UNIT 3
Projections of Planes
Plane figures or surfaces have only two dimensions, viz. Length and breadth. They do not have thickness. A plane figure may be assumed to be contained by a plane, and its projections can be drawn, if the position of that plane with respect to the principal planes of projection is known.
In this chapter, we shall discuss the following topics:
1. Types of planes and their projections.
2. Traces of planes.
Type of planes:
Planes may be divided into two main types:
(1) Perpendicular planes.
(2) Oblique planes.
Perpendicular planes:
These planes can be divided into the following sub-types:
(i) Perpendicular to both the reference planes.
(ii) Perpendicular to one plane and parallel to the other.
(iii) Perpendicular to one plane and inclined to the other.
Perpendicular to both the reference planes (fig 1):
A square ABCD is perpendicular to both the planes. Its H.T. And V.T. Are in a straight-line perpendicular to xy.
Figure 1
The front view b'c' and the top view ab of the square are both lines coinciding with the V.T. And the H.T. Respectively.
Perpendicular to one plane and parallel to the other plane:
a) Plane, perpendicular to the H.P. And parallel to the V.P. [fig. 12-2(i)]. A triangle PQR is perpendicular to the H.P. And is parallel to the V.P. Its H.T. Is parallel to xy. It has no V.T.
The front view p'q'r' shows the exact shape and size of the triangle. The top view pqr is a line parallel to xy. It coincides with the H.T.
(b) Plane, perpendicular to the V.P. And parallel to the H.P. [fig. 12-2(ii)]. A square ABCD is perpendicular to the V.P. And parallel to the H.P. Its V.T. Is parallel to xy. It has no H.T.
The top view abed shows the true shape and true size of the square. The front view a'b' is a line, parallel to xy. It coincides with the V.T.
Figure 2
Perpendicular to one plane and inclined to the other plane:
A square ABCD is perpendicular to the H.P. And inclined at an angle φ to the V.P. Its V.T. Is perpendicular to xy. Its H.T. Is inclined at φ to xy.
Its top view ab is a line inclined at φ to xy. The front view a'b'c'd' is smaller than ABCD.
Figure 3
(b) Plane, perpendicular to the V.P. And inclined to the H.P. (fig. 12-4).
A square ABCD is perpendicular to the V.P. And inclined at an angle θ to the H.P. Its H.T. Is perpendicular to xy. Its V.T. Makes the angle e with xy. Its front view a'b' is a line inclined at θ to xy. The top view abed is a rectangle which is smaller than the square ABCD.
Figure 4
Fig. 5 shows the projections and the traces of all these perpendicular planes by third-angle projection method.
Figure 5
Oblique planes:
Planes which are inclined to both the reference planes are called oblique planes.
Traces of planes:
A plane, extended if necessary, will meet the reference planes in lines, unless it is parallel to any one of them.
These lines are called the traces of the plane. The line in which the plane meets the H.P. Is called the horizontal trace or the H.T. Of the plane. The line in which it meets the V.P. Is called its vertical trace or the V.T. A plane is usually represented by its traces.
General conclusions:
(1) Traces:
(a) When a plane is perpendicular to both the reference planes, its traces lie on a straight-line perpendicular to xy.
(b) When a plane is perpendicular to one of the reference planes, its trace upon the other plane is perpendicular to xy (except when it is parallel to the other plane).
(c) When a plane is parallel to a reference plane, it has no trace on that plane. Its trace on the other reference plane, to which it is perpendicular, is parallel to xy.
(d) When a plane is inclined to the H.P. And perpendicular to the V.P., its inclination is shown by the angle which its V.T. Makes with xy. When it is inclined to the V.P. And perpendicular to the H.P., its inclination is shown by the angle which its H.T. Makes with xy.
(e) When a plane has two traces, they, produced if necessary, intersect in xy (except when both are parallel to xy as in case of some oblique planes).
(2) Projections:
(a) When a plane is perpendicular to a reference plane, its projection on that plane is a straight line.
(b) When a plane is parallel to a reference plane, its projection on that plane shows its true shape and size.
(c) When a plane is perpendicular to one of the reference planes and inclined to the other, its inclination is shown by the angle which its projection on the plane to which it is perpendicular, makes with xy. Its projection on the plane to which it is inclined, is smaller than the plane itself.
Problem 1. Show by means of traces, each of the following planes:
(a) Perpendicular to the H.P. And the V.P.
(b) Perpendicular to the H.P. And inclined at 30° to the V.P.
(c) Parallel to and 40 mm away from the V.P.
(d) Inclined at 45° to the H.P. And perpendicular to the V.P.
(e) Parallel to the H.P. And 25 mm away from it.
Figure 6
(a) The H.T. And the V.T. Are in a line perpendicular to xy.
(b) The H.T. Is inclined at 30° to xy; the V.T. Is normal to xy; both the traces intersect in xy.
(c) The H.T. Is parallel to and 40 mm away from xy. It has no V.T.
(d) The H.T. Is perpendicular to xy; the V.T. Makes 45° angle with xy; both intersect in xy.
(e) The V.T. Is parallel to and 25 mm away from xy. It has no H.T.
Projection of planes parallel to one of the reference planes:
The projection of a plane on the reference plane parallel to it will show its true shape. Hence, beginning should be made by drawing that view. The other view which will be a line, should then be projected from it.
Note: When the plane is parallel to the H.P.: The top view should be drawn first, and the front view projected from it.
Problem:
An equilateral triangle of 50 mm side has its V.T. Parallel to and 25 rnm above xy. It has no H.T. Draw its projections when one of its sides is inclined at 45° to the V.P.
As the V.T. Is parallel to xy and as there is no H.T. The triangle is parallel to the H.P. Therefore, begin with the top view.
Figure 7
(i) Draw an equilateral triangle abc of 50 mm side, keeping one side, say ac, inclined at 45° to xy.
(ii) Project the front view, parallel to and 25 mm above xy, as shown in fig. 7.
When the plane is parallel to V.P.
Beginning should be made with the front view and the top view projected from it.
Problem:
A square ABCD of 40 mm side has a corner on the H.P. And 20 mm in front of the V.P. All the sides of the square are equally inclined to the H.P. And parallel to the V.P. Draw its projections and show its traces.
Figure 8
As all the sides are parallel to the V.P., the surface of the square also is parallel to it. The front view will show the true shape and position of the square.
(i) Draw a square a'b'c'd' in the front view with one corner in xy and all its sides inclined at 45° to xy.
(ii) Project the top view keeping the line ac parallel to xy and 30 mm below it. The top view is its H.T. It has no V.T.
Projections of planes inclined to one reference plane and perpendicular to other:
When a plane is inclined to a reference plane, its projections may be obtained in two stages. In the initial stage, the plane is assumed to be parallel to that reference plane to which it must be made inclined. It is then tilted to the required inclination in the second stage.
1) Plane, Inclined to the H.P. And perpendicular to the V.P.:
When the plane is inclined to the H.P. And perpendicular to the V.P., in the initial stage, it is assumed to be parallel to the H.P. Its top view will show the true shape. The front view will be a line parallel to xy. The plane is then tilted so that it is inclined to the H.P. The new front view will be inclined to xy at the true inclination. In the top view the corners will move along their respective paths (parallel to xy).
Problem:
A regular pentagon of 25 mm side has one side on the ground. Its plane is inclined at 45° to the H.P and perpendicular to the V.P. Draw its projections and show its traces.
Figure 9
Assuming it to be parallel to the H.P.
(i) Draw the pentagon in the top view with one side perpendicular to xy [fig. 9 (i)]. Project the front view. It will be the line a'c' contained by xy.
(ii) Tilt the front view about the point a', so that it makes 45° angle with xy.
(iii) Project the new top view ab1c1d1e upwards from this front view and horizontally from the first top view. It will be more convenient if the front view is reproduced in the new position separately and the top view projected from it, as shown in fig. 9 (ii). The V.T. Coincides with the front view and the H.T. Is perpendicular to xy, through the point of intersection between xy and the front view-produced.
2) Plane, inclined to the V.P. And perpendicular to the H.P.:
In the initial stage, the plane may be assumed to be parallel to the V.P. And then tilted to the required position in the next stage.
Problem:
Draw the projections of a circle of 50 mm diameter, having its plane vertical and inclined at 30° to the V.P. Its center is 30 mm above the H.P. And 20 mm in front of the V.P. Show also its traces.
Figure 10
A circle has no corners to project one view from another. However, many points, say twelve, equal distances apart, may be marked on its circumference.
(i) Assuming the circle to be parallel to the V.P., draw its projections. The front view will be a circle [fig. 10 (i)], having its center 30 mm above xy. The top view will be a line, parallel to and 20 mm below xy.
(ii) Divide the circumference into twelve parts (with a 30°-60° set- square) and mark the points as shown. Project these points in the top view. The centre O will coincide with the point 4.
(iii) When the circle is tilted, to make 30° angle with the V.P., its top view will become inclined at 30° to xy. In the front view all the points will move along their respective paths (parallel to xy). Reproduce the top view keeping the centre o at the same distance, viz. 20 mm from xy and inclined at 30° to xy [fig. 10 (ii)].
(iv) For the final front view, project all the points upwards from this top view and horizontally from the first front view. Draw a freehand curve through the twelve points 1'1, 2'1 etc. This curve will be an ellipse.
Projection of oblique planes:
When a plane has its surface inclined to one plane and an edge or a diameter or a diagonal parallel to that plane and inclined to the other plane, its projections are drawn in three stages.
(1) If the surface of the plane is inclined to the H.P. And an edge (or a diameter or a diagonal) is parallel to the H.P. And inclined to the V.P.,
(i) in the initial position the plane is assumed to be parallel to the H.P. And an edge perpendicular to the V.P.
(ii) It is then tilted to make the required angle with the H.P. As already explained, its front view in this position will be a line, while its top view will be smaller in size.
(iii) In the final position, when the plane is turned to the required inclination with the V.P., only the position of the top view will change. Its shape and size will not be affected. In the final front view, the corresponding distances of all the corners from xy will remain the same as in the second front view.
If an edge is in the H.P. Or on the ground, in the initial position, the plane is assumed to be lying in the H.P. Or on the ground, with the edge perpendicular to the V.P. If a corner is in the H.P. Or on the ground, the line joining that corner with the center of the plane is kept parallel to the V.P.
(2) Similarly, if the surface of the plane is inclined to the V.P. And an edge (or a diameter or a diagonal) is parallel to the V.P. And inclined to the H.P.,
(i) in the initial position, the plane is assumed to be parallel to the V.P. And an edge perpendicular to the H.P.
(ii) It is then tilted to make the required angle with the V.P. Its top view in this position will be a line, while its front view will be smaller in size.
(iii) When the plane is turned to the required inclination with the H.P., only the position of the front view will change. Its shape and size will not be affected. In the final top view, the corresponding distances of all the corners from xy will remain the same as in the second top view.
If an edge is in the V.P., in the initial position the plane is assumed to be lying in the V.P. With an edge perpendicular to the H.P. If a corner is in the V.P., the line joining that corner with centre of the plane is kept parallel to the H.P.
Problem:
1. A square ABCD of 50 mm side has its corner A in the H.P., its diagonal AC inclined at 30° to the H.P. And the diagonal BO inclined at 45° to the V.P. And parallel to the H.P. Draw its projections.
Figure 11
In the initial stage, assume the square to be lying in the H.P. With AC parallel to the V.P.
(i) Draw the top view and the front view. When the square is tilted about the corner A so that AC makes 30° angle with the H.P., BO remains perpendicular to the V.P. And parallel to the H.P.
(ii) Draw the second front view with a'c' inclined at 30° to xy, keeping a' or c' in xy. Project the second top view. The square may now be turned so that BO makes 45° angle with the V.P. And remains parallel to the H.P. Only the position of the top view will change. Its shape and size will remain the same.
(iii) Reproduce the top view so that b1d1 is inclined at 45° to xy. Project the final front view upwards from this top view and horizontally from the second front view.
2. Draw the projections of a circle of 50 mm diameter resting in the H.P. On a point A on the circumference, its plane inclined at 45° to the H.P. And
(a) the top view of the diameter AB making 30° angle with the V.P.;
(b) the diameter AB making 30° angle with the V. P.
Figure 12
Draw the projections of the circle with A in the H.P. And its plane inclined at 45° to the H.P. And perpendicular to the V.P. [fig. 12(i) and fig. 12(ii)].
(a) In the second top view, the line a1b1 is the top view of the diameter AB. Reproduce this top view so that a1b1 makes 30° angle with xy [fig. 12(iii)]. Project the required front view.
(b) If the diameter AB, which makes 45° angle with the H.P., is inclined at 30° to the V.P. Also, its top view a1b1 will make an angle greater than 30° with xy. This apparent angle of inclination is determined as described below.
Draw any line a1b2 equal to AB and inclined at 30° to xy [fig. 12(iv)]. With a1 as center and radius equal to the top view of AB, viz. a1b1, draw an arc cutting rs (the path of B in the top view) at b3. Draw the line joining a1 with b3, and around it, reproduce the second top view. Project the final front view. It is evident that a1b3 is inclined to xy at an angle ϕ which is greater than 30°.
3. A thin 30°-60° set-square has its longest edge in the V.P. And inclined at 30° to the H.P. Its surface makes an angle of 45° with the V.P. Draw its projections.
In the initial stage, assume the set-square to be in the V.P. With its hypotenuse perpendicular to the H.P.
(i) Draw the front view a'b'c' and project the top view ac in xy.
(ii) Tilt ac around the end a so that it makes 45° angle with xy and project the front view a'1b'1c'1.
(iii) Reproduce the second front view a'1b'1c'1 so that a'1b'1 makes an angle of 30° with xy. Project the final top view a1b1c1.
Two views of an object, viz. The front view and the top view (projected on the
Principal planes of projection), are sometimes not sufficient to convey all the information
Regarding the object. Additional views, called auxiliary views, are therefore, projected
On other planes known as auxiliary planes. These views are often found necessary
In technical drawings. Auxiliary views may also be used for determining
(i) the true length of a line,
(ii) the point-view of a line,
(iii) the edge-view of a plane,
(iv) the true size and form of a plane etc.
They are thus especially useful in finding solutions of problems in practical solid geometry.
This chapter deals with the following topics:
1. Types of auxiliary planes and views.
2. Projection of a point on an auxiliary plane.
3. Projections of lines and planes using auxiliary planes.
4. To determine true length of a line.
5. To obtain point-view of a line and edge-view of a plane.
Types of Auxiliary planes and views:
Auxiliary planes are of two types:
(i) auxiliary vertical plane or A.V.P., and
(ii) auxiliary inclined plane or A.LP.
(i) Auxiliary vertical plane is perpendicular to the H.P. And inclined to the V.P.
Projection on an A.V.P. Is called auxiliary front view.
(ii) Auxiliary inclined plane is perpendicular to the V.P. And inclined to the H.P.
Projection on an A.LP. Is called auxiliary top view.
The orthographic views of the auxiliary projections are drawn by rotating the
Auxiliary plane about that principal plane to which it is perpendicular.
Projection of a point on an Auxiliary plane:
A point A [fig. 13(i)] is situated in front of the V.P. And above the H.P. A.V.P.is the auxiliary vertical plane inclined at an angle a to the V.P. The H.P. And theA.V.P. Meet at right angles in the line x1y1.
a' and a are respectively, the front view and the top view of the point A. a'1 isthe auxiliary front view obtained by drawing a projector Aa'1 perpendicular to the A.V.P.It can be clearly seen that a'1o1 (the distance of the auxiliary front view a'1 fromx1y1) = a'o (the distance of the front view a' from xy) = Aa (the distance of thepoint A from the H.P.).
Figure 13
Fig. 13(ii) shows the V. P. And the A.V.P. Rotated about the H.P. To which they are perpendicular. The line of intersection x1y1between the A.V.P. And the H.P.,is inclined at the angles a to xy. The line joining the top view a with the auxiliary front view a'1, isat right angles to x1y1 and intersectsit at q1. Note that a'1o1 = a'o.
To draw the orthographic viewsa1[fig. 13(iii)], start with the reference line xy and mark the front view a' and the top view a.
Draw a new reference line x1y1, making the angle α with xy. Through the topview a, draw a projector aa'1 perpendicular to and intersecting x1y1 at o1 and suchthat a'1o1 = a'o. a'1 is the required auxiliary front view.
The new reference line making the angle a with xy, can be drawn in fourdifferent positions, as shown in fig. 14 by lines x1y1, x2y2 etc. All the front viewsare projected from the top view a and their distances from their respective referencelines are equal, i.e. a'1o1 = a'2o2 ..... = a'o.
Figure 14
Projection of a point on an auxiliary inclined plane:
A point P [fig. 15(i)] is situated above the H.P. And in front of the V.P. A.I.P.is an auxiliary inclined plane making an angle β with the H.P. It meets the V.P. Atright angles and in a line x1y1.
p' and p are respectively the front view and the top view of the point P. p1 is theauxiliary top view obtained by drawing the projector Pp1 perpendicular to the A.LP. Itcan be seen that p1o1 (the distance of the auxiliary top view p1 from x1y1) = po (thedistance of the top view p from xy) = Pp' (the distance of the point P from the V.P.).
Figure 15
The H.P. And the A.LP. Are then rotated about the V.P. To which they areperpendicular [fig. 15(ii)]. x1y1, the line of intersection between the V.P. And theA.LP. Makes the angle 13 with xy. The line joining the front view p' and the auxiliarytop view p1 is at right angles to x1y1 and intersects it at o1. Note that p1o1 = po.
To draw the orthographic views [fig. 15(iii)], draw xy and mark p' and p.Draw x1y1 making the angle β with xy.Through the front view p',draw a projector p'p1 perpendicularto and intersecting x1y1 ato1 and such that P101 = po. P1is the required auxiliary front view.In this case also, the newreference line can be drawn infour different positions as shownin fig. 16 by lines X1Y1, X2Y2etc., each inclined at β to xy.
All the top views are projected from the front view p' and their distances from their respectivereference lines are equal, i.e. p1o1= P2o2..... = po.
Figure 16
If the inclination of the A.V.P. With the V.P. Is increased so that α = 90°, theA.V.P. Will be perpendicular to both the planes. Similarly, if the inclination of theA.LP. With the H.P. Is increased so that β = 90°, it will also be perpendicular toboth the H.P. And the V.P. This plane is called the profile plane (P.P.). It may berotated about any one of the two principal planes. The view on this plane can,therefore, be projected from either the top view or the front view and namedaccordingly.
