PROJECTIONS OF LINES,
PLANES AND
AUXILIARY PROJECTIONS
PROJECTIONS OF LINES
• A straight line may be generated by a
point moving in one direction.
• A straight line may also be defined as the
shortest distance between two points. A
straight line may be located in space either
by specifying the location of the end points
or by specifying the location of one end
point and the direction.
POSSIBLE POSITIONS OF
STRAIGHT LINES
1. Parallel to both the planes
2. Perpendicular to one plane
3. Inclined to one plane and parallel to
other.
4. Inclined to both the planes
5. Contained by a plane, perpendicular to
both the principal planes
Straight line parallel to both the planes
NOTE: when a line is parallel to any plane, its projection on that plane is a
straight line of the same length. This is because, in orthographic projections,
the line is imagined to be viewed from infinity. Hence, the rays of sight are
parallel to each other. When they pass through the end points of the straight
line, meet the plane of projection at two points, the distance between them is
equal to the length of the projected line itself.
Straight line perpendicular to H.P and parallel to V.P
Straight line perpendicular to V.P and parallel to H.P
Straight line inclined to H.P and parallel to V.P.
Problem: A line AB is 30 long and inclined at 300 to H.P and parallel to V.P. The
end A of the line is 15 above H.P and 20 in front of V.P. Draw the projections of the
line.
Straight line inclined to V.P and parallel to H.P.
Problem: A line AB is 30 long and inclined at 300 to V.P and parallel to H.P.
The end A of the line is 15 above H.P and 20 in front of V.P. Draw the
projections of the line.
Straight line inclined to both the planes
Problem: A line AB of 100 length, is inclined at an angle of 300 to H.P
and 450 to V.P. The point A is 15 above H.P and 20 in front of V.P. Draw
the projections of the oblique lines.
Traces of a straight line inclined to both the planes
Straight line contained by profile plane
Determination of true length and true inclinations
Problem: Fig. a shows the projections of a line AB. Determine the true length of
the line and its inclinations with H.P and V.P.
Fig. a
Fig. b
NOTE: 1. the method followed here is known as rotating line method
2. the other methods that can be followed are; Trapezoidal
method and Auxiliary plane method
Problem: A line AB of 80 long, has its end A, 15 from both H.P and V.P. the
other end B is 40 above H.P and 50 in front of V.P. Draw the projections of the
line and determine the inclinations of the line with H.P and V.P.
Problem: A line AB, which is inclined at 300 to H.P, has its ends A and B, at
25 and 60 in front of V.P respectively. The length of the top view is 65 and its
V.T is 15 above H.P. Draw the projections of the line.
PROJECTION OF LINE
True length
B
T
A
T
B
T
Equal
length
A
T
B
A
A B
F F
Point
B
R
A
R
A B
F F
A
R
True length
NORMAL LINE
B
R
PROJECTION OF LINE
True length
B
T
A
T
B
T
Equal
length
A
T
B
A
A
A
F
B
F
B
R
A
R
A
F
B
F
A
R
Foreshortened
Foreshortened
INCLINED LINE
B
R
PROJECTION OF LINE
Foreshortened
B
T
A
T
B
T
B
Equal
length
B
R
B
F
A
T
B
A
A
F
B
F
B
R
A
A
R
A
F
A
R
Foreshortened
Foreshortened
OBLIQUED LINE
PROJECTIONS OF PLANES
• In structural design, it is necessary to know
the representation of plane surfaces of
different shapes in all possible orientations.
• Plane surfaces have two dimensions, viz.,
length and breadth; the third dimension, the
thickness being zero. In principle, plane
surfaces may be considered of infinite sizes.
However, for convenience, segments of
planes only are considered in the treatment
here.
ORIENTATIONS OF PLANES
1. Plane parallel to one of the principal
planes and perpendicular to the other.
2. Plane inclined to one of the principal
planes and perpendicular to the other.
3. Plane perpendicular to both the
principal planes.
4. Plane inclined to both the principal
planes.
Plane parallel to H.P and perpendicular to V.P.
Problem: A square plane ABCD of side 30, is parallel to H.P and 20 away
from it. Draw the projections of the plane, when two of its sides are (i)
parallel to V.P and (ii) inclined at 300 to V.P.
Parallel to V.P
Inclined at 300 to V.P.
Plane parallel to V.P and perpendicular to H.P.
Problem: An equilateral triangular plane ABC of side 40, has its plane
parallel to V.P and 20 away from it. Draw the projections of the plane when
one of its sides is (i) perpendicular to H.P (ii) parallel to H.P (iii) inclined to
H.P at an angle of 450.
Perpendicular to H.P
Parallel to H.P
Inclined to H.P
Plane inclined to one of the principal planes and
perpendicular to the other
Conditions to be satisfied
• If a plane has an edge parallel to H.P, that edge
should be kept perpendicular to V.P. Similarly, if
the edge of a plane is parallel to V.P, that edge
should kept perpendicular to H.P.
• If a plane has a corner on H.P, the sides
containing that corner should be equally inclined
to V.P, whereas, if the corner is on V.P, the
sides containing the corner should be kept
equally inclined to H.P.
Plane inclined to H.P and perpendicular to V.P.
Problem: Draw the projections of a regular pentagon of 25 side, with its
surface making an angle of 450 with H.P. One of the sides of the pentagon
is parallel to H.P and 15 away from it.
Plane inclined to V.P and perpendicular to H.P.
Problem: A regular hexagonal plane of 30 side, has a corner at 20 from V.P
and 50 from H.P. Its surface is inclined at 450 to V.P and perpendicular to H.P.
Draw the projections of the plane.
Plane perpendicular to both H.P and V.P.
When a plane is positioned such that it is perpendicular to both the principal planes,
then it is said to be lying parallel to the profile plane. In this case, the side view
appears in its true shape and both the front and top views appear as straight lines on
a single projector.
Problem: A rectangular plane of 50*25 size, is perpendicular to both H.P and V.P.
The longer edges are parallel to H.P and the nearest one is 20 above it. The
shorter edge, nearer to V.P is 15 from it. The plane is 50 from the profile plane.
Draw the projections of the plane.
Oblique plane- plane inclined to both H.P and V.P.
Problem: A rectangular plane of size 60*30, has its shorter side on H.P and
inclined at 300 to V.P. Draw the projections of the plane, if its surface is
inclined at 450 to H.P.
Problem: A regular hexagonal plane of 45 side has a corner on H.P, and
its surface is inclined at 450 to H.P. draw projections, when the diagonal
through the, which is on H.P, makes 300 to V.P
PROJECTION OF PLANE
True size
C
T
B
T
C
T
A
T
Equal
length
B
T
C
B
C
R
A
B
F
A ,C
F F
A
T
B
F
A ,B
R R
A ,C
F F
A ,B
R R
Edge
Edge
NORMAL PLANE
C
R
PROJECTION OF PLANE
Foreshortened
C
T
B
T
C
T
A
T
C
Equal
length
C
R
C
F
B
T
A
T
C
B
C
C
R
F
A
B
F
A
F
Foreshortened
A ,B
R R
Edge
B
F
A
F
A ,B
R R
INCLINED PLANE
PROJECTION OF PLANE
Foreshortened
C
T
B
T
C
T
A
T
C
Equal
length
B
B
F
C
R
C
F
B
T
A
T
C
B
C
F
B
R
B
F
A
A
F
Foreshortened
B
R
A
R
Foreshortened
A
F
A
R
OBLIQUED PLANE
C
R
AUXILIARY PROJECTIONS
•
•
Three views, viz., front, top and side views of an object are some
times not sufficient to reveal complete information regarding the
size and shape of the object. To serve the purpose, additional
views known as auxiliary views; projected on auxiliary planes may
be used.
Auxiliary plane is a plane, perpendicular to one of the principal
planes of projection and inclined to the other.
Types of auxiliary planes and views:
1. Auxiliary vertical plane (A.V.P): It is a plane perpendicular to H.P
and inclined to V.P. the projection on to an A.V.P is called an
auxiliary front view (when the inclination of an A.V.P is 900, it then
becomes a profile plane)
2. Auxiliary inclined plane (A.I.P): It is a plane inclined to H.P and
perpendicular to V.P. the projection on an A.I.P is called an
auxiliary top view.
NOTE: For presenting the relative positions of the views, the auxiliary
plane should always be rotated about the plane to which it is
perpendicular.
Projections of points
Problem: A point A is 25 above H.P and 15 in front of V.P. Draw the front and
top views of the point . Also, obtain the auxiliary front view of the point on a
plane, which makes an angle of 600 with V.P and perpendicular to H.P.
NOTE: 1. the distance of the auxiliary front view from x1y1 is equal to
the distance of the front view from xy.
2. the top view and auxiliary front view lie on a single
projector which is perpendicular to x1y1.
Problem: A point B is 25 above H.P and 15 in front of V.P. Draw the front
view and top views of the point. Also obtain the auxiliary top view of the point
on a plane, which makes an angle of 450 with H.P and perpendicular to V.P.
NOTE: 1. the distance of the auxiliary top view from x1y1 is equal to the
distance of the top view from xy.
2. The front view and auxiliary top view lie on a single projector,
which is perpendicular to x1y1
Projections of straight lines
Problem: A straight line AB of 75 length, is inclined at 300 to H.P. The end A of
the line is 20 above H.P and 20 in front of V.P. Draw the projections, by
auxiliary plane method.
Problem: A straight line PQ of 50 length, is inclined at 450 to V.P. The end P
of the line is 20 above H.P and 15 in front of V.P. Draw the projections, by
auxiliary plane method.
Determination of true length and true inclinations
Problem: Fig. a shows the projections of a straight line, inclined to both
H.P and V.P. Determine the true length of the line and its true inclinations
with H.P and V.P.
Fig. a
Fig. b
NOTE: the true length of a line and its inclinations with the planes of projection
may be determined by making each of its projections, parallel to the reference line.
For this a new reference line, representing an auxiliary plane may be drawn,
parallel to the projection concerned. The projection on this plane will reveal the
true length and true inclinations of the line with the other principal plane.
Problem: a line AB of 60 long, has its end A, at 20 above H.P and 25 in
front of V.P. the line is inclined at 300 to H.P and 450 to V.P. Draw the
projections
fig 10.6, page 10.7
Projections of planes
Problem: A rectangular plane ABCD of size 100* 60, is inclined to V.P by an
angle of 450. shorter edge of which is making of an angle of 600 with H.P. Draw
the projections, by auxiliary plane method.