Answer (Detailed Solution Below) Option 1 : Below H.P and behind V.P.
10 Qs. 20 Marks 12 Mins
Explanation: The orthographic projection system is used to represent a 3D object in a 2D plane. The orthographic projection system utilizes parallel lines, to project 3D object views onto a 2D plane. According to the rule of orthographic projection. To draw a projection view of a 3D object on a 2D Plane. The horizontal plane is rotated in the clockwise direction. Types of Orthographic projection systems are first angle and third angle projection. 1. First Angle Projection: In the first angle projection, the object is placed in the 1st quadrant. The object is positioned at the front of a vertical plane and top of the horizontal plane. First angle projection is widely used in India and European countries. The object is placed between the observer and projection planes. The plane of projection is taken solid in 1st angle projection. Symbol – 2. Third Angle Projection: In the third angle projection, the object is placed in the third quadrant. The object is placed behind the vertical planes and bottom of the horizontal plane. Third angle projection is widely used in the United States. The projection planes come between the object and the observer. The plane of projection is taken as transparent in 3rd angle projection. Symbol – Additional Information Comparison between First angle and the Third angle projection
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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
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. 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 1st angle or 3rd 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
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 1st 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 1st angle and 3rd angle projection techniques. A summary of the difference between 1st and 3rd angle projections is shown if Table 1.
Figure 19 Differentiating between the 1st angle and 3rd 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 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.
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 ) Projection of Points A
POINT
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 1st 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.
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
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.
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.
(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 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. AB1is the new position of the line AB when it is inclined at q to HP and parallel to VP. Project AB1 on VP and HP. Since AB1 is parallel to VP, a΄b1΄, the projection of AB1 on VP is in true length inclined at q to the XY line, and ab1, the projection of AB1 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. 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 ab1, 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 b1΄. 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'b2' and in the top view the projection of the end B lies on the locus line passing through b2. Thus a΄b2΄ 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 b2. 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 ab1 and a'b2' respectively and (ii) The projections of the end
B, (i.e., b and b‘) should lie along the locus line passing through
b2and b1΄ respectively. Orthographic
projections
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 ab2 passing through a and incline at φ to the XY line. The length ab2 is equal to the true length of AB. The end points a and b2 are projected on to a line parallel to XY line and passing through a’ to get a'b2' which is the front view of the line when it is parallel to HP and inclined to VP. Draw the horizontal locus lines through b2, and b1'. With center a and radius ab1, draw an arc to cut the locus line drawn through b2 at b. Connect ab, the top view of the line AB. With center a' and radius a'b2΄, draw an arc to cut the locus line drawn through b1' 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 (i) Rotating line method
or Rotating line method The method of obtaining the top
and front views of a line, when its true length and true
inclinations are given.
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
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
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 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 The auxiliary view method may be applied
Types of auxiliary
planes
Auxiliary Vertical Plane
(AVP) 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 X1Y1 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.
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.
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 X1Y1 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 X1Y1 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 p1' will also be at a height m above the X1Y1 line. Therefore, mark P1’ by measuring o1p1’=op’ = m on the projector drawn from p perpendicular to the X1Y1 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
P1 is the projection
on AIP.
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
X1Y1, line, so that it becomes in-plane with
that of both HP and VP.
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
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. Problem
1: Solution
Figure 1. The FV and TV of the line AB when parallel to HP and VP Since the line is to be inclined
at 300 to HP, set up an AIP inclined at
300 to HP and perpendicular to
VP.
Figure 2. Projection of line on to the AIP. Since the top view of the line
appears inclined to VP at 600, draw the
X2Y2 line inclined at
600 to the auxiliary
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. Solution. The solution is
shown in figure . The method of obtaining the projections is
described below. Draw
X1Y1 line inclined at θ =
300 to the XY line. Mark AIP and VP. Project the
auxiliary top view ab The projections ab on the AIP and
a1'b1' 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.
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
X2Y2, the line of intersection
of AIP and AVP cannot be drawn directly at Φ
to ab. Shortest distance between
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. Shortest distance between two parallel lines
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:
Figure 6. Determination of shortest distance between two parallel lines. Draw the
X1Y1 line parallel to ab and pq at any
convenient distance from them. 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
A1B1 is made showing the true length of
AB. 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. 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.
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 X1Y1 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 X1Y1 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
X1Y1 line. To project the right view,
draw a 45° line at the point of intersection of the XY and
X1Y1 lines.
Figure 3. Orthographic projections of the lamina ABC B) Plane parallel to HP
and perpendicular to both VP and PP
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. C) Plane parallel to PP
and perpendicular to both HP and 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 X1Y1 line. The top and front views of
the lamina appear as lines perpendicular to XY
line. 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).
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 300 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 300 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 a1,b1, c1, d1, and e1 in the right bottom of Figure 1. Draw vertical projectors from points a1,b1, c1, d1, and e1. 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 a1’,b1’, c1’, d1’, and e1’ 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 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: Project the four corner points to get the Auxiliary top view s1 r1 p1 q1 (line view). Project the auxiliary Front View on to another Auxiliary vertical plane by drawing the X2Y2 line, parallel to s1r1p1q1 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
Figure 1. The projections of line AB in problem 1.
Drawing the top view and front view of line AB
The required inclinations
are: Problem
-2 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: Practice problems on projections of lines by auxiliary plane method Worked out problem in Auxiliary projections Problem
1.
Figure1. Solution for problem No. 2.
By measurement, the following
dimensions are obtained: Problem
2.
Figure 2. Solution for problem No. 2.
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