UNIT 5 | unit 5 sectioning of solids isometric projections

UNIT 5

Sectioning Of Solids, Isometric Projections

1. What is Isometric scale?

Ans –

As all the edges of the cube are equally foreshortened, the square faces are rhombuses. The rhombus ABCD (fig. 2) shows the isometric projection ofthe top square face of the cube in which BO is the true length of the diagonal.

Construct a square BQDP around BO as a diagonal. Then BP shows the truelength of BA.

In triangle ABO,

In triangle PBO,

The ratio, .

Thus, the isometric projection is reduced in the ratio , i.e. the isometric lengths are 0.815 of the true lengths.

2. Methods to calculate true length.

Ans –

a) Draw a horizontal line BO of any length (fig.). At the end B, draw lines BA and BP, such that L OBA = 30° and L OBP = 45°.Mark divisions of true length on the line BP and from each division-point, draw verticals to BO meeting BA at respective points. The divisions thus obtained on BA give lengths on isometric scale.

Figure

(b) The same scale may also be drawn with divisions of natural scale on a horizontal line AB (fig.). At the ends A and B, draw lines AC and BC making 15° and 45° angles with AB respectively, and intersecting each other at C.

Figure

From division-points of true lengths on AB, draw lines parallel to BC and meeting AC at respective points. The divisions along AC give lengths to isometric scale.

The lines BO and AC (fig.) represent equal diagonals of a square face of the cube, but are not equally shortened in isometric projection. BO retains its true length, while AC is considerably shortened. Thus, it is seen that lines which are not parallel to the isometric axes are not reduced according to any fixed ratio. Such lines are called non-isometric lines. The measurements should, therefore, be made on isometric axes and isometric lines only. The non-isometric lines are drawn by locating positions of their ends on isometric planes and then joining them.

3. How to draw isometric projections from given orthographic views.

Ans –

The orthographic view of a sphere seen from any direction is a circle of diameter equal to the diameter of the sphere. Hence, the isometric projection of a sphere is also a circle of the same diameter as explained below.

The front view and the top view of a sphere resting on the ground are shown in fig. C is its centre, D is the diameter and P is the point of its contact with the ground.

Figure

Assume a vertical section through the centre of the sphere. Its shape will bea circle of diameter D. The isometric projection of this circle is shown in fig. 25 (ii)by ellipses 1 and 2, drawn in two different vertical positions around the same centre C. The length of the major axis in each case is equal to D. The distance of the point P from the centre C is equal to the isometric radius of the sphere.

Again, assume a horizontal section through the centre of the sphere. The isometric projection of this circle is shown by the ellipse 3, drawn in a horizontal position around the same centre C. In this case also, the distance of the outermost points on the ellipse from the centre C is equal to O.5D.

Thus, in an isometric projection, the distances of all the points on the surface of a sphere from its centre, are equal to the radius of the sphere.

Hence, the isometric projection of a sphere is a circle whose diameter is equal to the true diameter of the sphere.

Also, the distance of the centre of the sphere from its point of contact with the ground is equal to the isometric radius of the sphere, viz. CP.

It is, therefore, of the utmost importance to note that, isometric scale must invariably be used, while drawing isometric projections of solids in conjunction with spheres or having spherical parts.

4. Draw the isometric projection of a sphere resting centrally on the top of a square prism, the front view of which is shown in fig.