In the preceding chapters 1 to 4 plane geometry, where the constructions of the geometrical figures having only two dimensions are discussed, solid geometry is delt with in the following chapters.
Engineering drawing, particularly solid geometry is the graphic language used in the industry to record the ideas and information necessary in the form of blue prints to make machines, buildings, structures etc., by engineers and technicians who design, develop, manufacture and market the products.
As per the optical physics, an object is seen when the light rays called visual rays coming from the object strike the observer’s eye. The size of the image formed in the retina depends on the distance of the observer from the object.
If an imaginary transparent plane is introduced such that the object is in between the observer and the plane, the image obtained on the screen is as shown in Fig.1. This is called perspective view of the object. Here, straight lines (rays) are drawn from various points on the contour of the object to meet the transparent plane, thus the object is said to be projected on that plane.
The figure or view formed by joining, in correct sequence, the points at which these lines meet the plane is called the projection of the object. The lines or rays drawn from the object to the plane are called projectors. The transparent plane on which the projections are drawn is known as plane of projection.
Types of Projections
1. Pictorial projections
- Perspective projection
- Isometric projection
- Oblique projection
2. Orthographic Projections
1. Pictorial Projections
The Projections in which the description of the object is completely understood in one view is known as pictorial projection. They have the advantage of conveying an immediate impression of the general shape and details of the object, but not its true dimensions or sizes.
2. Orthographic Projection
‘ORTHO’ means right angle and orthographic means right angled drawing. When the projectors are perpendicular to the plane on which the projection is obtained, it is known as orthographic projection.
Method of Obtaining Front View
Imagine an observer looking at the object from an infinite distance (Fig.2). The rays are parallel to each other and perpendicular to both the front surface of the object and the plane. When the observer is at a finite distance from the object, the rays converge to the eye as in the case of perspective projection. When the observer looks from the front surface F or the block, its true shape and size is seen. When the rays or projectors are extended further they meet the vertical plane(Y.P) located behind the object. By joining the projectors meeting the plane in correct sequence the Front view (Fig. 2) is obtained.
Front view shows only two dimensions of the object, Viz. length L and height H. It does not show the breadth B. Thus one view or projection is insufficient for the complete description of the object.
As Front view alone is insufficient for the complete description of the object, another plane called Horizontal plane (H.P) is assumed such that it is hinged and perpendicular to Y.P and the object is in front of the Y.P and above the H.P as shown in Fig.3a.
Method of Obtaining Top View
Looking from the top, the projection of the top surface is the Top view (Tv). Both top surface and Top view are of exactly the same shape and size. Thus, Top view gives the True length L and breadth B of the block but not the height H.
(1) Each projection shows that surface of the object which is nearer to the observer. and far away from the plane.
(2) Orthographic projection is the standard drawing form of the industrial world.
X Y Line: The line of intersection of V.P and H.P is called the reference line and is denoted as xy.
Obtaining the Projection on the Drawing Sheet
It is convention to rotate the H.P through 90 in the clockwise direction about xy line so that it lies in the extension of V.P as shown in Fig. 3a. The two projections Front view and Top view may be drawn on the two dimensional drawing sheet as shown in Fig.3b.
Thus, all details regarding the shape and size, Viz. Length (L), Height(H) and Breadth(B) of any object may be represented by means of orthographic projections i.e., Front view and Top view.
VP and H.P are called as Principal planes of projection or reference planes. They are always transparent and at right angles to each other. The projection on V.P is designated as Front view and the projection on H.P as Top view.
When the planes of projections are extended beyond their line of intersection, they form Four Quadrants. These quadrants are numbered as I, II, Ill and IV in clockwise direction when rotated about reference line xy as shown in Fig.4 and 6(a).
In the Figure 5 the object is in the first quadrant and the projections obtained are “First angle projections” Le., the object lies in between the observer and the planes of projection. Front view shows the length(L) and height(H) of the object, and Top view shows the length(L) and the breadth(B) of it.
The object may be situated in anyone of four quadrants, its position relative to the planes being described as in front of V.P and above H.P in the first quadrant and so on.
Figure 5 shows the two principle planes H.P and v.p and another Auxiliary vertical plane (AVP). AVP is perpendicular to both VP and H.P.
Front view is drawn by projecting the object on the V.P. Top view is drawn by projecting the object on the H.P. The projection on the AVP as seen from the left of the object and drawn on the right of the front view, is called left side view.
First Angle Projection
When the object is situated in First Quadrant, that is, in front of V.P and above H.P, the projections obtained on these planes is called First angle projection.
(i) The object lies in between the observer and the plane of projection.
(li) The front view is drawn above the xy line and the top view below xy. (above xy line is V.P and below xy line is H.P).
(iii) In the front view, H.P coincides with xy line and in top view v.p coincides with xy line.
(iv) Front view shows the length(L) and height(H) of the object and Top view shows the length(L) and breadth(B) or width(W) or thickens (T) of it.
Third Angle Projection
In this, the object is situated in Third Quadrant. The Planes of projection lie between the object and the observer. The front view comes below the xy line and the top view about it.
BIS Specification (SP46 : 2003)
BIS has recommended the use of First angle projection in line with the specifications of ISO adapted by all countries in the world.
Designation and Relative Position of Views
An object in space may be imagined as surrounded by six mutually perpendicular planes. So, it is possible to obtain six different views by viewing the object along the six directions, normal to the six planes. Fig.6 shows an object with the six possible directions to obtain the six different views which are designated as follows.
1. View in the direction a = front view
2. View in the direction b = top view
3. View in the direction c = left side view
4. View in the direction d = right side view
5. View in the direction e = bottom view
6. View in the direction f= rear view
The relative position of the views in First angle projection are shown in Fig.7.
Note: A study of the Figure 7 reveals that in both the methods of projection, the views are identical in shape and size but their location with respect to the front view only is different.
Projection of Points
A solid consists of a number of planes, a plane consists of a number of lines and a line in turn consists of number of points:-‘From this, it is obvious that a solid may be generated by a plane ABCD moving in space (Fig.8a), a plane may be generated by a straight line AD moving in space(Fig.8b) and a straight line in tum, may be generated by a point A moving in space (Fig.8c)
Points in Space
A point may lie in space in anyone of the four quadrants. The positions of a point are:
1. First quadrant, when it lies above H.P and in front of V.P.
2. Second quadrant, when it lies above HP and behind V.P.
3. Third quadrant, when it lies below H.P and behind V.P.
4. Fourth quadrant, when it lies below H.P and in front of V.P.
Knowing the distances of a point from H.P and V.P, projections on H.P and V.P are found by extending the projections perpendicular to both the planes. Projection on H.P is called Top view and projection on V.P is called Front view
1. Actual points in space are denoted by capital letters A, B, C.
2. Their front views are denoted by their corresponding lower case letters with dashes a1, b1, c1, etc., and their top views by the lower case letters a, b, c etc.
3. Projectors are always drawn as continuous thin lines.
1. Students are advised to make their own paper/card board/perplex model of H.P and V.P as shown in Fig.4. The model will facilitate developing a good concept of the relative position of the points lying in any of the four quadrants.
2. Since the projections of points, lines and planes are the basic chapters for the subsequent topics on solids viz, projection of solids, development, pictorial drawings and conversion of pictorial to orthographic and vice versa, the students should follow these basic chapters carefully to draw the projections.
Projection of Lines
The shortest distance between two points is called a straight line. The projectors of a straight line are drawn therefore by joining the projections of its end points. The possible projections of straight. lines with respect to V.P and H.P in the first quadrant are as follows:
I. Perpendicular to one plane and parallel to the other.
2. Parallel to both the planes.
3. Parallel to one plane and inclined to the other.
4. Inclined to both the planes.
1. Line perpendicular to H.P and parallel to V.P
The pictorial view of a straight line AB in the First Quadrant is shown in Fig.9a.
1. Looking from the front; the front view of AB, which is parallel to V.P and marked, a1b1, is obtained. True length of AB = a1b1.
2. Looking from the top; the top view of AB, which is perpendicular to H.P is obtained a and b coincide.
3. The Position of the line AB and its projections on H.P. and V.P are shown in Fig.9b.
4. The H.P is rotated through 900 in clock wise direction as shown in Fig.9b.
5. The projection of the line on V.P which is the front view and the projection on H.P, the top view are shown in Fig.9c.
Note: Only Fig.9c is drawn on the drawing sheet as a solution.
Line perpendicular to v.p and parallel to H.P.
2. Line parallel to both the planes
3. Line parallel to V.P and inclined to H.P.
4. Line inclined to both the planes
Projection of Planes
A plane figure has two dimensions viz. the length and breadth. It may be of any shape such as triangular, square, pentagonal, hexagonal, circular etc. The possible orientations of the planes with respect to the principal planes H.P and v.p of projection are:
1. Plane parallel to one of the principal planes and perpendicular to the other,
2. Plane perpendicular to both the principal planes,
3. Plane inclined to one of the principal planes and perpendicular to the other,
4. Plane inclined to both the principal planes.
1. Plane parallel to one of the principal planes and perpendicular to the other
When a plane is parallel to V.P the front view shows the true shape of the plane. The top view appears as a line parallel to xy. Figure 10a shows the projections of a square plane ABCD, when it is parallel to V.P and perpendicular to H.P. The distances of one of the edges above H.P and from the V.P are denoted by d1 and d1 respectively.
Figure 10b shows the projections of the plane. Figure 10c shows the projections of the plane, when its edges are equally inclined to H.P.
Figure 11 shows the projections of a circular plane, parallel to H.P and perpendicular to V.P.
2. Plane perpendicular to both H.P and V.P.
When a plane is perpendicular to both H.P. and V.P, the projections of the plane appear as straight lines. Figure 12 shows the projections of a rectangular plane ABCD, when one of its longer edges is parallel to H.P. Here, the lengths of the front and top views are equal to the true lengths of the edges.
3. Plane inclined to one of the principal planes and perpendicular to the other
When a plane is inclined to one plane and perpendicular to the other, the projections are obtained in two stages.
Stage I Assume the plane is parallel to H.P (lying on H.P) and perpendicular to V.P.
1. Draw the projections of the pentagon ABCDE, assuming the edge AE perpendicular to V.P. a1e1 b11d11c11 on xy is the front view and ab1 c1d1e is the top view.
Stage II Rotate the plane (front view) till it makes the given angle with H.P.
2. Rotate the front view till it makes the given angle e with xy which is the final front view.
3. obtain the final top view abcde by projection.
Plane inclined to both H.P and V.P
If a plane is inclined to both H.P and V.P, it is said to be an oblique plane. Projections of oblique planes are obtained in three stages .
Stage I: Assume the plane is parallel to H.P and a shorter edge of it is perpendicular to V.P.
1. Draw the projections of the plane.
Stage II : Rotate the plane till it makes the given angle with H.P.
2. Redraw the front view, making given angle e with xy and then project the top. view.
Stage Ill : Rotate the plane till its shorter edge makes the given angle Φ with V.P.
3. Redraw the top view abed such that the shorter edge ad, is inclined to xy by Φ .
4. Obtain the final front view a1b1c1d1, by projection.
Traces of a line
When a line is inclined to a plane, it will meet that plane when produced if necessary. The point at which the line or line produced meets the plane is called its trace.
The point of intersection of the line or line produced with H.P. is called Horizontal Trace (H.T) and that with V.P. is called Vertical Trace (V.T).