Projection of Straight lines and Planes
- Principle of Projection
If straight lines are drawn from various points on the contour of an object to meet a plane, the object is said to be projected on that plane. The figure formed by joining, in correct sequence, the points at which these lines meet the plane, is called the projection of the object. The lines from the object to the plane are called projectors.
Methods of Projection:
Following four methods of projection are commonly used,
1) Orthographic projection.
2) Isometric projection.
3) Oblique projection.
4) Perspective projection.
In the above methods 2, 3 and 4 represent the object by a pictorial view as eyes see it. In these methods of projection, a three-dimensional object is represented on a projection plane by one view only, while in the orthographic projection an object is represented by two or three views on the mutual perpendicular projection planes. Each projection view represents two dimensions of an object. For the complete description of the three-dimensional object at least two or three views are required.
- Orthographic Projection:
Theory of orthographic projection:
Let us suppose that a transparent plane has been set up between an object and the station point of an observer's eye (Fig. 1). The intersection of this plane with the rays formed by lines of sight from the eye to all points of the object would give a picture that is practically the same as the image formed in the eye of the observer. This is perspective projection.
Figure 1 Perspective projection. The rays of the projection converge at the station point from which the object is observed.
If the observer would then walk backward from the station point until he reached a theoretically infinite distance, the rays formed by lines of sight from his eye to the object would grow longer and finally become infinite in length, parallel to each other, and perpendicular to the picture plane. The image so formed on the picture plane is what is known as "orthographic projection." See Fig.
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Basically, orthographic projections could be defined as any single projection made by dropping perpendiculars to a plane. However, it has been accepted through long usage to mean the combination of two or more such views, hence the following definition has been put forward: Orthographic projection is the method of representing the exact shape of an object by dropping perpendiculars from two or more sides of the object to planes, generally at right angles to each other; collectively, the views on these planes describe the object completely. (The term "orthogonal" is sometimes used for this system of drawing.)
- Orthographic views:
The rays from the picture plane to infinity may be discarded and the picture, or "view," thought of as being found by extending perpendiculars to the plane from all points of the object, as in Fig. This picture, or projection on a frontal plane, shows the shape of the object when viewed from the front, but it does not tell the shape or distance from front to rear. Accordingly, more than one projection are required to describe the object.
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In addition to the frontal plane, imagine another transparent plane placed horizontally above the object, as in Fig. 4. The projection on this plane, found by extending perpendiculars to it from the object, will give the appearance of the object as if viewed from directly above and will show the distance from front to rear.
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The frontal and horizontal planes of projections. Projection on the horizontal plane produces the top view of the object.
If this horizontal plane is now rotated into coincidence with the frontal plane, as in the two views of the object will be in the same plane, as if on a sheet of paper.
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Now imagine a third plane, perpendicular to the first two. This plane is called a "profile plane," and a third view can be projected on it. This view shows the shape of the object when viewed from the side and the distance from bottom to top and front to rear.
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The three planes of projection: frontal, horizontal and profile. Each is perpendicular to other two.
The horizontal and profile planes are shown rotated into the same plane as the frontal plane (again thought of as the plane of the drawing paper). Thus, related in the same plane, they give correctly the three-dimensional shape of the object.
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The horizontal and profile planes rotated into the same plane as the frontal plane. This makes it possible to draw three views of the object.
In orthographic projection the picture planes are called "planes of projection"; and the perpendiculars, "projecting lines" or "projectors."
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A point may be situated, in space, in any one of the four quadrants formed by the two principal planes of projection or may lie in any one or both. Its projections are obtained by extending projectors perpendicular to the planes.
One of the planes is then rotated so that the first and third quadrants are opened out. The projections are shown on a flat surface in their respective positions either above or below or in xy.
- If a point is situated in the first quadrant
The pictorial view [fig. 14 (i)] shows a point A situated above the H.P. and in front of the V.P., i.e. in the first quadrant. a' is its front view and the top view. After rotation of the plane, these projections will be shown in fig.
The front view a' is above xy and the top view a below it. The line joining a' and a (which also is called a projector), intersects xy at right angles at a point o. It is quite evident from the pictorial view that a'o = Aa, i.e. the distance of the front view from xy = the distance of A from the H.P. viz. h. Similarly, ao = Aa',i.e. the distance of the top view from xy = the distance of A from the V.P. viz. d.
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- If a point is situated in second quadrant,
A point B (fig. 15) is above the H.P. and behind the V.P., i.e. in the second quadrant. b' is the front view and b the top view. When the planes are rotated, both the views are seen above xy. Note that b'o = Bb and bo = Bb'.
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- If a point is situated in third quadrant
A point C (fig. 16) is below the H.P. and behind the Y.P., i.e. in the third quadrant. Its front view c' is below xy and the top view c above xy. Also, c'o = Cc and co = Cc'.
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- If a point is situated in 4th quadrant,
A point f (fig. 17) is below the H.P. and in front of the V.P., i.e. in the fourth quadrant. Both its projections are below xy, and e'o = Ee and eo = Ee'.
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General conclusions:
1. The fine joining the top view and the front view of a point is always perpendicular to xy. It is called a projector.
2. When a point is above the H.P., its front view is above xy; when it is below the H.P., the front view is below xy. The distance of a point from the H.P. is shown by the length of the projector from its front view to xy.
3. When a point is in front of the V.P., its top view is below xy; when it is behind the V.P., the top view is above xy. The distance of a point from the V.P. is shown by the length of the projector from its top view to xy.
4. When a point is in a reference plane, its projection on the other reference plane is in xy.
A straight line is the shortest distance between two points. Hence, the projections of a straight line may be drawn by joining the respective projections of its ends which are points.
The position of a straight line may also be described with respect to the two reference planes. It may be:
1. Parallel to one or both the planes.
2. Contained by one or both the planes.
3. Perpendicular to one of the planes.
4. Inclined to one plane and parallel to the other.
5. Inclined to both the planes.
6. Projections of lines inclined to both the planes.
7. Line contained by a plane perpendicular to both the reference planes.
8. True length of a straight line and its inclinations with the reference planes.
9. Traces of a line.
10. Methods of determining traces of a line.
11. Traces of a line, the projections of which are perpendicular to xy.
12. Positions of traces of a line.
Line parallel to one or both the planes:
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(a) Line AB is parallel to the H.P.
a and b are the top views of the ends A and B respectively. It can be clearly seen that the figure ABba is a rectangle. Hence, the top view ab is equal to AB.
a'b' is the front view of AB and is parallel to xy.
(b) Line CD is parallel to the V.P. The line c'd' is the front view and is equal to CD; the top view cd is parallel to xy.
(c) Line ff is parallel to the H.P. and the V.P. ef is the top view and e'f' is the front view; both are equal to ff and parallel to xy.
Hence, when a line is parallel to a plane, its projection on that plane is equal to its true length; while its projection on the other plane is parallel to the reference line.
Line contained by one or both the planes:
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Line AB is in the H.P. Its top view ab is equal to AB; its front view a' b' is in xy.
Line CD is in the V.P. Its front view c'd' is equal to CO; its top view cd is in xy.
Line ff is in both the planes. Its front view e' f' and the top view ef coincide with each other in xy.
Hence, when a line is contained by a plane, its projection on that plane is equal to its true length; while its projection on the other plane is in the reference line.
Line perpendicular to one of the planes:
When a line is perpendicular to one reference plane, it will be parallel to the other.
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(a) Line AB is perpendicular to the H.P. The top views of its ends coincide in the point a. Hence, the top view of the line AB is the point a. Its front view a' b' is equal to AB and perpendicular to xy.
(b) Line CD is perpendicular to the V.P. The point d' is its front view and the line cd is the top view. cd is equal to CD and perpendicular to xy.
Hence, when a line is perpendicular to a plane its projection on that plane is a point; while its projection on the other plane is a line equal to its true length and perpendicular to the reference line.
In first-angle projection method, when top views of two or more points coincide, the point which is comparatively farther away from xy in the front view will be visible; and when their front views coincide, that which is farther away from xy in the top view will be visible.
In third-angle projection method, it is just the reverse. When top views of two or more points coincide the point, which is comparatively nearer xy in the front view will be visible; and when their front views coincide, the point which is nearer xy in the top view will be visible.
Line inclined to one plane and parallel to the other:
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The inclination of a line to a plane is the angle which the line makes with its projection on that plane.
Line PQ1 is inclined at an angle 8 to the H.P. and is parallel to the V.P. The inclination is shown by the angle 8 which PQ1 makes with its own projection on the H.P., viz. the top view pq1.
The projections [fig. 22 (ii)] may be drawn by first assuming the line to be parallel to both the H.P. and the V.P. Its front view p'q' and the top view pq will both be parallel to xy and equal to the true length. When the line is turned about the end P to the position PQ1 so that it makes the angle 8 with the H.P. while remaining parallel to the V.P., in the front view the point q' will move along an arc drawn with p' as center and p'q' as radius to a point q'1 so that p'q'1 makes the angle 8 with xy. In the top view, q will move towards p along pq to a point q1 on the projector through q'1. p'q'1 and pq1 are the front view and the top view respectively of the line PQ1.
A line AB (fig. 23) is inclined at θ to the H.P. and is parallel to the V.P.The end A is in the H.P. AB is shown as the hypotenuse of a right-angled triangle, making the angle θ with the base.
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The top view ab is shorter than AB and parallel to xy. The front view a'b' is equal to AB and makes the angle θ with xy.
Keeping the end, A fixed and the angle θ with the H.P. constant, if the end B is moved to any position, say B1, the line becomes inclined to the V.P. also.
In the top view, b will move along an arc, drawn with a as center and ab as radius, to a position b1. The new top view ab1 is equal to ab but shorter than AB.
In the front view, b' will move to a point b'1 keeping its distance from xy constant and equal to b'o; i.e. it will move along the line pq, drawn through b' and parallel to xy. This line pq is the locus or path of the end B in the front view. b'1 will lie on the projector through b1. The new front view a'b'1 is shorter than a'b' (i.e. AB) and makes an angle a with xy. a is greater than θ.
Thus, if the inclination θ of AB with the H.P.is constant, even when it is inclined to the V.P.
(i) its length in the top view, viz. ab remains constant; and
(ii) the distance between the paths of its ends in the front view,viz. b'o remains constant.
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b) The same line AB (fig. 24) is inclined at ϕ to the V.P. and is parallel to the H.P. Its end A is in the V.P. AB is shown as the hypotenuse of a right-angled triangle making the angle ϕ with the base.
The front view a'b'2 is shorter than AB and parallel to xy. The top view ab2 is equal to AB and makes an angle ϕ with xy.
Keeping the end, A fixed and the angle ϕ with the V.P. constant, if B is moved to any position, say B3, the line will become inclined to the H.P. also.
In the front view, b'2, will move along the arc, drawn with a' as center and a'b'2 as radius, to a position b'3. The new front view a'b'3 is equal to a'b'2 but is shorter than AB.
In the top view, b2 will move to a point b3 along the line rs, drawn through b2 and parallel to xy, thus keeping its distance from the path of a, viz. b2o constant. rs is the locus or path of the end B in the top view. The point b3 lies on the projector through b'3. The new top view ab3 is shorter than ab2 (i.e. AB) and makes an angle β with xy. β is greater than ϕ.
Here also we find that, if the inclination of AB with the V.P. does not change, even when it becomes inclined to the H.P.
(i) its length in the front view, viz. a'b'2 remains constant; and
(ii) the distance between the paths of its ends in the top view, viz. b2o remains constant.
Hence, when a line is inclined to both the planes, its projections are shorter than the twe length and inclined to xy at angles greater than the true inclinations. These angles viz. α and β are called apparent angles of inclination.
Traces of a line:
When a line is inclined to a plane, it will meet that plane, produced if necessary. The point in which the line or line-produced meets the plane is called its trace.
The point of intersection of the line with the H.P. is called the horizontal trace, usually denoted as H.T. and that with the V.P. is called the vertical trace or V.T.
Refer to fig. 25.
(i) A line AB is parallel to the H.P. and the V.P. It has no trace.
(ii) A line CD is inclined to the H.P. and parallel to the V.P. It has only the H.T. but no V.T.
(iii) A line ff is inclined to the V.P. and parallel to the H.P. It has only the V.T. but no H.T.
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Thus, when a line is parallel to a plane it has no trace upon that plane.
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(i) A line PQ is perpendicular to the H.P. Its H.T. coincides with its top view which is a point. It has no V.T.
(ii) A line RS is perpendicular to the V.P. Its V.T. coincides with its front view which is a point. It has no H.T.
Hence, when a line is perpendicular to a plane, its trace on that plane coincides with its projection on that plane. It has no trace on the other plane.
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(i) A line AB has its end A in the H.P. and the end B in the V.P. Its H.T. coincides with the top view of A and the V.T. coincides with b' the front view of B.
(ii) A line CD has its end C in both the H.P. and the V.P. Its H.T. and V.T. coincide with c and c' (the projections of C) in xy. Hence, when a line has an end in a plane, its trace upon that plane coincides with the projection of that end on that plane.
Methods of determining traces of a line:
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Method I:
Fig. 28 (i) shows a line AB inclined to both the reference planes. Its end A is in the H.P. and 8 is in the V.P. a'b' and ab are the front view and the top view respectively [fig. 28 (ii)].
The H.T. of the line is on the projector through a' and coincides with a. The V.T. is on the projector through b and coincides with b'
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Let us now assume that AB is shortened from both its ends, its inclination with the planes remaining constant. The H.T. and V.T. of the new line CD are still the same as can be seen clearly in fig. 29 (i). c'd' and cd are the projections of CD [fig. 29 (ii)]. Its traces may be determined as described below.
(i) Produce the front view c'd' to meet xy at a point h.
(ii) Through h, draw a projector to meet the top view cd-produced, at the H.T. of the line.
(iii) Similarly, produce the top view cd to meet xy at a point v.
(iv) Through v, draw a projector to meet the front view c'd'-produced, at the V.T. of the line.
Method II:
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c'd' and cd are the projections of the line CD [fig. 30 (ii)]. Determine the true length C1D1 from the front view c'd' by trapezoid method. The point of intersection between c'd'-produced and C1D1-produced is the V.T. of the line.
Similarly, determine the true length C2D2 from the top view ed. Produce them to intersect at the H.T. of the line.
Examples:
1. A line AB 50 mm long, has its end A in both the H.P. and the V.P. It is inclined at 30° to the H.P. and at 45° to the V.P. Draw its projections.
As the end A is in both the planes, its top view and the front view will coincide in xy.
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(i) Assuming AB to be parallel to the V.P. and inclined at θ (equal to 30°) to the H.P., draw its front view ab' (equal to AB) and project the top view ab.
(ii) Again, assuming AB to be parallel to the H.P. and inclined at ϕ (equal to 45°) to the V.P., draw its top view ab1 (equal to AB). Project the front view ab'1.
ab and ab'1 are the lengths of AB in the top view and the front view respectively, and pq and rs are the loci of the end B in the front view and the top view respectively.
(iii) With a as center and radius equal to ab'1, draw an arc cutting pq in b'2. With the same center and radius equal to ab, draw an arc cutting rs in b2.
Draw lines joining a with b'2 and b2. ab'2 and ab2 are the required projections.
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Shows in pictorial and orthographic views, the solution obtained with all the above steps combined in one figure only.
Draw the projections of a pentagonal pyramid, base 30 mm edge and axis 50 mm long, having its base on the H.P. and an edge of the base parallel to the V.P. Also draw its side view.
(i) Assume the side DE which is nearer the V.P., to be parallel to the V.P. as shown
in the pictorial view.
(ii) In the top view, draw a regular pentagon abcde with ed parallel to and nearer
xy. Locate its centre o and join it with the corners to indicate the slant edges.
(iii) Through o, project the axis in the front view and mark the apex o', 50 mm
above xy. Project all the corners of the base on xy. Draw lines o'a', o'b'
and o'c' to show the visible edges. Show o'd' and o'e' for the hidden edges
as dashed lines.
(iv) For the side view looking from the left, draw a new reference line x1y1
perpendicular to xy and to the right of the front view. Project the side
view on it, horizontally from the front view as shown. The respective
distances of all the points in the side view from x1y1, should be equal to
their distances in the top view from xy. This is done systematically as
explained below:
(v) From each point in the top view, draw horizontal lines upto x1y1 . Then
draw lines inclined at 45° to x1y1 (or xy) as shown. Or, with q, the point
of intersection between xy and x1y1 as centre, draw quarter circles. Project
up all the points to intersect the corresponding horizontal lines from the
front view and complete the side view as shown in the figure. Lines o1d1
and o1c1 coincide with o1e1 and o1a1 respectively.
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Q. Draw the projections of (i) a cylinder, base 40 mm diameter and axis 50 mm Jong, and (ii) a cone, base 40 mm diameter and axis 50 mm long, resting on the H.P. on their respective bases.
(i) Draw a circle of 40 mm diameter in the top view and project the front view
which will be a rectangle [fig.(ii)].
(ii) Draw the top view [fig. (iii)]. Through the centre o, project the apex o', 50 mm
above xy. Complete the triangle in the front view as shown.
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Q. Draw the projections of a hexagonal pyramid, base 30 mm side and axis 60 mm long, having its base on the H.P. and one of the edges of the base inclined at 45° to the V.P.
(i) In the top view, draw a line af 30 mm long and inclined at 45° to xy. Construct a regular hexagon on af. Mark its centre o and complete the top view by drawing lines joining it with the corners.
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Q. A hexagonal prism has one of its rectangular faces parallel to the H.P. Its axis is perpendicular to the V.P. and 3.5 cm above the ground.
Draw its projections when the nearer end is 2 cm in front of the V.P. Side of base 2.5 cm long: axis 5 cm long.
(i) Begin with the front view. Construct a regular hexagon of 2.5 cm long
sides with its centre 3.5 cm above xy and one side parallel to it.
(ii) Project down the top view, keeping the line for nearer end, viz. 1-4, 2 cm
below xy.
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Q. A square pyramid, base 40 mm side and axis 65 mm long, has its base in the V.P. One edge of the base is inclined at 30° to the H.P. and a corner contained by that edge is on the H.P. Draw its projections.
(i) Draw a square in the front view with the corner d' in xy and the side d'c'
inclined at 30° to it. Locate the centre o' and join it with the corners of the
square.
(ii) Project down all the corners in xy (because the base is in the V.P.). Mark the
apex o on a projector through o'. Draw lines for the slant edges and complete
the top view.
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