Figure 6.1 shows three views of a cube in orthographic projection; the phantom line indicates the original position of the cube, and the full line indicates the position after rotation about the diagonal AB. The cube has been rotated so that the angle of 45° between side AC1 and diagonal AB now appears to be 30° in the front elevation, C1 having been rotated to position C. It can clearly be seen in the end view that to obtain this result the angle of rotation is greater than 30°. Also, note that, although DF in the front elevation appears to be vertical, a cross check with the end elevation will confirm that the line slopes, and that
El D
point F lies to the rear of point D. However, the front elevation now shows a three dimensional view, and when taken in isolation it is known as an isometric projection.
This type of view is commonly used in pictorial presentations, for example in car and motor-cycle service manuals and model kits, where an assembly has been 'exploded' to indicate the correct order and position of the component parts.
It will be noted that, in the isometric cube, line AC1 is drawn as line AC, and the length of the line is reduced. Figure 6.2 shows an isometric scale which in principle is obtained from lines at 45° and 30° to a horizontal axis. The 45° line XY is calibrated in millimetres commencing from point X, and the dimensions are projected vertically on to the line XZ. By similar c, c,
Fig. 6.2 Isometric scale triangles, all dimensions are reduced by the same amount, and isometric lengths can be measured from point X when required. The reduction in length is in the ratio isometric length = cos 45° = 0.7071 = 0 8165
Now, to reduce the length of each line by the use of an isometric scale is an interesting academic exercise, but commercially an isometric projection would be drawn using the true dimensions and would then be enlarged or reduced to the size required.
Note that, in the isometric projection, lines AE and DB are equal in length to line AD; hence an equal reduction in length takes place along the apparent vertical and the two axes at 30° to the horizontal. Note also that the length of the diagonal AB does not change from orthographic to isometric, but that of diagonal C1D1 clearly does. When setting out an isometric projection, therefore, measurements must be made only along the isometric axes EF, DF, and GF.
Figure 6.3 shows a wedge which has been produced from a solid cylinder, and dimensions A, B, and C indicate typical measurements to be taken along the principal axes when setting out the isometric projection. Any curve can be produced by plotting a succession of points in space after taking ordinates from the X, Y, and Z axes.
Figure 6.4(a) shows a cross-section through an extruded alloy bar: the views (b), (c), and (d) give alternative isometric presentations drawn in the three principal planes of projection. In every case, the lengths of ordinates Op, OQ, P1, and Q2, etc. are the same, but are positioned either vertically or inclined at 30° to the horizontal.
Figure 6.5 shows an approximate method for the construction of isometric circles in each of the three major planes. Note the position of the points of intersection of radii RA and RB.
The construction shown in Fig. 6.5 can be used partly for producing corner radii. Fig. 6.6 shows a small block with radiused corners together with isometric projection which emphasises the construction to find the centres for the corner radii; this should be the first part of the drawing to be attempted. The thickness of the block is obtained from projecting back these radii a distance equal to the block thickness and at 30°. Line in those parts of the corners visible behind the front face, and complete the pictorial view by adding the connecting straight lines for the outside of the profile.
In the approximate construction shown, a small inaccuracy occurs along the major axis of the ellipse, and Fig. 6.7 shows the extent of the error in conjunction with a plotted circle. In the vast majority of applications where complete but small circles are used, for example spindles, pins, parts of nuts, bolts, and fixing holes, this error is of little importance and can be neglected.
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