UNIT 3
Projection of Solid
- Draw the projections of a triangular prism, base 40 mm side and axis 50 mm long, resting on one of its bases on the H.P. With a vertical face perpendicular to the V.P.
(i) As the axis is perpendicular to the ground i.e. the H.P. Begin with the top view. It will be an equilateral triangle of sides 40 mm long, with one of its sides perpendicular to xy. Name the corners as shown, thus completing the top view. The corners d, e and fare hidden and coincide with the top corners a, b and c respectively
(ii) Project the front view, which will be a rectangle. Name the corners. The line b'e' coincides with a'd'.
2. Draw the projections of a pentagonal pyramid, base 30 mm edge and axis 50 mm long, having its base on the H.P. And an edge of the base parallel to the V.P. Also draw its side view.
(i) Assume the side DE which is nearer the V.P., to be parallel to the V.P. As shown in the pictorial view.
(ii) In the top view, draw a regular pentagon abcde with ed parallel to and nearer xy. Locate its centre o and join it with the corners to indicate the slant edges.
(iii) Through o, project the axis in the front view and mark the apex o', 50 mm above xy. Project all the corners of the base on xy. Draw lines o'a', o'b' and o'c' to show the visible edges. Show o'd' and o'e' for the hidden edges as dashed lines.
(iv) For the side view looking from the left, draw a new reference line x1y1 perpendicular to xy and to the right of the front view. Project the side view on it, horizontally from the front view as shown. The respective distances of all the points in the side view from x1y1, should be equal to their distances in the top view from xy. This is done systematically as explained below:
(v) From each point in the top view, draw horizontal lines up to x1y1. Then draw lines inclined at 45° to x1y1 (or xy) as shown. Or, with q, the point of intersection between xy and x1y1 as centre, draw quarter circles. Project up all the points to intersect the corresponding horizontal lines from the front view and complete the side view as shown in the figure. Lines o1d1 and o1c1 coincide with o1e1 and o1a1 respectively.
3. A hexagonal pyramid, base 25 mm side and axis 50 mm long, has an edge of its base on the ground. Its axis is inclined at 30° to the ground and parallel to the V.P. Draw its projections. In the initial position assume the axis to be perpendicular to the H.P.
Draw the projections with the base in xy and its one edge perpendicular to the V.P. Fig. 18 (i)
If the pyramid is now tilted about the edge AF (or CD) the axis will become inclined to the H.P. But will remain parallel to the V.P. The distances of all the corners from the V.P. Will remain constant.
The front view will not be affected except in its position in relation to xy. The new top view will have its corners at same distances from xy, as before.
Method I: [fig. 18 (ii)]:
(i) Reproduce the front view so that the axis makes 30° angle with xy and the point a' remains in xy.
(ii) Project all the points vertically from this front view and horizontally from the first top view. Complete the new top view by drawing (a) lines joining the apex o'1 with the corners of the base and (b) lines for the edges of the base.
The base will be partly hidden as shown by dashed line a1b1, e1f1 and f1a1. Similarly, o1f1 and o1a1 are also dashed lines.
Method II:
(i) Through a' draw a new reference line x1y1 inclined at 30° to the axis, to represent an auxiliary inclined plane.
(ii) From the front view project, the required top view on x1y1, keeping the distance of each point from x1y1 equal to the distance of its first top view from xy, viz. e1q = eb' etc.
4. Draw the projections of a cone, base 75 mm diameter and axis 100 mm long, lying on the H.P. On one of its generators with the axis parallel to the V.P.
(i) Assuming the cone to be resting on its base on the ground, draw its projections.
(ii) Re-draw the front view so that the line o'7' (or o'1 ') is in xy. Project the required top view as shown. The lines from o1 should be tangents to the ellipse.
The top view obtained by auxiliary-plane method is shown in fig. 13-24(ii). The new reference line x1y1 is so drawn as to contain the generator o'1' instead of o'7' (for sake of convenience). The cone is thus lying on the generator o'1 '. Note that 1 '1 1 = 1 '1, o'o1 = 4'o etc. Also note that the base is fully visible in both the methods.
Projections of the solids with axis inclined to both the H.P. And the V.P.
The projections of a solid with its axis inclined to both the planes are drawn in three stages:
(i) Simple position
(ii) Axis inclined to one plane and parallel to the other
(iii) Final position.
The second and final positions may be obtained either by the alteration of the positions of the solid, i.e. the views, or by the alteration of reference lines.
5. A square prism, base 40 mm side and height 65 mm, has its axis inclined at 45° to the H.P. And has an edge of its base, on the H.P and inclined at 30° to the V.P. Draw its projections.
i) Assuming the prism to be resting on its base on the ground with an edge of the base perpendicular to the V.P., draw its projections. Assume the prism to be tilted about the edge which is perpendicular to the V.P., so that the axis makes 45° angle with the H.P.
(ii) Hence, change the position of the front view so that the axis is inclined at 45° to xy and f' (or e') is in xy. Project the second top view. Again, assume the prism to be turned so that the edge on which it rests, makes an angle of 30° with the V.P., keeping the inclination of the axis with the ground constant. The shape and size of the second top view will remain the same; only its position will change. In the front view, the distances of all the corners from xy will remain the same as in the second front view.
(iii) Therefore, reproduce the second top view making f1g1 inclined at 30° to xy. Project the final front view upwards from this top view and horizontally from the second front view, e.g. a vertical from a1 and a horizontal from a' intersecting at a'1. As the top end is further away from xy in the top view it will be fully visible in the front view. Complete the front view showing the hidden edges by dashed lines.
(iv) The second top view may be turned in the opposite direction as shown. In this position, the lower end of the prism, viz. e'lf'1g'1h'1 will be fully visible in the front view.
Method II:
(i) Draw the top view and front view in simple position.
(ii) Through f', draw a new reference line x1y1 making 45° angle with the axis. On it, project the auxiliary top view.
(iii) Draw another reference line x2y2 inclined at 30° to the line fig 1. From the auxiliary top view, project the required front view, keeping the distance of each point from x2y2, equal to its distance (in the first front view) from x1y1 i.e. a'1q1 = a'q etc.
6. What do you mean by development of solid surface and write its principle?
The following are the principal methods of development:
1. Parallel-line development:
It is employed in case of prisms and cylinders in which stretch-out-line principle is used. Lines A-A and A1-A1in fig.2 are called the stretch-out Iines.
2. Radial-line development:
Itis used for pyramids and cones in which the true length of the slant edge or the generator is used as radius.
3. Triangulation development:
This is used to develop transition pieces. This is simply a method of dividing a surface into many triangles and transferring them into the development.
4. Approximate method:
It is used to develop objects of double curved or warped surfaces as sphere, paraboloid, ellipsoid, hyperboloid and helicoid.
Developments of lateral surfaces of right solids:
The methods of drawing developments of surfaces of various solids are explained by means of the following typical problems. Only the lateral surfaces of the solids (except the cube) have been developed. The ends or bases have been omitted. They can be easily incorporated if required.
Cube
The development of the surface of a cube consists of six equal squares, the length of the side of the squares being equal to the length of the edge of the cube.
7. Draw the development of the surface of the part P of the cube, the front view of which is shown in fig. Name all the corners of the cube and the points at which the edges are cut.
(i) Draw the stretch-out lines A-A and E-E directly in line with the front view, and assuming the cube to be whole, draw four squares for the vertical faces, one square for the top and another for the bottom as shown in fig. 3(ii).
(ii) Name all the corners. Draw a horizontal line through 1' to cut AE at 1 and DH at 4. a' b' is the true length of the edge. Hence, mark a point 2 on AB and 3 on CD such that A 2 = a' 2' and C 3 = c' 3'. Mark the point3 on CD in the top square also.
(iii) Draw lines 1-2, 2-3, 3-4 and 4-1, and complete the development as shown. Keep lines for the removed portion, viz. A1, A2, 3D, D4 and DA thin and fainter.
8. Draw the development of the lateral surface of the part P of the pentagonal prism shown in fig.
Name the corners of the prism and the points at which the edges are cut.
(i) Draw the development assuming the prism to be whole [fig. 5(ii)].It is made up of five equal rectangles.
(ii) Draw horizontal lines through points 1’, 2’ etc. to cut the lines for the corresponding edges in the development at points 1, 2 etc.
(iii)Draw lines joining these points and complete the development as shown.
9. Draw the development of the lateral surface of the part P of the hexagonal prism shown in fig.
Name the points at which the edges are cut and draw the development assuming the prism to be whole [fig. 6(ii)].
(i) Obtain all the points except 5 and 6 by drawing horizontal lines. Note that points 3 and 8 lie on vertical lines drawn through the mid-points of BC and EF.
(ii) Mark points 5 and 6 such that SD1 = 5d1 and D16 = d16.
(iii) Draw lines joining points 1, 2, 3 etc. in correct sequence and complete the required development as shown.
10. Draw the development of the lateral surface of the frustum of the square pyramid shown in fig.
(i) Determine the position of the apex. None of the lines in the front view shows the true length of the slant edge. Therefore, draw the top view and make any one line (for the slant edge) horizontal, i.e. parallel to xy and determine the true length o'1 '1. Through a', draw a line parallel to the baseand obtain the true length o'a".
(ii) With O as centre and radius o'1 '1, draw an arc and obtain the development of the lateral surface of the whole pyramid [fig. 9(ii)].
(iii) With centre O and radius o' a", draw an arc cutting O1, O2 etc. at points, B etc. respectively.
(iv) Draw lines AB, BC, CD and DA and complete the required development. Note that these lines are respectively parallel to lines 1-2, 2-3 etc.
3.A frustum of a square pyramid has its base 50 mm side; top 25 mm side and height 75 mm. Draw the development of its lateral surface.
(i) Mark the mid-point P of CD and Q of A1B1. Draw a line joining P and Q and cutting CC1 at R and BB1 at S. Transfer these points to the front view and the top view. For example, with o' as centre and radius o'R, draw an arc cutting o' A1 at R1. Through R1, draw a line parallel to the base and cutting c'c'1 at r'. Project r' to r on cc1 in the top view. r' and r are the projections of R.
(ii) Similarly, obtain s' and s on b'b'1 and bb1 respectively. Draw lines pr, rs and sq which will show the top view of the line PQ. p'r's'q' will be thepath of the line PQ in the front view.
Cone
The development of the curved surface of a cone is a sector of a circle, the radius and the length of the arc of which are respectively equal to the slant height and the circumference of the base-circle of the cone.
