to it, An is the side of the square. But it is evident the angle 41an =120o and the angle eAn=45°; so that if a triangle be constructed, in any convenient position, having an angle of 120° and an angle of 45°, the ratio between the true length of any line parallel to one of the axes and its isometric projection is as the side opposite the angle of 120° to the side opposite the angle of 45°. In this way an isometric scale can at once be constructed from the natural scale. The method of drawing an isometric projection is so similar to that already described for any three axes that it is unnecessary to enter into further details. It is in fact a simple case of Axometric Projection.

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