Theories of projections
Theory of Projections
Projection theory
In engineering, 3-dimensonal objects and structures are represented graphically on a 2-dimensional media. The act of obtaining the image of an object is termed “projection”. The image obtained by projection is known as a “view”. A simple projection system is shown in figure 1.
All projection theory are based on two variables:
Plane of Projection
A plane of projection (i.e, an image or picture plane) is an imaginary flat plane upon which the image created by the line of sight is projected. The image is produced by connecting the points where the lines of sight pierce the projection plane. In effect, 3-D object is transformed into a 2-D representation, also called projections. The paper or computer screen on which a drawing is created is a plane of projection.
Figure 1 : A simple Projection system
Projection Methods
Projection methods are very important techniques in engineering drawing.
Two projection methods used are:
Figure 2 shows a photograph of a series of building and this view represents a perspective projection on to the camera. The observer is assumed to be stationed at finite distance from the object. The height of the buildings appears to be reducing as we move away from the observer. In perspective projection, all lines of sight start at a single point and is schematically shown in figure 3. .
Figure 2. Photographic image of a series of buildings.
Figure 3. A schematic representation of a Perspective projection
In parallel projection, all lines of sight are parallel and is schematically represented in figure. 4. The observer is assumed to be stationed at infinite distance from the object.
Figure 4. A schematic representation of a Parallel projection
Parallel vs Perspective Projection
Parallel projection
√ Distance from the observer to the object is infinite projection lines are parallel – object is positioned at infinity.
√ Less realistic but easier to draw.
Perspective projection
Orthographic Projection
Orthographic projection is a parallel projection technique in which the plane of projection is perpendicular to the parallel line of sight. Orthographic projection technique can produce either pictorial drawings that show all three dimensions of an object in one view or multi-views that show only two dimensions of an object in a single view. These views are shown in figure 5.
Figure 5. Orthographic projections of a solid showing isometric, oblique and multi-view drawings.
Transparent viewing box
Assume that the object is placed in a transparent box, the faces of which are orthogonal to each other, as shown in figure 6. Here we view the object faces normal to the three planes of the transparent box.
Figure 6. The object placed inside a transparent box.
When the viewing planes are
parallel to these principal planes, we obtain the Orthographic
views
The picture we obtain when the line
of sight is projected on to each plane is called as the respective
view of the object. The image obtained on the projection planes ,
i.e., on the top face, Front Face, and Right side face are
respectively the Top View, Front view and Right side view of
the object and is shown in figure 7.
Figure 7 showing the Front view, Top
View and Side view of an object
Multi-view Projection
In an orthographic projection, the object is oriented in such a way that only two of its dimensions are shown. The dimensions obtained are the true dimensions of the object .
Frontal plane of projection
Frontal plane of projection is
the plane onto which the Front View (FV) of the multi-view drawing
is projected.
Figure 8 illustrates the method of
obtaining the Front view of an object. Front view of an object
shows the width and height dimensions.
Figure 8 illustrates the method of obtaining the Front view of an object.
Horizontal plane of projection
Horizontal plane of projection is the plane onto which the Top View of the multi-view drawing is projected and is shownin Figure 9. The Top view of an object shows the width and depth dimensions of the object.
Figure 9 illustrates the method of obtaining the Top view of an object.
Profile plane of projection
In multi-view drawings, the right side view is the standard side view used and is illustrated in figure 10. The right side view of an object shows the depth and the height dimensions. The right side view is projected onto the profile plane of projection, which is a plane that is parallel to the right side of the object.
Figure 10 illustrates the method of obtaining the Side View of an object.
Orientation of views from projection planes
Multi-view drawings gives the complete description of an object. For conveying the complete information, all the three views, i.e., the Front view, Top view and side view of the object is required. To obtain all the technical information, at least two out of the three views are required. It is also necessary to position the three views in a particular order. Top view is always positioned and aligned with the front view, and side view is always positioned to the side of the Front view and aligned with the front view. The positions of each view is shown in figure 11. Depending on whether 1^{st} angle or 3^{rd} angle projection techniques are used, the top view and Front view will be interchanged. Also the position of the side view will be either towards the Right or left of the Front view.
Figure 11. Relative positions and alignment of the views in a multi-view drawing.
Six Principal views
The plane of projection can be oriented to produce an infinite number of views of an object. However, some views are more important than others. These principal views are the six mutually perpendicular views that are produced by six mutually perpendicular planes of projection and is shown in figure 12. Imagine suspending an object in a glass box with major surfaces of the object positioned so that they are parallel to the sides of the box, six sides of the box become projection planes, showing the six views – front, top, left, right, bottom and rear.
Object is suspended in a glass box producing six principal views: each view is perpendicular to and aligned with the adjacent views.
Figure 12. Shows the six perpendicular views of an object
The glass box is now slowly unfolded as shown in figure 13. After complete unfolding of the box on to a single plane, we get the six views of the object in a single plane as shown in figure 14. The top, front and bottom views are all aligned vertically and share the same width dimension where as the rear, left side, front and right side views are all aligned horizontally and share the same height dimension.
Figure 13. Illustration of the views after the box has been partially unfolded.
Figure 14 shows the views of the object with their relative positions after the box has been unfolded completely on to a single plane.
Conventional view
placement
The three-view multi-view drawing is
the standard used in engineering and technology, because many times
the other three principal views are mirror images and do not add to
the knowledge about the object. Figure 15 shows the
standard views used in a three-view drawing i.e., the
top, front and the right side views
Figure 15 showing the three standard views of a multi-view drawing.
The width dimensions are aligned between the front and top views, using vertical projection lines. The height dimensions are aligned between the front and the profile views, using horizontal projection lines. Because of the relative positioning of the three views, the depth dimension cannot be aligned using projection lines. Instead, the depth dimension is measured in either the top or right side view.
Projection methods: 1st angle and 3rd angle projections.
Projection Methods
Universally either the 1^{st} angle projection or the third angle projection methods is followed for obtaining engineering drawings. The principal projection planes and quadrants used to create drawings are shown in figure 16. The object can be considered to be in any of the four quadrant.
Figure 16. The principal projection planes and quadrants for creation of drawings.
First Angle Projection
In this the object in assumed to be positioned in the first quadrant and is shown in figure 17 The object is assumed to be positioned in between the projection planes and the observer. The views are obtained by projecting the images on the respective planes. Note that the right hand side view is projected on the plane placed at the left of the object. After projecting on to the respective planes, the bottom plane and left plane is unfolded on to the front view plane. i.e. the left plane is unfolded towards the left side to obtain the Right hand side view on the left side of the Front view and aligned with the Front view. The bottom plane is unfolded towards the bottom to obtain the Top view below the Front view and aligned with the Front View.
Figure 17. Illustrating the views obtained using first angle projection technique.
Third Angle Projection
In the third angle projection method, the object is assumed to be in the third quadrant. i.e. the object behind vertical plane and below the horizontal plane. In this projection technique, Placing the object in the third quadrant puts the projection planes between the viewer and the object and is shown in figure 18.
Figure 18. Illustrating the views obtained using first angle projection technique
Figure 19 illustrates the difference between the 1^{st} angle and 3^{rd} angle projection techniques. A summary of the difference between 1st and 3rd angle projections is shown if Table 1.
Figure 19 Differentiating between the 1^{st} angle and 3^{rd} angle projection techniques.
Table 1. Difference between first- and third-angle projections
Either first angle projection or third angle projection are used for engineering drawing. Second angle projection and fourth angle projections are not used since the drawing becomes complicated. This is being explained with illustrations in the lecture on Projections of points (lecture 18).
Symbol of projection
The type of projection obtained should be indicated symbolically in the space provided for the purpose in the title box of the drawing sheet. The symbol recommended by BIS is to draw the two sides of a frustum of a cone placed with its axis horizontal The left view is drawn.
Conventions and projections of simple solids
Orthographic Projections
Lines are used to construct a drawing. Various type of lines are used to construct meaningful drawings. Each line in a drawing is used to convey some specific information. The types of lines generally used in engineerign drawing is shown in Table-1.
Table -1. Types of lines generally used in drawings
All visible edges are to be represented by visible lines. This includes the boundary of the object and intersection between two planes. All hidden edges and features should be represented by dashed lines. Figure 1 shows the orthographic front view (line of sight in the direction of arrow)of an object. The external boundary of the object is a rectangle and is shown by visible lines. In Figure-1(a), the step part of the object is hidden and hence shown as dashed lines while for the position of the object shown in figure-1(b) , the step part is directly visible and hence shown by the two solid lines.
Figure 1 shows the pictorial view and front view of the object when the middle stepped region is (a) hidden and (b) visible.
Figure 2 shows the front view (view along the direction indicated by the arrow) of a solid and hollow cylindrical object. The front view of the solid cylinder is seen as a rectangle (figure 2(a)). For the hollow cylinder in addition to the rectangle representing the boundary of the object, two dashed lines are shown to present the boundary of the hole, which is a hidden feature in the object.
Figure 3 shows the Front view
of three objects. Figure 3(a) is the view of one part of a
hollow cylinder which has been split in to two equal parts. The
wall thickness can be represented by the two visible lines. Figure
3(b) is one part of solid cylinder which has been
sectioned in to two equal part. Where as figure 3(c) is
one part of a solid cylindrical part which has been split in to two
unequal parts. The edge formed by the intersection of two surfaces
are represented by solid lines. In case of cylindrical objects or
when holes are present in a component, the centre of the holes or
centre lines of cylinder will have to be represented in the drawing
by means of centre lines as shown in figure 4. Figure 5 shows
the FV, TV, and RHSV of an object showing visiblke edges, hidden
edges (or holes), and centre lines.
Figure 2 shows the pictorial view and front view of (a) a hollow cylindrical object and (b) solid cylindrical object.
Figure 3 shows the pictorial view and front view of sectioned part of (a) a hollow cylindrical object (b) solid cylindrical object and (c) solid cylinder split in to two unequal parts.
Figure 4 shows the centre lines for cylindrical objects
Figure 5. Showing TV, FV and RHSV of an object showing the three types of lines mentioned above. The pictorial view of the object is shown at the top hight hand side.
Conventions used for lines
In orthographic projections, many times different types of lines may fall at the same regions. In such cases, the following rules for precedence of lines are to be followed:
When a visible line and a hidden line are to be drawn at the same area, It will be shown by the visible line only and no hidden line will be shown. Similarly, in case of hidden line and centre line, onlu hidden line will be shown. In such case, the centre line will be shown only if it is extending beyond the length of the hidden line.
Intersecting Lines in Orthographic Projections
The conventions used when different lines intersect is shown in figure - 6(a) & (b).
Figure 6(a): The conventions practiced for intersection lines.
Figure 6(b): The conventions practiced for intersection lines.
Some ortho graphic projections of
solids showing the different lines and their precedence are
shown as examples below. The 3-D view of the respective objects are
also shown in the figures with the direction of arrow representing
the line of sight in the front view.
A few examples of the projections
showing the conventions in drawing are presented below.
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8 (application of Precedence rule)
Example 9 (Objects with circular features : holes, flanges, etc )
Projections of points
Projection of Points
A
POINT
The position of a point in
engineering drawing is defined with respect to its distance from
the three principle planes i.e., with respect to the VP, HP, &
PP.
The point is assumed to be in the
respective quadrant shown in figure 1(a). The point at which
the line of sight (line of sight is normal to the respective plane
of projection) intersects the three planes are
obtained. The horizontal plane and the side planes are
rotated so such that they lie on the plane containing the vertical
plane. The direction of rotation of the horizontal plane is shown
in figure 1 (b).
Figure 1(a). The relative positions of projection planes and the quadrants
Figure 1(b). The direction of rotation of the Horizontal plane.
Conventions used while drawing the projections of points
With respect to the 1^{st} angle projection of point “P’ shown in figure 2,
Figure 2. Showing the three planes and the projectionof the point P after the planes have been rotated on to the vertical plane.
Point in the First quadrant
Figure 3 shown the projections of a point P which is 40 mm in front of VP, 50 mm above HP, 30 mm in front of left profile plane (PP)
Figure 3. Projection of the point “P” on to the three projection planes before the planes are rotated.
Figure 4 shows the planes and the position of the points when the planes are partially rotated. The arrows indicate the direction of rotation of the planes. The three views after complete rotation of the planes is shown in figure 2.
Figure 4. Projection of the point “P” on to the three projection planes after the planes are partially rotated.
The procedure of drawing the three views of the point “P” is shown in figure-4.
Figure 5 First angle multi-view drawing of the point “P”
Projections of points in 2nd, 3rd and 4th quadrant
Point in the Second quadrant
Point P is 30 mm above HP, 50 mm behind VP and 45 mm in front of left PP. Since point P is located behind VP, the VP is assumed transparent. The position of the point w.r.t the three planes are shown in Figure 1. The direction of viewing are shown by arrows. After projecting the point on to the three planes, the HP and PP are rotated such that they lie along the VP. The direction of rotation of the HP and PP is shown in figure 2. As shown in figure 3, after rotation of the PP and HP, it is found that the VP and HP is overlapping. The multiview drawing for the point P lying in the second quadrant is shown in figure 4. Though for the projection of a single point, this may not be a problem, the multiview drawing of solids, where a number of lines are to be drawn, will be very complicated. Hence second angle projection technic is not followed anywhere for engineering drawing.
Figure 1. The projection of point P on to the three projection planes.
Figure 2. The direction of rotation of HP.
Figure 3. The projection of point P after complete rotation of the HP and PP.
Figure 4. The multiview drawing of the point P lying in the second quadrant.
Point in the Third quadrant
Projection of a point P in the third quadrant where P is 40 mm behind VP, 50 mm below HP and 30 mm behind the right PP is shown in figure 5.
Since the three planes of
projections lie in between the observer and the point P, they are
assumed as transparent planes. After the point P is projected
on to the three planes, the HP and VP are rotated along the
direction shown in figure 6, such that the HP and PP is in plane
with the VP. The orthographic projection of the point P lying in
the third quadrant is shown in figure 7.
Figure 5. Projection of a point P placed in the third quadrant
In the third angle projection, the Top view is always above the front view and the Right side view will be towards the right of the Front view.
Figure 6. shows the sense of direction of rotation of PP and HP.
Figure 7. Multi-view drawing of the point lying in the third quadrant.
In the third angle projection, the Top view is always above the front view and the Right side view will be towards the right of the Front view.
Point in the Fourth quadrant
If A point is lying in the fourth
quadrant, the point will be below the HP and infront of the VP. The
point is projected on to the respective projection planes. After
rotation of the HP and PP on to the VP, it will be observed that
that the HP and VP are overlapping, similar to the second angle
projection. The multi-view drawing of objects in such case
would be very confusing and hence fourth angle projection technique
is not followed by engineers
Projection of lines
Projections of lines
Straight line
A line is a geometric primitive that has length and direction, but no thickness. Straight line is the Locus of a point, which moves linearly. Straight line is also the shortest distance between any two given points.
The location of a line in projection quadrants is described by specifying the distances of its end points from the VP, HP and PP. A line may be:
Projection of a line
The projection of a line can be obtained by projecting its end points on planes of projections and then connecting the points of projections. The projected length and inclination of a line, can be different compared to its true length and inclination.
Case 1. Line parallel to a plane
When a line is parallel to a plane, the projection of the line on to that plane will be its true length. The projection of line AB lying parallel to the Vertical plane (VP) is shown in figure 1 as a’b’.
Figure 1. Projection of line on VP. Line AB is parallel to VP.
Case 2. Line inclined to a plane
When a line is parallel to one plane and inclined to the other, The projection of the line on the plane to which it is parallel will show its true length. The projected length on the plane to which it is inclined will always be shorter than the true length. In figure 2, the line AB is parallel to VP and is inclined to HP. The angle of inclination of AB with HP is being θ degrees. Projection of line AB on VP is a’b’ and is the true length of AB. The projection of line AB on HP is indicated as line ab. Length ab is shorter than the true length AB of the line.
Figure 2. Projection of line AB parallel to VO and inclined to HP.
Case 3. Projection of a line parallel to both HP and VP
A line AB having length 80 mm is parallel to both HP and VP. The line is 70 mm above HP, 60 mm in front of VP. End B is 30 mm in front of right PP. To draw the projection of line AB, assume the line in the first quadrant. The projection points of AB on the vertical plane VP, horizontal plane HP and Right Profile plane PP is shown in figure 3(a). Since the line is parallel to both HP and VP, both the front view a'b' and the top view ab are in true lengths. Since the line is perpendicular to the right PP, the left side view of the line will be a point a΄΄(b΄΄). After projection on to the projection planes, the planes are rotated such that all the three projection planes lie in the same planes. The multi-view drawing of line AB is shown in Figure 3(b).
Figure 3. Projection of line parallel to both HP and VP.
Case 4. Line perpendicular to HP & parallel to VP
A line AB of length 80 mm is parallel to VP and perpendicular to HP. The line is 80 mm in front of VP and 80 mm in front of right PP. The lower end of the line is 30 mm above HP. The projections of line AB shown in figure 4 can be obtained by the following method.
Draw a line XY which is the
intersection between VP and HP. Draw the front view a'b' = 80 mm
perpendicular to the XY line, with the lower end b' lying 30 mm
above the XY line. Project the top view of the line which
will be a point a(b) at a distance of 60 mm below XY line.
Since the line is 70 mm in front of the right PP draw the
X_{1}Y_{1} line at a distance of 70 mm on the
right- side of the front view.
Through O the point of intersection
of XY and X_{1}Y_{1}, lines draw a 45° line.
Draw the horizontal projector through a(b) to cut the 45
degree line at m. Draw the horizontal projectors through a'
and b' to intersect the vertical projector drawn through m at
a΄΄ and b΄΄.
a΄΄b΄΄ is the left view of the line
AB.
Figure 4. Projections of a line AB perpendicular to HP and parallel to VP.
Line parallel to one plane and inclined to the other
Case 5. Line parallel to VP and inclined to HP
A line AB, 90 mm long is inclined at 30° to HP and parallel to VP. The line is 80 mm in front of VP. The lower end A is 30 mm above HP. The upper end B is 50 mm in front of the right PP. The projections of line AB shown in figure 5 can be obtained in the following manner. Mark a', the front view of the end A, 30 mm above HP. Draw the front view a΄b΄ = 90 mm inclined at 30° to XY line.
Project the top
view ab parallel to XY line. The top
view is 80 mm in front of VP. Draw the
X_{1}Y_{1}line at a distance of 50 mm
from b'. Draw a 45° line
through O. Draw the horizontal projector
through the top view ab to cut the 45 °
line at m. Draw a vertical projector
through m. Draw the horizontal
projectors
through a' and b' to
intersect the vertical projector drawn through m
ata” and b”.
Connect a΄΄ b΄΄ which is the left side
view.
(b)
Figure 5. Projections of line AB parallel to VP and inclined to HP.
Projection of lines inclined to HP and VP
Case
6. Line inclined to HP and
VP
When a line is inclined to both HP
and VP, the apparent inclination of the line to
both the projection planes will be different from the actual
inclinations. Similarly the projected length of the lines on to the
planes will not be the same as the true length f the line. The
following notation will be used for the inclinations and length of
the lines for this entire lecture series:
Actual inclinations are θ degrees to
HP and φ degrees to VP.
Apparent Inclinations are a and b to
HP and VP respectively.
The Apparent Lengths of line AB are
ab and a΄b΄in the top view and front view respectively.
Example: Draw the projections of a line AB inclined to both HP and VP, whose true length and true inclinations and locations of one of the end points, say A are given.
The projections of the line AB are illustrated in figure 1. Since the line AB is inclined at θ to HP and φ to VP – its top view ab and the front view a΄b΄ are not in true lengths and they are also not inclined at angles θ to HP and φ to VP in the Front view and top view respectively. Figure 2 illustrates the projections of the line AB when the line is rotated about A and made parallel to VP and HP respectively. A clear understanding of these can be understood if the procedure followed in the subsequent sub-sections are followed:
Figure 1: The projections of a line inclined to both HP and VP
Step 1: Rotate the line AB to make it parallel to VP.
Rotate the line AB about the end A, keeping θ, the inclination of AB with HP constant till it becomes parallel to VP. This rotation of the line will bring the end B to the new position B1. AB_{1}is the new position of the line AB when it is inclined at q to HP and parallel to VP. Project AB_{1} on VP and HP. Since AB_{1} is parallel to VP, a΄b_{1}΄, the projection of AB1 on VP is in true length inclined at q to the XY line, and ab1, the projection of AB_{1} on HP is parallel to the XY line. Now the line is rotated back to its original position AB.
Figure 2. Illustrates the locus of end B of the line AB when the line is rotated about end A
Step 2: Rotate the line AB to make it
parallel to HP.
Rotate the line AB about the end A
keeping φ the inclination of AB with VP constant, till it becomes
parallel to HP as shown in figure 2. This rotation of
the line will bring the end B to the second new Position
B_{2}. AB_{2} is the new position of the line
AB, when it is inclined at f to VP and parallel to
HP.
Project AB_{2} on HP
and VP. Since AB_{2} is parallel to HP,
ab_{2}, the projection of AB_{2} on HP is in
true length inclined at f to XY line, and a΄b_{2}΄ the
projection of AB_{2} on VP is parallel to XY line. Now
the line is rotated back to its original position AB.
Step 3: Locus of end B in the front view
Referring to figure 2, when the line AB is swept around about the end A by one complete rotation, while keeping θ the inclination of the line with the HP constant, the end B will always be at the same vertical height above HP, and the locus of the end B will be a circle which appears in the front view as a horizontal line passing through b'.
As long as the line is inclined at θ to HP, whatever may be the position of the line (i.e., whatever may be the inclination of the line with VP) the length of the top view will always be equal to ab1 and in the front view the projection of the end B lies on the locus line passing through b1’.
Thus ab_{1}, the top view of the line when it is inclined at θ to HP and parallel to VP will be equal to ab and b΄, the projection of the end B in the front view will lie on the locus line passing through b_{1}΄.
Step 4: Locus of end B in the top view
It is evident from figure 2, that when the line AB is swept around about the end A by one complete rotation, keeping f the inclination of the line with the VP constant, the end B will always be at the same distance in front of VP and the locus of the end B will be a circle which appears in the top view as a line, parallel to XY, passing through b.
As long as the line is inclined at φ to VP, whatever may be the position of the line (i.e., whatever may be the inclination of the line with HP), the length of the front view will always be equal to a'b_{2}' and in the top view the projection of the end B lies on the locus line passing through b_{2}.
Thus a΄b_{2}΄ the front view of the line when it is inclined at f to VP and parallel to HP, will be equal to a'b' and also b, the projection of the end B in the top view lies on the locus line passing through b_{2}.
Step 5: To obtain the top and front views of AB
From the above two cases of rotation it can be said that
(i)the length of the line AB in top and front views will be equal to ab_{1} and a'b_{2}' respectively and
(ii) The projections of the end
B, (i.e., b and b‘) should lie along the locus line passing through
b_{2}and b_{1}΄ respectively.
With center a, and radius
ab_{2} draw an arc to intersect the locus line through
b_{2} at b. Connect ab the top view of the line
AB.
Similarly with center a', and radius
a'b_{2}' draw an arc to intersect the locus line through
b_{1}' at b΄. Connect a'b' the front view of the line
AB.
Orthographic
projections
As the location of one of the end
points (i.e. A) with respect to HP and VP, is given,
mark a anda΄, the top and the front views of point
A.
If the line AB is assumed to be made
parallel to VP and inclined at θ to HP. The front view of the
line will be equal to the true length and true
inclination of the line with HP. Draw a'b_{1}'
passing through a' at θ to XY line and equal to the true length of
AB. a'b_{1}' is projected down to get ab_{1}, the
top view parallel to the XY line. This is illustrated in
figure 3.
Figure 3. Illustrates the true length and true inclination of the line when it is made parallel to VP.
Now the line AB is assumed to be made parallel to HP and inclined at φ to VP. This is shown in figure 4. The top view of the line will be equal to the true length of the line and also φ, the inclination of the line with VP is seen in the top view. For this, draw ab_{2} passing through a and incline at φ to the XY line. The length ab_{2} is equal to the true length of AB. The end points a and b_{2} are projected on to a line parallel to XY line and passing through a’ to get a'b_{2}' which is the front view of the line when it is parallel to HP and inclined to VP. Draw the horizontal locus lines through b_{2}, and b_{1}'. With center a and radius ab1, draw an arc to cut the locus line drawn through b_{2} at b. Connect ab, the top view of the line AB. With center a' and radius a'b_{2}΄, draw an arc to cut the locus line drawn through b_{1}' at b'. Connect a'b', the front view of the line AB. Orthographic projections of line AB inclined to both VP and HP, illustrating the projected length, true lengths apparent inclinations and true inclinations are shown in figure 5.
Figure 4. Illustrates the true length and true inclination of the line when it is made parallel to HP.
Figure 5. Illustrates the true length, apparent lengths, tue inclination and apparent inclination of the line AB inclined to HP and VP..
Projections of line: True length, True inclinations, Traces of lines
To Find True length and
true inclinations of a
line
Many times if
the top and front views of a line are given,
the true length and true inclinations of a line is required to be
determined.
The top and front views of the
object can be drawn from if any of the following data are
available:
(a) Distance between the end
projectors,
(b) Distance of one or both the end
points from HP and VP and
(c) Apparent inclinations of the
line.
The problems may be solved
by
(i) Rotating line method
or
(ii) Rotating trapezoidal plane
method or
(iii) Auxiliary plane
method.
Rotating line method
The method of obtaining the top
and front views of a line, when its true length and true
inclinations are given.
When a view of a line is parallel to
the XY line, its other view will be in true length and at true
inclination.
By following the procedure mentioned
previously, in the reverse order, the true length and true
inclinations of a line from the given set of top and front views
can be found. The step by step procedure is shown below in
figure 1.
Figure 1. determinationof ture length and true inclinations of a line.
Traces of a line
Trace of a line
perpendicular to one plane and parallel to the
other
Since the line is perpendicular to
one plane and parallel to the other, the trace of the line is
obtained only on the plane to which it is perpendicular, and no
trace of the line is obtained on the other plane to which it is
parallel. Figures 2 and 3 illustrates the trace of a line
parallel tp0VP and perpendicular to HP and parallel to HP and
perpendicular to VP respectively.
Figure 2. Trace of line parallel to VP and perpendicular to HP
Figure 3. Trace of a line perpendicular to the VP and parallel to HP
Traces of a line inclined
to one plane and parallel to the
other
When the line is inclined to one
plane and parallel to the other, the trace of the line is obtained
only on the plane to which it is inclined, and no trace
is obtained on the plane to which it is parallel.
Figure 4 shows the horizontal trace of line AB which is in lined HP
and parallel to VP
Figure 4 Horizontal trace of line AB
Figure 5 shows the vertical trace of line AB which is inclined to VP and parallel to HP
Figure 5 Vertical trace of line AB
Traces of a line inclined
to both the planes
Figure 6 shows the Vertical trace
(V) and Horizontal Trace (H) of Line AB inclined at q to HP
and Φ to VP.
The line when extended intersects HP
at H, the horizontal trace, but will never intersect the portion of
VP above XY line, i.e. within the portion of the VP in the
1^{st} quadrant. Therefore VP is extended below HP
such that when the line AB is produced it will intersect in the
extended portion of VP at V, the vertical trace.
In this case both horizontal trace
(H) and Vertical Trace (V) of the line AB lie below XY
line.
Figure 6 Vertical trace and horizontal trace of line AB which is inclined to both vertical plane and horizontal plane.
Auxiliary projection Technique
Projections on Auxiliary
Planes
Sometimes none of the three
principal orthographic views of an object show the different edges
and faces of an object in their true sizes, since these edges and
faces, are not parallel to any one of the three principal planes of
projection. In order to show such edges and faces in their
true sizes, it becomes necessary to set up additional
planes of projection other than the three principal
planes of projection in the positions which will show them in true
sizes. If an edge or a face is to be shown in true size, it
should be parallel to the plane of projection. Hence the
additional planes are set up so as to be parallel to the edges and
faces which should be shown in true sizes. These additional
planes of projection which are set up to obtain the true sizes are
called Auxiliary Planes. The views projected on
these auxiliary planes are called Auxiliary
Views.
The auxiliary view method may be applied
Types of auxiliary
planes
Usually the auxiliary planes are set
up such that they are parallel to the edge or face which is to be
shown in true size and perpendicular to any one of the three
principal planes of projection. Therefore, the selection of
the auxiliary plane as to which of the principal planes of
projection it should be perpendicular, obviously depends on the
shape of the object whose edge or face that is to be shown in true
size.
Auxiliary Vertical Plane
(AVP)
An AVP is placed in the first
quadrant with its surface perpendicular to HP and inclined at Φ to
VP. The object is assumed to be placed in the space in between HP,
VP and AVP. The AVP intersects HP along the
X_{1}Y_{1} line. The direction of sight
to project the auxiliary front view will be normal to AVP.
The position of the auxiliary vertical plane w.r.t HP
and VP is shown in figure 1.
After obtaining the top view, front view and auxiliary front view on HP, VP and AVP, the HP, with the AVP being held perpendicular to it, is rotated so as to be in-plane with that of VP, and then the AVP is rotated about the X_{1}Y_{1} line so as to be in plane with that of already rotated HP.
Figure 1. The position of the auxiliary vertical plane w.r.t HP and VP
Auxiliary Inclined Plane (AIP)
AIP is placed in the first
quadrant with its surface perpendicular to VP and inclined at q to
HP. The object is to be placed in the space between HP, VP and AIP.
The AIP intersects the VP along the X1Y1 line.
The direction of sight to project the auxiliary top view will be
normal to the AIP. The position of the AIP w.r.t HP and VP is
shown in figure 2.
After obtaining the top view, front
view and auxiliary top view on HP, VP and AIP,
HP is rotated about the XY line independently (detaching
the AIP from HP). The AIP is then rotated about X1Y1 line
independently so as to be in-planewith that of VP.
Figure 2. The position of the AIP w.r.t HP and VP
Projection of Points on Auxiliary Planes
Projection on Auxiliary Vertical Plane
Point P is situated in the first
quadrant at a height m above HP. An auxiliary vertical
plane AVP is set up perpendicular to HP and
inclined at Φ to VP. The point P is projected
on VP,
HP and AVP.
As shown in figure
3, p' is the projection on VP,
p is the projection on HP and P_{1}' is the
projection on AVP.
Since point is at a height m above
HP, both p' and p_{1}’ are at a height m above the XY and
X_{1}Y_{1}lines, respectively
Figure 3. Projection of Point P on VP, HP and AVP
HP is rotated by 90 degree to bring it in plane of VP (figure 4(a) . Subsequently, the AVP is rotated about the X_{1}Y_{1} line (figure 4(b), such that it becomes in-plane with that of both HP andVP.
Figure 4. The rotation of (a) HP and (b) AVP to make HP and AVP in plane with VP.
The orthographic projections (projections of point P on HP, VP and AVP) of point P can be obtained be the following steps.
Draw the XY line and mark p and p', the top and front views of the point P. Since AVP is inclined at Φ to VP, draw the X_{1}Y_{1} line inclined at Φ to the XY line at any convenient distance from p. Since point P is at a height m above HP, the auxiliary front view p_{1}' will also be at a height m above the X_{1}Y_{1} line. Therefore, mark P_{1}’ by measuring o_{1}p_{1}’=op’ = m on the projector drawn from p perpendicular to the X_{1}Y_{1} line.
Figure 5. Orthographic projection of the point P by Auxiliary projection method.
Projection on Auxiliary Inclined Plane
Point P is situated in first quadrant at a distance n from VP. An auxiliary planeAIP is set up perpendicular to VP and inclined at θ to HP. The point P is projected on VP, HP and AIP.
p' is the
projection on VP, p is the projection on HP and
P_{1} is the projection
on AIP.
Since the point is at a distance n
from VP, both p and p_{1} are at a distance n above
the XY and X_{1}Y_{1} lines,
respectively
Figure 6. Orthographic projection of point P by auxiliary projection on AIP.
HP is now rotated by 90°
about XY line to bring it in plane with VP, as shown in
figure 7(a). After the HP lies in-plane with VP,
the AIP is rotated about the
X_{1}Y_{1}, line, so that it becomes in-plane with
that of both HP and VP.
p and p’ lie on a vertical projector
perpendicular to the XY line, and p’ and p_{1} lie on
a projector perpendicular to the
X_{1}Y_{1} line which itself is inclined
at θ to XY line.
Figure 7. Orthographic projection of point P by auxiliary projection on AIP.
The orthographic projections (projections of point P on HP, VP and AIP) of point P can be obtained be the following steps.
Step by step procedure to draw auxiliary views
Auxiliary front view |
Auxiliary top view |
Draw the top and front views. |
Draw the top and front views |
Draw X_{1}Y_{1} line inclined at f (the inclination of AVP with VP) to the XY line. |
Draw X_{1}Y_{1} line inclined at q (the inclination of AIP with HP) to XY line. |
Draw the projectors through the top views of the points perpendicular to the X_{1}Y_{1} line. |
Draw the projectors through the front views of the points perpendicular to the X_{1}Y_{1} line. |
The auxiliary front view of a point is obtained by stepping off a distance from the X_{1}Y_{1} line equal to the distance of the front view of the given point from the XY line. |
The auxiliary top view of a point is obtained by stepping off a distance from X_{1}Y_{1} line equal to the distance of the top view of the given point from the XY line |
Examples on Projections by Auxiliary plane method
Projection of lines by auxiliary plane method
The problems on projection of
lines inclined to both the planes may also be solved by the
auxiliary plane methods.
In this method, the line is always
placed parallel to both HP and VP, and then two auxiliary planes
are set up: one auxiliary plane will be perpendicular to VP and
inclined at q to HP, i.e., AIP, and the other will be perpendicular
to HP and inclined at f (true inclination) or b (apparent
inclination) to VP. Projection of lines by auxiliary plane
method are illustrated by problems shown below:
Problem
1:
Draw the projections of a line 80 mm
long inclined at 300 to HP and its top view appears to be inclined
at 600 to VP. One of the ends of the line is 45 mm above HP and 60
mm in front of VP. Draw its projections by auxiliary plane
method
Solution
Draw the top and front views of one
of the ends, say A, 45 mm above HP and 60 mm in front of
VP.
Assume that the line is parallel to
both HP and VP and draw its top and front views as shown in
figure 1.
Figure 1. The FV and TV of the line AB when parallel to HP and VP
Since the line is to be inclined
at 30^{0} to HP, set up an AIP inclined at
30^{0} to HP and perpendicular to
VP.
Draw
X_{1}Y_{1} line inclined at
30^{0} to XY line at any convenient distance from it
as shown in figure 2.
To project an auxiliary top view on
AIP, draw projections from a_{1}’ and b_{1}’
perpendicular to X_{1}Y_{1}line, and on them step
off 1a_{1}=3a and 2b_{1}=4b from the
X_{1}Y_{1} line.
Connect ab which will be the
auxiliary top view.
Figure 2. Projection of line on to the AIP.
Since the top view of the line
appears inclined to VP at 60^{0}, draw the
X_{2}Y_{2} line inclined at
60^{0} to the auxiliary
top view ab at any convenient
distance from it as shown in figure 3. Draw the projections
from a and b perpendicular to
X_{2}Y_{2} and
on them step off 5a’ =
3a_{1}’and 6b’=4b_{1}’. Connect a’b’
which will be the auxiliary front view.
Figure 3. Auxiliary front view of the the line Ab.
Problem 2
A line AB 60 mm
long has one of its extremities 60 mm infront of VP and 45 mm above
HP.
The line is inclined at
30^{0} to HP and 45^{0} to VP. Draw the
projections of the line by the auxiliary plane
method.
Solution. The solution is
shown in figure . The method of obtaining the projections is
described below.
Let A be one of
the extremities of the line AB at distance 60 mm in front of
VP and 45 mm above HP.
Mark a_{1} and
a_{1}’ the top and the front views of the extremity
A.
Initially the line is assumed to be
parallel to HP and VP.
a_{1}b_{1} and
a_{1}’ b_{1}’ are the projections of the line in
this position.
Then instead of rotating the line so
as to make it inclined to both the planes, an AIP is set up at an
angle θ, which the line is supposed to make with HP and the
auxiliary top view is projected on it.
To draw the Auxiliary Top View on
AIP
Draw
X_{1}Y_{1} line inclined at θ =
30^{0} to the XY line. Mark AIP and VP. Project the
auxiliary top view ab The projections ab on the AIP and
a_{1}'b_{1}' on VP are the auxiliary view and
the front view of the line when it is inclined at θ to HP and
parallel to VP.
Since the line is inclined at true
inclination Φ to VP, to project the auxiliary front view an AVP
inclined at Φ to VP should be setup.
Figure 4. Projection of line AB (problem 2) by auxiliary projection mentod.
To draw the Auxiliary F.V. on AVP
Already the line is inclined at θ
to AIP and parallel to VP. If the line is to be
inclined at Φ to VP, anAVP inclined at Φ to the given
line should be setup. But we know that when a
line is inclined to both the planes, they will not be inclined at
true inclinations to the XY line, instead they
will be at apparent inclinations with the XY line. Therefore
X_{2}Y_{2}, the line of intersection
of AIP and AVP cannot be drawn directly at Φ
to ab.
The apparent inclination bof
abwith the X_{2}Y_{2} line should be
found out. To find b, through a draw ab_{2} equal to
60 mm, the true length of AB inclined at Φ =
45^{0} to ab.
Through b_{2}, draw the
locus of B parallel to
X_{1}Y_{1} line. With
center aandradius ab strike an arc to
intersect the locus of B at b_{3}.
Connect ab_{3} and
measure its inclination b with ab. Now draw
the X_{2}Y_{2 }line inclined
at b to ab. Mark AVP and AIP on either
side of X_{2}Y_{2}.
Project the auxiliary front view
a’b’. ab and a’b’ are the required
projections.
Shortest distance between
two lines
Two lines may be parallel, or
intersecting, or non-parallel and
non-intersecting.
When the lines are
intersecting, the point of intersection lies on both the lines and
hence these lines have no shortest distance between
them.
Non-parallel and non-intersecting
lines are called Skew Lines.
The parallel lines and the skew
lines have a shortest distance between
them.
The shortest distance between the
two lines is the shortest perpendicular drawn between the two
lines.
Shortest distance between two parallel lines
The shortest distance between two
parallel lines is equal to the length of the perpendicular drawn
between them.
If its true length is to be
measured, then the two given parallel lines should be shown in
their point views.
If the point views of the lines are
required, then first they have to be shown in their true lengths in
one of the orthographic views.
If none of the orthographic
views show the given lines in their true lengths, an auxiliary
plane parallel to the two given lines should be set up to project
them in their true lengths on it.
Even the auxiliary view which shows
the lines in their true lengths may not show the perpendicular
distance between them in true length. Hence another auxiliary plane
perpendicular to the two given lines should be set up. Then the
lines appear as points on this auxiliary plane and the distance
between these point views will be the shortest distance between
them.
Shortest distance between two parallel lines
Problem:
3
Projections of a pair of parallel lines AB and PQ are shown in figure 5. ab and a'b' are the top and front views of the line AB. pq and p'q' are the top and front views of the line PQ. Determine the Shortest distance between the two lines.
Figure 5. The projections of lines AB and PQ for problem 3.
Solution:
Since the top and front views of the
lines are inclined to the XY line, neither the top view nor the
front view show the lines in their true lengths. To show
these lines in their true lengths, an auxiliary plane, parallel to
the two given lines, should be set up parallel to the projections
of the lines either in the top view or front view. In this
case the auxiliary plane is set up so as to be parallel to the two
given lines in top view. The method of determining the shortest
distance between the two lines is shown in figure
6.
Figure 6. Determination of shortest distance between two parallel lines.
Draw the
X_{1}Y_{1} line parallel to ab and pq at any
convenient distance from them.
Through the points a, b, p and q,
draw projector lines perpendicular to
X_{1}Y_{1} line.
Measure
5a_{1}’=1a_{1}’ along the projector drawn through a
from the X_{1}Y_{1} line, and
6b_{1}’=2b’ along the projector drawn through b from the
X_{1}Y_{1} line.
Connect a_{1}'b_{1}'
which will be equal to the true length of the line
AB.
Similarly by measuring
7p_{1}'= 3p' and 8q_{1}'
= 4q' obtain p_{1}'q_{1}' the true length
view of the line PQ.
The line AB and PQare shown in their
true lengths, and now an another auxiliary plane perpendicular to
the two given lines should be set up to project their point views
on it.
Draw the line
X_{2}Y_{2} perpendicular
to a_{1}’b_{1}'
and p_{1}'q_{1}' at any convenient distance from
them.
Produce a_{1}'b_{1}'
and p_{1}'q_{1}'.
Measure a5 = b6 =
9a_{1}along a_{1}'b_{1}' produced from
X_{2}Y_{2}. Similarly obtain the point,view
p_{1}(q_{1}) by measuring
p_{1}(10)=p_{7} = q8.
Connect p_{1}a_{1} the
required shortest distance between the lines AB and PQ in its true
length .
Shortest distance between two skew lines
Projections of two skew lines AB and CD are shown as A’B’, C’D’ and AB and CD.
Determine the shortest distance EF between the line segments
First an Auxiliary
A_{1}B_{1} is made showing the true length of
AB.
A second auxiliary view showing the
point view of AB is projected.
For this draw the reference line
normal to A_{1}B_{1} and draw the projectors
C_{2} D_{2} (of C_{1} and
D_{1}).
The shortest distance
F_{2}E_{2} can be established perpendicular to
CD.
To project FE back to the Front and
Top Views, FE is first projected in first auxiliary plane by first
projecting point E, which is on CD, from the second to the first
auxiliary view and then back to the front and top views.
Projections of planes
Projections of Planes
Plane surface (plane/lamina/plate)
A plane is as two dimensional surface having length and breadth with negligible thickness. They are formed when any three non-collinear points are joined. Planes are bounded by straight/curved lines and may be either regular or an irregular. Regular plane surface are in which all the sides are equal. Irregular plane surface are in which the lengths of the sides are unequal.
Positioning of a Plane surface
A plane surface may be positioned in space with reference to the three principal planes of projection in any of the following positions:
Projections of a Plane surface
A plane surface when held
parallel to a plane of projection, it will be perpendicular to the
other two planes of projection. The view of the plane surface
projected on the plane of projection to which it will be
perpendicular will be a line, called the line view of a plane
surface. When the plane surface is held with its surface parallel
to one of the planes of projection, the view of the plane surface
projected on it will be in true shape because all the sides or the
edges of the plane surface will be parallel to the plane of
projection on which the plane surface is
projected.
When a plane surface is inclined to
any plane of projection, the view of the plane surface projected on
it will be its apparent shape.
A few examples of projections of
plane surfaces are illustrated below:
A: Plane surface parallel to one plane and perpendicular to the other two
Consider A triangular lamina
placed in the first quadrant with its surface parallel to VP
and perpendicular to both HP and left PP. The lamina and its
projections on the three projection planes are shown in figure
1.
a'b'c' is the
front view, abc the top view
and a’’b’’c’’ the side
view
Since the plane is parallel
to VP , the front view a'b'c'
shows the true shape of the lamina. Since the lamina is
perpendicular to both HP and PP, the top view and side views
are seen as lines.
Figure 1. Projections of a triangular lamina on the projection planes
After projecting the triangular lamina on VP, HP and PP, both HP and PP are rotated about XY and X_{1}Y_{1} lines, as shown in figure 2, till they lie in-plane with that of VP
Figure 2. Rotation of PP and HP after projection.
The orthographic projections of the plane, shown in figure 3 can be obtained be the following steps.
Draw XY and X_{1}Y_{1} lines and mark HP, VP and left PP. Draw the triangle a'b'c' in true shape to represent the front view at any convenient distance above the XY line. In the top view the triangular lamina appears as a lineparallel to the XY line. Obtain the top view acb as a line by projecting from the front view at any convenient distance below the XY line.
Since the triangular lamina is
also perpendicular to left PP, the right view will be
a line parallel to the
X_{1}Y_{1} line. To project the right view,
draw a 45° line at the point of intersection of the XY and
X_{1}Y_{1} lines.
Draw the horizontal projector
through the corner a in the top view to cut the
45° line at m. Through m draw a vertical projector. From the
corners c' and a' in the front view draw the
horizontal projectors to cut the vertical projector drawn through m
at c’’ and b’’. In the right view the corner A coincides with B and
hence is invisible.
Figure 3. Orthographic projections of the lamina ABC
B) Plane parallel to HP
and perpendicular to both VP and PP
A square lamina (plane surface) is
placed in the first quadrant with its surface
parallel to HP and perpendicular to both VP and left PP.
Figure 4 (a) shows the views of the object when projected on to the
three planes. Top view is shown as abcd, the
front view as a’(d’)b’(c’) and the side view as
b”(a”)c”(d”). Since the plane is parallel to the HP,
its top view abcd will be in its true
shape. Since the plane is perpendicular to VP and PP, its front and
side views will be
linesa’(d’)b’(c’) and b”(a”)c”(d”) respectively.
After projecting the square lamina
on VP, HP and PP, both HP and PP are rotated about XY and
X_{1}Y_{1} lines , as shown in figure 4(b) ,
till they lie in-plane with that of VP.
Figure 4. Projections of the lamina with its surface parallel to HO and perpendicular to both VP and PP.
The orthographic projections of
the plane, shown in figure 4(c) can be obtained be the following
steps.
Draw XY and X1Y1 lines and mark HP,
VP and left PP.
Draw the
square abcd in true shape to represent the top
view at any convenient distance below the XY
line.
In the front view, the square lamina
appears as a line parallel to the XY line. Obtain the
front view as a line a'(d')b'(c') by projecting
from the top view, parallel to the XY line at any convenient
distance above it. In the front view, the rear corners D and C
coincide with the front corners A and B,
hence d' and c' are indicated
within brackets.
Since the square lamina is also
perpendicular to left PP, the right view projected on it will also
be a line perpendicular to X_{1}Y_{1} line.
Project the right view as explained in the previous case. In right
view, the corners A and D coincide with the corners B and C
respectively, hence (a') and (d'), are indicated within
brackets.
C) Plane parallel to PP
and perpendicular to both HP and VP
A pentagon lamina (plane surface) is
placed in the first quadrant with its surface is parallel to
left PP and perpendicular to both VP and HP.
Figure 5 (a) shows the views of the
object when projected on to the three planes. Side view is shown
as a”b”c’’d”e”, the front view as
b’(c’)a’(d’)e’ and the top view as
a(b)e(c)d .Since the plane is parallel
to the PP, its side view a”b”c’’d”e” will be in
its true shape. Since the plane is perpendicular to VP and HP, its
front and side views will be projected as lines.
After projecting the pentagon lamina
on VP, HP and PP, both HP and PP are rotated about XY and
X_{1}Y_{1} lines , as shown in figure 5(b),
till they lie in-plane with that of VP.
Figure 5 Projections of a pentagonal lamina with its surface parallel to PP and perpendicular to HP and VP.
The orthographic projections of
the plane, shown in figure 5(c) can be obtained be the following
steps. Draw XY and X1Y1 lines, and mark HP,
VP and left PP .Draw the
pentagon a”b”c”d”e” in true shape to represent
the side view at any convenient distance above the XY line and left
of X_{1}Y_{1} line. The top and front views of
the lamina appear as lines perpendicular to XY
line.
Obtain the front
view b’(c’)a’(d’)e’ as a line by projecting from
the right view at any convenient distance from the X1Y1 line. In
the front view, the rear corners D and C coincide with A and B
respectively, hence d’ and c’ are indicated within brackets. The
orthographic projections of the plane, shown in figure 4(c) can be
obtained be the following step. Since the pentagon lamina is also
perpendicular to HP, the top view also appears as a line. Project
the top view from the right and front views.
D) Plane surface perpendicular to one plane and inclined to the other two
Draw the projections of a triangular lamina (plane surface) placed in the first quadrant with its surface is inclined at f to VP and perpendicular to the HP.
Since the lamina is inclined to
VP, it is also inclined to left PP at (90 - Φ).
The triangular lamina ABC is
projected onto VP, HP and left PP.
a’b’c’ – is the front
view projected on on VP.
a”b”c” – is the right
view projected on left PP.
Since lamina is inclined to VP and
PP, front and side views are not in true shape.
Since lamina is perpendicular to HP,
its top view is projected as a line acb
Figure 6 (c) shows the multiview
drawing of the lamina.
Figure 6. The projections of the triangular lamina
Examples on projections of planes
Problem 1:A regular pentagon lamina of 30 mm side rests on HP with its plane surface vertical and inclined at 30^{0} to VP. Draw its top and front views when one of its sides is perpendicular to HP.
Solution: The projections The pentagonal lamina has its surface vertical (i.e., perpendicular to HP) and inclined at 30^{0} t oVP.Since the lamina is inclined to VP, initially it is assumed to be parallel to VP. In this position one of the sides of the pentagon should be perpendicular to HP. Therefore, draw a regular pentagon a'b'c'd'e' in the VP to represent the front view with its side a'e' perpendicular to HP. Since the lamina is perpendicular to HP, the top view will be a line, a(e)b(d)c. Assume that edge a’ e’ perpendicular to HP in the final position. The top view of the lamina is now rotated about a(e) such that the line is inclined at 30° to XY line, as shown by points a_{1},b_{1}, c_{1}, d_{1}, and e_{1} in the right bottom of Figure 1. Draw vertical projectors from points a_{1},b_{1}, c_{1}, d_{1}, and e_{1}. Draw horizontal projectors from points a’, b’, c’, d’, and e’. The intersection gives the respective positions of the points In the Front view. Join a_{1}’,b_{1}’, c_{1}’, d_{1}’, and e_{1}’ to obtain the Front view of the lamina.
Figure 1. Orthographic projections of the pentagonal lamina.
Problem 2. Draw the front view, top view and side view of a square lamina. The surface of the lamina is inclined at θ to HP and perpendicular to VP.
Solution. The thre views of the square lamina is shown in figure 2. Since the lamina is perpendicular to VP, its front view will be a line [a’(b’) c’ (d’)] having length as the true length of the edge of the square and inclined at θ to XY line. The corners B and C coincide with A and D in the front view. Since the lamina is inclined to HP at θ, it is also inclined to the left PP at (90- θ). The square lamina is projected on to VP, HP and left PP. Draw vertical projectors from points a’, b’, c’ and d’. On any position on these lines construct the rectangle a-b-c-d such that length ab and cd are equal to the true length of the square edge. The rectangle a-b-c-d is the top view of the lamina. The side view of the lamina a”,b”,c” and d” can be obtained by drawing projectors from points a’,b’,c’and d’ and a, b, c, and d.
Figure 2. The projection sof the square lamina as mentioned in problem 2.
Problem 3. Draw the Top view and front view of a circular lamina if the surface of the lamina is perpendicular to HP and inclined at 30° to VP.
Solution: The projections of the circular lamina is shown in figure 3. Let us first assume that the plane is perpendicular to HP and parallel to VP. The Front view will be a circle and with diameter equal to the diameter of the lamina. Divide the circle in to 12 equal parts and label then as 1’, 2’, 3’, …., 12’. The top view will be a straight line 1-7 , parallel to XY line and can be obtained by drawing projectors from 1’, 2’, …. and 12’. Since the circle is inclined at 30° to VP and perpendicular to HP, reconstruct the top view such that the straight line is inclined at 30° to XY line. Let the respective points be 11, 21, 31, …. 121. Draw vertical projectors from points 11, 21, 31, …. 121 to meet the horizontal projectors from points 1’, 2’, 3’, … 12’ to obtain the points 11’, 21’, 31’, …. 121’ in the Front view. Draw a smooth curve passing through points 11’, 21’, 31’, …. 121’ to obtain the Front view of the circular lamina.
Figure 3. Projections of the circular lamina mentioned in problem 3.
To find the True shape of
a plane surface
The true shape of plane surface when
its top an front views are given may be determined by setting up
two auxiliary planes and projecting on to these. The example below
will demonstrate the method of finding the true shape of a
quadrilateral.
Problem 4:The corners of a quadrilateral PQRS area as follows: P is 25 mm above HP and 50 mm in front of VP, Q is in HP and 80 mm in front of VP. R is 50 mm above HP and 40 mm in front of VP. S is 65 mm above HP and 20 mm in front of VP. The distances between the vertical projectors parallel to the XY lines are as follows: Between P and S is 20 mm, between P and Q is 35 mm, between P and R is 60 mm. Draw the top and front views of the quadrilateral and find its true shape.
Solution:
The method os obtaining the true
shape of the lamina is shown in figure 4. First draw the front view
and top view of the lamina as per the conditions mentioned in the
problem. Through any one corner in any of the two view, say p in
the top view, draw a line parallel to the XY line to intersect the
edge qr at t. Project t to the top view to get t’ on q’r’.
Connect p’t’. Since pt is horizontal, p’t’ is in true
length. The point view of line p’t’ can be obtained by
projecting on to an AIP (by drawing the reference line X1 Y1
perpendicular to p’ t’).
Project the four corner points to get the Auxiliary top view s_{1} r_{1} p_{1} q_{1} (line view). Project the auxiliary Front View on to another Auxiliary vertical plane by drawing the X_{2}Y_{2} line, parallel to s_{1}r_{1}p_{1}q_{1} line.
The Auxiliary Front view will be the true shape of the object.
Figure 4. Solution of
Practice problems on projections of lines
Projections of lines (Drawing practice)
Problem
-1
A straight line AB of true length
100 mm has its end A 20 mm above HP and 30 mm in front of VP. The
top view of the line is 80 mm and front view is 70 mm. Draw the
projections (TV and FV) of the line AB and obtain the true
inclinations of the line AB with HP and VP.
Solution: The
solution to the problem is shown in figure 1. The step wise
procedure for the solution is discussed below:
Figure 1. The projections of line AB in problem 1.
Drawing the top view and front view of line AB
The required inclinations
are:
Angle of inclination with HP =
350
Angle of inclination with VP =
450
Problem
-2
Straight line AB is 40 mm long. End
A is 10mm above HP and 15 mm in front of VP. FV of the line is
inclined at 45° and TV is inclined at 60° to XY line. Draw
the projections of line AB (FV and TV) and obtain the true
inclination of line AB with HP and VP.
Solution: The solution for problem 2 is shown in figure 2. The step wise procedure for the solution is discussed below:
Figure 2. shows the solution of Problem 2.
The required inclinations
of line AB are:
Angle of inclination with HP ≈
270
Angle of inclination with VP ≈
500
Practice problems on projections of lines by auxiliary plane method
Worked out problem in Auxiliary projections
Problem
1.
The end projectors of a line AB is
40 mm. The point A is 24 mm above HP and 10 mm in front of VP.
Point B is 46 mm above HP and 46 mm in front of VP. Determine
the True length of the line and its True inclinations with both the
reference planes.
Solution: The
solution to the problem is shown in figure 1.
Figure1. Solution for problem No. 2.
By measurement, the following
dimensions are obtained:
True length of the line is 57
mm
True inclination of the line with HP
= 22°
True inclination of the line with VP
= 37°
Problem
2.
A line AB 60 mm long has one of its
extremities 60 mm infront of VP and 45 mm above HP. The line is
inclined at 300 to HP and 450 to VP. Draw the projections of the
line by the auxiliary plane method.
Solution: The
solution to the problem is shown in figure 2.
Figure 2. Solution for problem No. 2.