Notes-Engineering-Graphics-and-Design-1-.pdf

Engineering Drawing
Lecture # 01
Introduction
1

Syllabus
1. Importance of engineering drawing; Drawing techniques
2. Manual drawing instruments and their uses – Drawing board; Mini-
drafter; Set squares; Drawing instrument box; Scales; Protractor;
French curves; Drawing papers; Drawing pencils; Eraser; Drawing
pins/clips; Sand paper block; Duster.
3. Conventions - ISO and BIS; Layout of drawing sheets; Border lines;
Title block; Folding of drawing sheets; Lines, lettering and
dimensioning.
4. Scales – Plane, diagonal and vernier
5. Curves used in engineering practice:
6. Orthographic projection – Theory of projection
2

7. Projection of points
8. Projection of straight lines
9. Projection of planes
10.Projection of solids
11.Auxiliary projections
12.Sections of solids
13.Development of surfaces
14.Intersections of solids
15.Isometric projections
3

Books/references
1. Dhananjay A Jolhe, Engineering drawing, TMH, 2008
2. M.B. Shah and B.C. Rana, Engineering Drawing, Pearsonson,
2009.
3. N D Bhatt and V M Panchal, Engineering Drawing, 43rd edition,
Charator Publishing House, 2001
4. T E French, C J Vierck and R J Foster, Graphic Science and
Design, 4th edition, McGraw Hill, 1984
5. W J Luzadder and J M Duff, Fundamentals of Engineering
Drawing, 11th edition, Prentice-Hall of India, 1995.
6. K Venugpoal, Engineering Drawing and Graphics, 3nd edition,
New Age International, 1998.
4

Note to the students
1. Practical assignments are to be completed in the Drawing Hall during
the respective practice period itself.
2. No make-up class for the completion of the incomplete assignments.
3. Only one make-up class for a missed class, that too only under
medical ground. Students having attendance (lecture + tutorial) less
than 75%, or for both lecture and tutorial independently will be
debarred from appearing in the end semester examination.
4. No entry to the lecture hall 5 minutes after the start of the class.
5

ENGINEERING DRAWING
Graphical means of expression of
technical details without the
barrier of a language.
Universal language for Engineers
7

What will you learn from this course?
How to communicate technical information.
• Visualization – the ability to mentally understand visual
information.
• Graphics theory – geometry and projection techniques
used for preparation of drawings.
• Standards – set of rules for preparation of technical
drawings. Conventions – commonly accepted practices
in technical drawings.
• Tools – devices used to create technical drawings and
models.
• Applications – the various uses for technical drawings.
8

Graphic language: mode of communication through SKETCHES
Drawing: graphical representation of an OBJECT
Engineering Drawing
Drawing of an object contains all the necessary information,
required for the construction/fabrication of the object, like
➢ actual shape,
➢ accurate sizes,
➢ manufacturing methods,
➢ materials to be used etc.,
9

List of tools required for the drawing practice session
Sl. No. Item Quantity
1 Mini-drafter (or T-Square) 1
2 Engineering Drawing Box 1
3 French curves 1 set
4 Set-square 1 set
5 Protractor 1
6 Drawing Clip 1 set
7 Lead pencil/clutch pencil 2-3
8 Lead (HB, H & 2H) 1 each set
9 Eraser 1
10 Sand paper/cello tape 1
11 Blade / pencil sharpener 1
12 Drawing Sheet 1 per session
➢ Students without Engineering Drawing Box will not be
allowed t
o
attend the practical session.
➢ School
Instrument box is not allowed. 10

Mini-drafter fixed on a drawing table
12

Set-square
Drawing Clips
French Curves
13

Dimensions of Engineer’s Drawing Boards
Designation Length x Width
(mm)
Recommended
for use with sheet
sizes
D0 1500 x 1000 A0
D1 1000 x 700 A1
D2 700 x 500 A2
D3 500 x 500 A3
D0 and D1 for drawing offices, for students use – D2
18

Standard sizes of drawing sheets as per BIS
Designation Size
(mm)
A0 841 x 1189
A1 594 x 841
A2 420 x 594
A3 297 x 420
A4 210 x 297
19

LINES AND LETTERING*
LINES
Lines are the basic feature of a drawing. A line may
be straight, curved, continuous, segmented, thin,
thick, etc., each having its own specific sense.
Line strokes refer to the directions of drawing
straight and curved lines
*standard given in BIS : SP-46, 2003
23

Line Strokes
Vertical and inclined lines are drawn from top to bottom,
horizontal lines are drawn from left to right. Curved lines
are drawn from left to right or top to bottom.
24

Line types
Illustration Application
Thick Outlines, visible edges, surface
boundaries of objects, margin lines
Continuous thin Dimension lines, extension lines,
section lines leader or pointer lines,
construction lines, boarder lines
Continuous thin wavy Short break lines or irregular
boundary lines – drawn freehand
Continuous thin with zig-zag
Long break lines
Short dashes, gap 1, length 3 mm
Invisible or interior surfaces
25

Line types
Illustration Application
Short dashes Center lines, locus lines
Alternate long and short
dashes in a proportion of 6:1,
Long chain thick at end and
Cutting plane lines
Continuous thick border line
Border
26

Uses of different types of lines in a given drawing
27

Units of Measurement
➢ International systems of units (SI) – which is
b
a
s
e
d
on the meter.
➢ Millimeter (mm) - The common SI unit of
measure o
n
engineering drawing.
➢ Individual identification of linear units is not
required if all dimensions on a drawing are in the
same unit (mm).
➢ The drawing shall however contain a note:
A
L
L
DIMENSIONS ARE IN MM. (Bottom left corner
outside the title box)
28

Dimensioning
➢ Indicating on a drawing, the size of the object
a
n
dother details essential for its construction and
function, using lines, numerals, symbols, notes, etc.
➢ Dimensions indicated on a drawing should be
t
h
o
s
e that are essential for the production, inspection
and functioning of the object.
➢ Dimensions indicated should not be mistaken
a
s
those that are required to make the drawing of an
object.
29

➢ E
xt
ension line
– a thin, solid line
perpendicular to a dimension line,
indicating which feature is associated with
the dimension.
➢ V
isible
gap – there should be a visible
gap of 1.5 mm between the feature’s
corners and the end of the extension line.

Leader line
❑ A thin, solid line used to indicate the
feature with which a dimension, note, or symbol
is associated.
❑ Generally a straight line drawn at an angle
that i
s
neither horizontal nor vertical.
❑ Terminated with an arrow touching the
parto
r
detail.
❑ On the end opposite the arrow,
t
h
eleader line will have a short, horizontal
shoulder. Text is extended from this
shoulder such that the text height is centered
with the shoulder line
32

8/11/2011 33
Arrows
3 mm wide and should be 1/3rd as wide as they are
long - symbols placed at the end of dimension lines to
show the limits of the dimension. Arrows are uniform
in size and style, regardless of the size of the drawing.

Orientation of Dimensioning Text
Placing of Dimensions
35

8/11/2011 36
Dimensioning of angles

011
Dimensioning of Circular Features
A circle should be dimensioned by giving its diameter
instead of radius. The dimension indicating a
diameter should always be preceded by the symbol ø,
37

Dimensioning a Length
Depends on Available Space
38

Dimensioning Radii
Arcs of Circle Precede with ‘R’ to distinguish from length
39

RULES OF DIMENSIONING
1. Between any two extension lines, there must be one and only one
dimension line bearing one dimension.
2. As far as possible, all the dimensions should be placed outside the
views. Inside dimensions are preferred only if they are clearer and
more easily readable.
3. All the dimensions on a drawing must be shown using either Aligned
System or Unidirectional System. In no case should, the two systems
be mixed on the same drawing.
4. The same unit of length should be used for all the dimensions on a
drawing. The unit should not be written after each dimension, but a
note mentioning the unit should be placed below the drawing.
5. Dimension lines should not cross each other. Dimension lines should
also not cross any other lines of the object.
6. All dimensions must be given.
7. Each dimension should be given only once. No dimension should be
redundant.

8. Do not use an outline or a centre line as a dimension line. A centre line
may be extended to serve as an extension line.
9. Avoid dimensioning hidden lines.
10. For dimensions in series, adopt any one of the following ways.
i. Chain dimensioning (Continuous dimensioning) All the
dimensions are aligned in such a way that an arrowhead of one
dimension touches tip-to-tip the arrowhead of the adjacent
dimension. The overall dimension is placed outside the other
smaller dimensions.
ii. Parallel dimensioning (Progressive dimensioning) All the
dimensions are shown from a common reference line. Obviously,
all these dimensions share a common extension line. This method
is adopted when dimensions have to be established from a
particular datum surface
iii. Combined dimensioning When both the methods, i.e., chain
dimensioning and parallel dimensioning are used on the same
drawing, the method of dimensioning is called combined
dimensioning.

Dimensioning Guidelines
Avoid crossing extension lines
Multiple extension
line crossings may
be confused for
the outside corner
of the part.
42

8/11/2011
•Single stroke refers to the thickness obtained in one stroke of a pencil
or ink pen .
•It does not mean that the pencil or pen should not be lifted while
completing a particular letter.
Lettering types
• Lettering A – Height of the capital letter is divided into 14 equal parts
• Lettering B – Height of the capital letter is divided into 10 equalparts
43

Ratio of height to width varies, but in most cases is 6:5
Specifications of A -Type Lettering
Specifications Value Size (mm)
Capital letter height h 2.5 3.5 5 7 10 14 20
Lowercase letter height a = (5/7)h - 2.5 3.5 5 7 10 14
Thickness of lines b = (1/14)h 0.18 0.25 0.35 0.5 0.7 1 1.4
Spacing between
characters
c = (1/7)h 0.35 0.5 0.7 1 1.4 2 2.8
Min. spacing b/n words d = (3/7)h 1.05 1.5 2.1 3 4.2 6 8.4
Min. spacing b/n baselines e = (10/7)h 3.5 5 7 10 14 20 28
Ratio of height to width varies, but in most cases is 6:5

Engineering Drawing
Lecture 2
Geometric Constructions

Geometric Construction
• Construction of primitive geometric
forms (points, lines and planes etc.)
that serve as the building blocks for
more complicated geometric shapes.
• Defining the position of the object in
space

Primitive geometric forms
• Point
• Line
• Plane
• Solid
• ……etc

The basic 2-D geometric primitives, from which
other more complex geometric forms are
derived.
➢ Points,
➢ Lines,
➢ Circles, and
➢ Arcs.

Point
➢ A theoretical location
that has neither width,
height, nor depth.
➢ Describes exact location in space.
➢ A point is represented in technical d
r
a
w
i
n
g
as a small cross made of dashes that are
approximately 3 mm long.

A point is used to mark the locations of centers
and loci, the intersection ends, middle of entities.

Line
➢ A geometric primitive that has
length and direction, but no thickness.
➢ It may be straight, c
u
r
v
e
d
ora combinationof these.
➢
conditions, such as parallel, intersecting, and
tangent.
➢ Lines – specific length and non-specific
length.
➢ Ray – Straight line that extends to infinity
f
r
o
ma specified point.

Relationship of one line to another line or arc

Dividing a line into equal parts
• Draw a line MO at any convenient angle (preferably an acute
angle) from point M.
• From M and along MO, cut off with a divider equal divisions
(say three) of any convenient length.
• Draw a line joining RN.
• Draw lines parallel to RN through the remaining points on line
MO. The intersection of these lines with line MN will divide the
line into (three) equal parts.

Planar tangent condition exists when two
geometric forms meet at a single point and do
not intersect.

Locating tangent points on circle and arcs

Drawing an arc tangent to a given point on the line
Steps
• Given line AB and
tangent point T.
Construct a line
perpendicular to line AB
and through point T.
• Locate the center of the
arc by making the radius
on the perpendicular
line. Put the point of the
compass at the center of
the arc, set the compass
for the radius of the arc,
and draw the arc which
will be tangent to the line
through the point T.

Drawing an arc, tangent to two lines

Drawing an arc, tangent to a line and an arc
(a) that do not intersect (b) that intersect

Construction of Regular Polygon of given length AB
(A) (B)
Draw a line of length AB. With A as centre and radius AB, draw a
semicircle.
With the divider, divide the semicircle into the number of sides of the
polygon.
Draw a line joining A with the second division-point2.

Construction of Regular Polygon of given length AB…...
(A) (B)
The perpendicular bisectors of A2 and AB meet at O. Draw a circle
with centre O and radius OA. With length A2, mark points F, E, D & C
on the circumferences starting from 2 (Inscribe circle method)
With centre B and radius AB draw an arc cutting the lineA6 produced
at C. Repeat this for other points D, E & F (Arc method)

General method of drawing any polygon
Draw AB = given length of polygon
At B, Draw BP perpendicular & = AB
Draw Straight line AP
With center B and radius AB, draw arc AP.
The perpendicular bisector of AB meets st.
line AP and arc AP in 4 and 6 respectively.
Draw circles with centers as 4, 5,&6 and
radii as 4B, 5B, & 6B and inscribe a square,
pentagon, & hexagon in the respective
circles.
Mark point 7, 8, etc with 6-7,7-8,etc. = 4-5 to
get the centers of circles of heptagon and
octagon, etc.

Inscribe a circle inside a regular polygon
• Bisect any two
adjacent internal
angles of the
polygon.
• From the
intersection of
these lines, draw a
perpendicular to
any one side of the
polygon (say OP).
• With OP as radius,
draw the circle with
O as center.

Inscribe a regular polygon of any number of sides
(say n = 5), in a circle
Draw the circle with diameter
AB.
Divide AB in to “n” equal parts
Number them.
With center A & B and radius
AB, draw arcs to intersect at P.
Draw line P2 and produce it to
meet the circle at C.
AC is the length of the side of
the polygon.

Inside a regular polygon, draw the same number of equal
circles as the side of the polygon, each circle touching
one side of the polygon and two of the other circles.
● Draw bisectors of all
the angles of the
polygon, meeting at O,
thus dividing the
polygon into the same
number of triangles.
● In each triangle
inscribe a circle.

Inside a regular polygon, draw the same number of
equal circles as the side of the polygon, each circle
touching two adjacent sides of the polygon and two of
the other circles.
● Draw the perpendicular
bisectors of the sides of
the polygon to obtain
same number of
quadrilaterals as the
number of sides of the
polygon.
● Inscribe a circle inside
each quadrilateral.

To draw a circle touching three lines inclined to
each other but not forming a triangle.
● Let AB, BC, and AD be
the lines.
● Draw bisectors of the
two angles, intersecting
at O.
● From O draw a
perpendicular to any one
line intersecting it at P.
● With O as center and OP
as radius draw the
desired circle.

Outside a regular polygon, draw the same number of
equal circles as the side of the polygon, each circle
touching one side of the polygon and two of the other
circles.
● Draw bisectors of two
adjacent angles and
produce them outside
the polygon.
● Draw a circle touching
the extended
bisectors and the side
AB (in this case) and
repeat the same for
other sides.

Construction of an arc tangent of given radius to two given
arcs
• Given - Arcs of radii M and N. Draw an arc of radius AB units which is
tangent to both the given arcs. Centers of the given arcs are inside the
required tangent arc.
Steps:
From centers C and D of the
given arcs, draw construction
arcs of radii (AB – M) and (AB
N), respectively.
With the intersection point as
the center, draw an arc of radius
AB.
This arc will be tangent to the
two given arcs.
Locate the tangent points T1 and
T2.

Construction of line tangents to two circles (Open belt)
Given: Circles of radii R1 and R with centers O and P, respectively.
Steps:
With P as center and a radius
equal to (R-R1) draw an arc.
Locate the midpoint of OP as
perpendicular bisector of OP as
“M”.
With as centre and as
radius draw a semicircle.
Locate the intersection point T
between the semicircle and the
circle with radius (R-R1).
draw a line PT and extend it to locate T1.
Draw OT2 parallel to PT1.
The line T1 to T2 is the required tangent

Construction of line tangents to two circles (crossed belt)
Given: Two circles of radii R1 and R with centers O and P, respectively.
Steps:
Using P as a center and a
radius equal to (R+ R1) draw
an arc.
Through O draw a tangent to
this arc.
Draw a line PT cutting the
circle at T1
Through O draw a line OT2
parallel to PT1.
The line T1T2 is the required
tangent.

Engineering Drawing
Lecture 3
SCALES AND Engineering Curves

Definition
A scale is defined as the ratio of the linear
dimensions of the object as represented in a
drawing to the actual dimensions of the
same.

Necessity
• Drawings drawn with the same size as the objects are
called full sized drawing.
• It is not convenient, always, to draw drawings of the
object to its actual size. e.g. Buildings, Heavy
• Hence scales are used to prepare drawing at
• Full size
• Reduced size
• Enlarged size

BIS Recommended Scales
Reducing scales 1:2 1:5 1:10
1:20 1:50 1:100
1:Y (Y>1) 1:200 1:500 1:1000
1:2000 1:5000 1:10000
Enlarging scales 50:1 20:1 10:1
X:1 (X>1) 5:1 2:1
Full size scales 1:1
Intermediate scales can be used in exceptional cases
where recommended scales can not be applied for
functional reasons.

Types of Scale
• Engineers Scale :
The relation between the dimension on the drawing
and the actual dimension of the object is mentioned
numerically (like 10 mm = 15 m).
• Graphical Scale:
Scale is drawn on the drawing itself. This takes care
of the shrinkage of the engineer’s scale when the
drawing becomes old.

Types of Graphical Scale
• Plain Scale
• Diagonal Scale
• Vernier Scale
• Comparative scale

Representative fraction (R.F.)
When a 1 cm long line in a drawing represents 1
meter length of the object,
R.F  1cm

1cm

1
1m 1x 100cm 100

Plain scale
• A plain scale consists of a line divided into suitable
number of equal units. The first unit is subdivided
into smaller parts.
• The zero should be placed at the end of the 1st main
unit.
• From the zero mark, the units should be numbered
to the right and the sub-divisions to the left.
• The units and the subdivisions should be labeled
clearly.
• The R.F. should be mentioned below the scale.

Construct a scale of 1:4, to show centimeters and long
enough to measure up to 5 decimeters.
• R.F. = ¼
• Length of the scale = R.F. × max. length = ¼ × 5 dm = 12.5 cm.
• Draw a line 12.5 cm long and divide it in to 5 equal divisions, each
representing 1 dm.
• Mark 0 at the end of the first division and 1, 2, 3 and 4 at the end
of each subsequent division to its right.
• Divide the first division into 10 equal sub-divisions, each
representing 1 cm.
• Mark cm to the left of 0 as shown.

Question: Construct a scale of 1:4, to show centimeters and
long enough to measure up to 5 decimeters
•
instead of only a line.
•
Draw the division lines showing decimeters throughout the width
of the scale.
•
Draw thick and dark horizontal lines in the middle of all
alternate divisions and sub-divisions.
•
Below the scale, print DECIMETERS on the right hand side,

CENTIMERTERS on the left hand side, and R.F. in the middle.

Diagonal Scale
• Through Diagonal scale, measurements can be up
to second decimal (e.g. 4.35).
• Diagonal scales are used to measure distances in a
unit and its immediate two subdivisions; e.g. dm,
cm & mm, or yard, foot & inch.
• Diagonal scale can measure more accurately than
the plain scale.

Diagonal scale…..Concept
•
At end B of line AB, draw a perpendicular.
•
Step-off ten equal divisions of any length along
the perpendicular starting from B and ending
at C.
•
Number the division points 9,8,7,…..1.
•
Through the points 1, 2, 3, etc., draw lines
parallel to AB and cutting AC at 1´, 2´, 3´, etc.
•
Since the triangles are similar; 1´1 = 0.1 AB,
2´2 = 0.2AB, …. 9´9 = 0.9AB.

•
Gives divisions of a given short line AB in
multiples of 1/10 its length, e.g. 0.1AB, 0.2AB,
0.3AB, etc.

Construct a Diagonal scale of RF = 3:200 (i.e. 1:66 2/3) showing
meters, decimeters and centimeters. The scale should measure up
to 6 meters. Show a distance of 4.56 meters
• Length of the scale = (3/200) x 6 m = 9 cm
• Draw a line AB = 9 cm . Divide it in to 6 equal parts.
• Divide the first part A0 into 10 equal divisions.
• At A draw a perpendicular and step-off along it 10 equal
divisions, ending at D.

Diagonal Scale
•
• Draw perpendiculars at meter-divisions i.e. 1, 2, 3, and 4.
• Draw horizontal lines through the division points on AD. Join D
with the end of the first division along A0 (i.e. 9).
• Through the remaining points i.e. 8, 7, 6, … draw lines // to D9.
• PQ = 4.56 meters
Complete the rectangle ABCD.

Vernier Scales
• Similar to Diagonal scale, Vernier scale is used for measuring up
to second decimal.
• A Vernier scale consists of (i) a primary scale and (ii) avernier.
• The primary scale is a plain scale fully divided in to minor
divisions.
• The graduations on the vernier are derived from those on the
primary scale.
Least count (LC) is the minimum distance that can be
measured.
Forward Vernier Scale :
MSD>VSD; LC = MSD-VSD
Backward Vernier scale:
VSD>MSD; LC = VSD - MSD

Vernier scale…. Concept
• Length A0 represents 10 cm and is divided in to 10 equal parts
each representing 1 cm.
• B0 = 11 (i.e. 10+1) such equal parts = 11 cm.
• Divide B0 into 10 equal divisions. Each division of B0 will be
equal to 11/10 = 1.1 cm or 11 mm.
• Difference between 1 part of A0 and one part of B0 = 1.1 cm -1.0
cm = 0.1cm or 1 mm.

Question: Draw a Vernier scale of R.F. = 1/25 to read up
to 4 meters. On it show lengths 2.39 m and 0.91 m
• Length of Scale = (1/25) × (4 × 100) = 16 cm
• Draw a 16 cm long line and divide it into 4 equal parts. Each part is
1 meter. Divide each of these parts in to 10 equal parts to show
decimeter (10 cm).
• Take 11 parts of dm length and divide it in to 10 equal parts. Each of
these parts will show a length of 1.1 dm or 11 cm.
• To measure 2.39 m, place one leg of the divider at A on 99 cm mark
and other leg at B on 1.4 mark. (0.99 + 1.4 = 2.39).
• To measure 0.91 m, place the divider at C and D (0.8 +0.11 = 0.91).

2
Common Engineering Curves
Parabolic shape
Elliptical shape
Hyperbola
spiral 0

Conic curves (conics)
Curves formed by the intersection of a plane with a
right circular cone. e.g. Parabola, hyperbola and ellipse
Right circular cone
is a cone that has a
circular base and
the axis is inclined
at 900 to the base
and passes through
the center of the
base.
21

22
Conic sections are always
"smooth". More precisely, they
never contain any inflection
points. This is important for
many applications, such as
aerodynamics, civil engg.,
mechanical engg, etc.

Conic
Conic is defined as the locus of a point moving in a plane such that
the ratio of its distance from a fixed point and a fixed straight line is
always constant.
Fixed point is called Focus
Fixed line is called Directrix
23

Eccentrici ty  Distance of the point from the focus
Distance of the point from the directric
When eccentricity
< 1 € Ellipse
=1 € Parabola
> 1 € Hyperbola
eg. when e=1/2, the curve is an Ellipse, when e=1, it is a parabola
and when e=2, it is a hyperbola. 24

25
Ellipse
An ellipse is
obtained when a
section plane,
inclined to the axis,
cuts all the
generators of the
cone.

26
Focus-Directrix or Eccentricity Method
Given : the distance of focus from the directrix and eccentricity
Example : Draw an ellipse if the distance of focus from the directrix is 70
mm and the eccentricity is 3/4.
1. Draw the directrix AB and
axis CC’
2. Mark F on CC’ such that
CF = 70 mm.
3. Divide CF into 7 equal
parts and mark V at the
fourth division from C.
Now, e = FV/ CV = 3/4.
4. At V, erect a perpendicular
VB = VF. Join CB. Through
F, draw a line at 45° to meet
CB produced at D. Through
D, drop a perpendicular
DV’ on CC’. Mark O at the
midpoint of V– V’.

Focus-Directrix or Eccentricity Method ( Continued)
5. With F as a centre and radius =
1–1’, cut two arcs on the
perpendicular through 1 to
locate P1 and P1’. Similarly,
with F as centre and radii = 2–
2’, 3–3’, etc., cut arcs on the
corresponding perpendiculars
to locate P2 and P2’, P3 and
P3’, etc. Also, cut similar arcs
on the perpendicular through
O to locate V1 and V1’.
6. Draw a smooth closed curve
passing through V,P1, P/2, P/3,
…, V1, …, V’, …, V1’, … P/3’,
P/2’, P1’.
7. Mark F’ on CC’ such that V’ F’
= VF.
27

ME 111: Engineering Drawing
Lecture 4
08-08-2011
Engineering Curves and Theory of Projection
Indian Institute of Technology Guwahati
Guwahati – 781039

Eccentrici ty  Distance of the point from the focus
Distance of the point from the directric
When eccentricity
< 1 € Ellipse
=1 € Parabola
> 1 € Hyperbola
eg. when e=1/2, the curve is an Ellipse, when e=1, it is a parabola
and when e=2, it is a hyperbola. 2

3
Focus-Directrix or Eccentricity Method
Given : the distance of focus from the directrix and eccentricity
Example : Draw an ellipse if the distance of focus from the directrix is 70
mm and the eccentricity is 3/4.
1. Draw the directrix AB and
axis CC’
2. Mark F on CC’ such that
CF = 70 mm.
3. Divide CF into 7 equal
parts and mark V at the
fourth division from C.
Now, e = FV/ CV = 3/4.
4. At V, erect a perpendicular
VB = VF. Join CB. Through
F, draw a line at 45° to meet
CB produced at D. Through
D, drop a perpendicular
DV’ on CC’. Mark O at the
midpoint of V– V’.

Focus-Directrix or Eccentricity Method ( Continued)
5. With F as a centre and radius =
1–1’, cut two arcs on the
perpendicular through 1 to
locate P1 and P1’. Similarly,
with F as centre and radii = 2–
2’, 3–3’, etc., cut arcs on the
corresponding perpendiculars
to locate P2 and P2’, P3 and
P3’, etc. Also, cut similar arcs
on the perpendicular through
O to locate V1 and V1’.
6. Draw a smooth closed curve
passing through V,P1, P/2, P/3,
…, V1, …, V’, …, V1’, … P/3’,
P/2’, P1’.
7. Mark F’ on CC’ such that V’ F’
= VF.
4

5
Constructing a Parabola (Eccentricity Method)
Example. Draw a parabola if the distance of the focus from
the directrix is 60 mm.
1. Draw directrix AB and axis CC’ asshown.
2. Mark F on CC’ such that CF = 60mm.
3. Mark V at the midpoint of CF. Therefore, e =
VF/ VC = 1.
4. At V,erect a perpendicular VB = VF. JoinCB.
5. Mark a few points, say, 1, 2, 3, … on VC’ and
erect perpendiculars through them meeting
CB produced at 1’, 2’, 3’, …
6. With F as a centre and radius = 1–1’, cut two
arcs on the perpendicular through 1 to locate
P1 and P1’. Similarly, with F as a centre and
radii = 2–2’, 3–3’, etc., cut arcs on the
corresponding perpendiculars to locate P2
and P2’, P3 and P3’, etc.
7. Draw a smooth curve passing through V, P1,
P2, P3 … P3’, P2’, P1’.

Constructing a Hyperbola (Eccentricity Method)
Draw a hyperbola of
e = 3/2 if the distance
of the focus from the
directrix = 50 mm.
Construction similar
to ellipse and
parabola
6

Drawing Tangent and Normal to any conic
When a tangent at any point on the curve (P) is produced to meet the
directrix, the line joining the focus with this meeting point (FT) will be at
right angle to the line joining the focus with the point of contact (PF).
The normal to the curve at any point is perpendicular to the tangentat
that point.
7

Another definition of the ellipse
An ellipse is the set of all points in a plane for which the sum of
the distances from the two fixed points (the foci) in the plane is
constant.
8

9
Arcs of Circle Method
Given conditions: (1) the major axis and minor axis are known OR
(2) the major axis and the distance between the foci are known
Draw AB & CD perpendicular to each
other as the major diameter minor
diameter respectively.
With centre as C or D, and half the major
diameter as radius draw arcs to intersect
the major diameter to obtain the foci at X
and Y.
Mark a numbe of points along line
segment XY and number them. Points need
not be equidistant.
Set the compass to radius B-1 and draw
two arcs, with Y as center. Set the compass
to radius A1, and draw two arcs with X as
center. Intersection points of the two arcs
are points on the ellipse. Repeat this step
for all the remaining points.
Use the French curve to connect the points,
thus drawing the ellipse.

Constructing an Ellipse (Concentric Circle Method)
Given:
Major axis and
minor axis
• With center C, draw two concentric circles with diameters equal to major and minor
diameters of the ellipse. Draw the major and minor diameters.
• Construct a line AB at any angle through C. Mark points D and E where the line
intersects the smaller circle.
• From points A and B, draw lines parallel to the minor diameter. Draw lines parallel to
the major diameter through D & E.
• The intersection of the lines from A and D is point F, and from B and E is point G.
Points F & G lies on the ellipse.
• Extend lines FD & BG and lines AF and GE to obtain two more points in the other
quadrants.
• Repeat steps 2-6 to create more points in each quadrant and then draw a smo
10
oth
curve through the points.

Constructing a Parabola (Parallelogram Method)
Example: Draw a parabola of base 100 mm and axis 50 mm if the axis
makes 70° to the base.
1. Draw the base RS = 100 mm and through its midpoint K, draw the axis KV = 50 mm, inclined
at 70° to RS. Draw a parallelogram RSMN such that SM is parallel and equal to KV.
2. Divide RN and RK into the same number of equal parts, say 5. Number the divisions as 1, 2, 3,
4 and 1’, 2’, 3’, 4’, starting from R.
3. Join V–1, V–2, V–3 and V–4. Through 1’, 2’, 3’ and 4’, draw lines parallel to KV to meet V–1 at
P1, V–2 at P2, V–3 at P3 and V–4 at P4, respectively.
4. Obtain P5, P6, P7 and P8 in the other half of the rectangle in a similar way. Alternatively, these
points can be obtained by drawing lines parallel to RS through P1, P2, P3 and P4. For example,
draw P1– P8 such that P1– x = x– P8. Join P1, P2, P3 … P8 to obtain the parabola. 11

Hyperbola
A Hyperbola is obtained
when a section plane,
parallel/inclined to the axis
cuts the cone on one side of
the axis.
A Rectangular Hyperbola is
obtained when a section,
parallel to the axis cuts the
cone on one side of the axis.
12

Hyperbola Mathematical definition
A hyperbola is
defined as the set of
points in a plane
whose distances
from two fixed
points called foci, in
the plane have a
constant difference.
13

Constructing a Hyperbola
Given: Distance between Foci and Distance between vertices
Draw the axis of symmetry and
construct a perpendicular through
the axis. Locate focal point F
equidistant from the perpendicular
and on either side of it. Locate
points A and B on the axis
equidistant from the perpendicular.
AB is the distance between vertices
With F as center and radius R1, and
draw the arcs. With R1 + AB, radius,
and F as center, draw a second set
of arcs. The intersection of the two
arcs on each side of the
perpendicular are points on the
hyperbola
Select a new radius R2 and repeat
step 2. Continue this process until
several points on the hyperbola are
marked
14

Roulettes
• Roulettes are curves generated by the rolling
contact of one curve or line on another curve or
line, without slipping.
• There are various types of roulettes.
• The most common types of roulettes used in
engineering practice are: Cycloids,
Trochoids, and Involutes.

Cycloid
Generating circle
Base line
A Cycloid is generated by a point on the circumference of a
circle rolling along a straight line without slipping
The rolling circle is called the Generating circle
The straight line is called the Directing line or Base line

➢ Generating circle has its center at C and has a radius of C-P’. Straight line
P
P
’
i
s equal in length to the circumference of the circle and is tangent to the circle at
point P’.
➢ Divide the circle into a number of equal segments, such as 12. Number t
h
e
intersections of the radii and the circle.
➢ From each point of intersection on the circle, draw a construction line parallel
to l
i
n
ePP’ and extending up to line P’C’.
➢ Divide the line CC’ into the same number of equal parts, and number
them. D
r
a
wvertical lines from each point to intersect the extended horizontal
centerline of the circle. Label each point as C1, C2, C3, …. C12.
Constructing a cycloid

Constructing a cycloid (contd.)
Using point C1 as the center and radius of the circle C-P’, draw an arc that
intersects the horizontal line extended from point 1 at P1. Set the compass at point
C2, then draw an arc that intersects the horizontal line passing through point 2 at
P2. Repeat this process using points C3, C4, …. C12, to locate points along the
horizontal line extended from points 3, 4, 5, etc..
Draw a smooth curve connecting P1, P2, P3, etc to form the cycloid
Draw normal NN and Tangent TT

Epicycloid
The cycloid is called Epicycloid when the generating circle
rolls along another circle outside it.

Constructing an Epicycloid
1) With O as centre
and OC as radius,
draw an arc to
represent locus of
centre.
2) Divide arc PQ in
to 12 equal parts
and name them as
1’, 2’, …., 12’.
3) Join O1’, O2’, … and produce them to cut the locus of centres at C1, C2, ….
4) Taking C1 as centre, and radius equal to 20 mm, draw an arc cutting the arc
through 1 at P1. Similarly obtain points P2, P3,…., P12.
5) Join P1, P2….. With French curve

Hypocycloid
Hypocycloid is obtained when the generating circle rolls
along another circle inside it.

Constructing an Hypocycloid
Construction is similar to epicycloid. The generating
circle is to be drawn below the base circle

Trochoid
• Trochoid is a curve generated by a point outside or inside the
circle rolling along a straight line.
• If the point is outside the circle the curve obtained is called
Superior Trochoid
• If the point is inside the circle, the curve obtained is called
Inferior Trochoid

Classification of Cycloidal curves
Generating Circle
On the
directing line
Outside the
directing line
Inside the
directing line
Generating
point
On the
generating
circle
Cycloid Epicycloid Hypocycloid
Outside the
generating
circle
Superior
trochoid
Superior
epitrochoid
Superior
Hypotrochoid
Inside the
generating
circle
Inferior
trochoid
Inferior
epitrochoid
Inferior
hypotrochoid

Involute
An Involute is a curve traced by the free end of a thread
unwound from a circle or a polygon in such a way that
the thread is always tight and tangential to the circle or
side of the polygon

Construction of Involute of circle
Draw the circle with c as center
and CP as radius.
Draw line PQ = 2CP, tangent to
the circle at P
Divide the circle into 12 equal
parts. Number them as 1, 2…
Divide the line PQ into 12 equal
parts and number as 1´, 2´…..
Draw tangents to the circle at 1,
2,3….
Locate points P1, P2 such that 1-
P1 = P1´, 2-P2 = P2´….
Join P, P1, P2….
The tangent to the circle at any point on it is always normal to the its involute.
Join CN. Draw a semicircle with CN as diameter, cutting the circle at M.
MN is the normal.

Theory of Projections
27
Engineering Drawing

3-D objects and structures are
represented graphically on 2-D media.
All projection theory are based on two
variables:
➢ Line of sight
➢ Plane of projection.
Projection theory

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

Projection Methods
Projection methods are very important
techniques in engineering drawing.
➢ Perspective and
➢ Parallel.

In perspective
projection, all
lines of sight
start at a
single point.

In parallel projection, all lines of sight are parallel.

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
Distance from the observer to the object is finite and
the object is viewed from a single point – projectors
are not parallel.
Perspective projections mimic what the human eyes
see, however, they are difficult to draw.
Parallel vs Perspective Projection

Engineering Drawing
Lecture 5
Projection of Points
1

Orthographic Projection
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

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.
Front view of an object shows the width and height
dimensions.

Horizontal plane of projection
Horizontal plane of projection is the plane onto
which the Top View of the multi-view drawing is
projected.
Top view of an object shows the width and depth
dimensions.

Profile plane of projection
In multi-view drawings, the right side view is the standard
side view used. 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.

Orientation of views from projection
planes
Top view is always
positioned and
aligned with the
front view, and side
view is always
positioned to the
side of and aligned
with the front view.

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 six mutually perpendicular
views that are produced by six mutually perpendicular
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.

Six Principal Views
Object is
suspended in a
glass box
producing six
principal views:
each view is
perpendicular to
and aligned with
the adjacent
views.

Unfolding the glass
box to produce six-
view drawing
Top, front and bottom
views are all aligned
vertically and share
the same width
dimension.
Rear, left side, front
and right side views
are all aligned
horizontally and share
the same height
dimension.

Conventional view placement
The three-view multiview
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.
The standard views used in
a three-view drawing are
the top, front and the right
side views

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.

The principal
projection
planes and
quadrants
used to
create first-
and third-
angle
projection
drawings

Orthographic projection and
Projection of points
18

FV Y1 RHSV
TV
X1
Example
X Y
1. Visible
2. Hidden
3. Center

Convention
• Projectors and the lines of the intersection of planes
of projections are shown as thin lines.

Precedence of Lines
• Visible lines take precedence over all other lines
0.70 mm
0.35 mm
• Center lines have lowest precedence
0.35 mm

Example: Application of Precedence

Intersecting Lines in Orthographic Projections
Solid Line Intersections
Dashed Line Special Case Intersections

Projection of Points
(Orthographic)

A POINT
Define its position with respect to the coordinates.
With respect to the VP, HP, & PP
P
P

Direction of rotation of the HP

Convention
• Top views are represented by only small letters eg. a.
• Their front views are conventionally represented by
small letters with dashes eg. a´
• Profile or side views are represented by small letters
with double dashes eg. a´´

Convention
• The line of
intersection of HP and
VP is denoted as XY.
• The line of
intersection of VP and
PP is denoted as X1Y1

Point in the First quadrant
Point P is 40 mm in front of VP, 50 mm above HP, 30 mm in front of leftprofile
plane (PP)

Y1
Procedure
• Draw a thin horizontal
line, XY, to represent
the line of intersection
• Draw X1Y1 line to
represent the line of
intersection of VP and
PP.
• Draw the Top View (p).
• Draw the projector line
• Draw the Front View
(p´)

Procedure
• To project the right
view on the left PP,
draw a horizontal
projector through p to
intersect the 45 degree
line at m.
• through m draw a
vertical projector to
intersect the horizontal
projector drawn
through p´ at p´´.
• p´´ is the right view of
point P

Engineering Drawing
Lecture 6
Projection of Lines
1

T he im ag e c an not be dis played. Y our c om p ut er m ay not have en ou g h m em ory t o ope n t he im ag e, or t he im age m ay have bee n c orrupt ed. Res t art your c om put er, and t hen ope n t he file ag ain. If t he red x st ill ap pears , you m ay hav e t o delet e t he im age an d t he n ins ert it again .

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.

Point in the Third quadrant
point P is 40 mm behind VP, 60 mm below HP and 30 mm
behind the right PP.
Since the three planes of projections lie in between the observer and the point P,
they are assumed as transparent planes.

Point in the Fourth quadrant
point P is 60 mm below HP, 50 mm in front of VP, 45 mm in
front of the left PP.

First Angle Projection
Object in the first quadrant

Placing the object in
the third quadrant
puts the projection
planes between the
viewer and the
object.
When placed in the
first quadrant, the
object is between
the viewer and the
projection planes.

Difference between first- and third-angle projections
First angle projection Third-angle projection
Object is kept in the first quadrant. Object is assumed to be kept in
the third quadrant.
Object lies between observer and the
plane of projection.
The plane of projection is assumed to
be non-transparent.
Front (elevation) view is drawn above
the XY line
Top (plan) view is drawn below the XY
line
Left view is projected on the right
plane and vise versa
Plane of projection lies between
the observer and the object.
The plane of projection is
assumed to be transparent.
Front (elevation) view is drawn
below the XY line
Top (plan) view is drawn above
the XY line
Left view is projected on the left
plane itself.
Followed in India, European countries Followed in USA

Symbol of projection
The method of projection used should be indicated 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.

Straight line
Locus of a point, which moves linearly the
shortest distance between any two given points.
Location of a line
The location of a line in projection quadrants is
described by specifying the distances of its end
points from the VP, HP and PP.
• Parallel to both the planes.
• Parallel to one plane and perpendicular to the
other.
• Parallel to one plane and inclined to the other.
• Inclined to both the planes.

Projection of a line
• 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.

Line parallel to both HP & VP
Line of 80 mm length is
placed 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.

Line parallel to both HP & VP…
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´´).

Line perpendicular to HP & parallel to VP
Line of 80 mm length
placed parallel to VP
and perpendicular to
HP.
The line is 60 mm in
front of VP and 70
mm in front of right
PP.
The lower end of the
line is 30 mm above
HP.

Line perpendicular to HP & parallel toVP…
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 X1Y1 line at a distance of 70 mm on
the right- side of the front view.
Through 0 the point of intersection of XY and
X1Y1, 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.

Line parallel to one plane and inclined to
the other

Line parallel to VP and inclined to HP….
A line AB, 90 mm long
is inclined at 30
degrees to HP and is
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.

Line parallel to VP and inclined toHP….
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 Y line at a distance of 50 mm
from a'.
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 at a´´ and b´´. Connect a´´b´´
which is the left side view.

Line inclined to HP and VP
Apparent Inclinations:  and 
Apparent Lengths: ab, a´b´

Line inclined to HP and VP…….
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 aregiven.
Since the line AB is inclined
at  to HP and  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  to
HP and  to VP.

Engineering Drawing
Lecture Slides
1

Engineering Drawing
Lecture 7
Projection of Lines Contd.
2

Step 1: Rotate the line AB to make it parallel to VP.
Rotate the line AB about the end A,
keeping , 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.
AB1 is the new position of the line
AB when it is inclined at  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  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.

Step 2: Rotate the line AB to make it parallel to HP.
Rotate the line AB about the end A
keeping  the inclination of AB with VP
constant, till it becomes parallel to HP.
This rotation of the line will bring the
end B to the second new Position B2.
AB2 is the new position of the line AB,
when it is inclined at  to VP and
parallel to HP.
Project AB2 on HP and VP. Since AB2
is parallel to HP, ab2, the projection of
AB2 on HP is in true length inclined at
 to XY line, and a´b2´ the projection of
AB2 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
When the line AB is swept around about the
end A keeping , the inclination of the line
with the HP constant, by one complete
rotation, 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  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  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

Step 4: Locus of end B in the top view
When the line AB is swept around about the end A
keeping  the inclination of the line with the VP
constant, by one complete rotation, 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  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  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
b2 and b1´ respectively.
• With center a, and radius ab2 draw an
arc to intersect the locus line through b2
at b. Connect ab the top view of the line
AB.
• Similarly with center a', and radius a'b2'
draw an arc to intersect the locus line
through b1' at b´. Connect a'b' the front
view of the line AB.

Line inclined to HP and VP
Orthographic projection
1. As the location of one of the end points
(i.e. A) with respect to HP and VP, is
given, mark a and a´, the top and the
front views of point A.
2. Suppose the line AB is assumed to be
made parallel to VP and inclined at  to
HP. The front view of the line will be
equal to the true length of the line and
also, the inclination of the line with HP
is seen in the front view. Draw a'b1'
passing through a' at  to XY line and
equal to the true length of AB. a'b1' is
projected down to get ab1, the top view
parallel to the XY line.

Orthographic projection…..
3. Now the line AB is assumed to be made parallel
to HP and inclined at  to VP. The top view of
the line will be equal to the true length of the
line and also , the inclination of the line with
VP is seen in the top view.
4. Draw ab2 passing through a at  to the XY line
and equal to the true length of AB. ab2, is
projected up to get a'b2', the front view parallel
to the XY line.
5. Draw the horizontal locus lines through b2, and
b1'.
6. 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.
7. 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.

To Find True length and true inclinations of a line
• Given : The top and front views of a line are given
• This is of great importance since some of the engineering problems
may be solved by this principle.
• The problems may be solved by
(i) Rotating line method or
(ii) Rotating trapezoidal plane method or
(iii) Auxiliary plane method.
• The top and front views of the object can be drawn from the
following data: (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.

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 above
procedure, 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.

Step by Step Procedure
Top and front views
Draw the top view ab and the
front view a'b' as given
2. Rotation of the top view
With center a and radius ab
rotate the top view to the new
position ab1 to make it parallel
to the XY line. Since ab1 is
parallel to the XY line, its
corresponding front view will be
in true length and at true
inclination.

3. Rotation of the front view
Similarly, with center a' and radius
a'b' rotate the front view to the new
position a'b2' parallel to the
XY line. Since a'b2' is
parallel to the XY line, its
corresponding top view will be in true
length and at true inclination.
In this position, the line will be
parallel to HP and inclined at  to VP.
Through b draw the locus of B in the
top view. Project b2' to get b2, in the
top view. Connect ab2 which will be in
true length and true inclination 
which the given line AB makes with
VP.

Traces of a line
The trace of a line is defined as a point at which the
given line, if produced, meets or intersects a plane.
When a line meets HP, (or if necessary on the extended
portion-of HP), the point at which the line meets or
intersects the horizontal plane, is called horizontal
trace (HT)of the line and denoted by the letter H.
When a line meets VP (or if necessary on the extended
portion of VP), the point at which the line meets or
intersects the vertical plane, is called vertical trace (VT)
of the line and denoted by the letter V.

When the line is parallel to both HP and VP, there will be
no traces on the said planes. Therefore the traces of lines
are determined in the following positions of the lines.
Lines perpendicular to one plane and parallel to the
other.
Lines inclined to one plane and parallel to the other.
Lines inclined to both the planes.

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.

TRACE of the line perpendicular to the VP

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.
A. Line inclined at  to HP and parallel to VP

Line inclined at  to VP and parallel to HP

Traces of a line inclined to both the planes
Line inclined at  to HP and  to VP.
The line when extended intersects HP at H, the horizontal trace, but will never intersectthe
portion of VP above XY line, i.e. within the portion of the VP in the 1st 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 HT and VT of the line AB lie below XYline.

TERMS USED IN PROJECTIONS OF PLANES
• True Shape The actual shape of a plane is called its true shape.
• Inclination with the HP: It is the acute angle the plane makes with theHP.
• Inclination with the VP It is the acute angle the plane makes with the VP.
• Traces of the Plane The traces of a plane are the lines of intersections of
the plane with the RPs.
• A plane may have a horizontal trace or vertical trace or both.
• Horizontal Trace (HT) The real or imaginary line of intersection of a plane
with the HP is called horizontal trace of the plane. HT is always located in
the TV.
• Vertical Trace (VT) The real or imaginary line of intersection of a plane with
the VP is called vertical trace of the plane. VT is always located in the FV.
• Line View or Edge View The view of a plane seen as a line is called line
view or edge view of the plane. One view of a perpendicular plane is
always an edge view.
29

Example.1
A rectangle ABCD of size 30 mm x 20mm is parallel to
the HP and has a shorter side AB perpendicular to the
VP. Draw its projections
37

Engineering Drawing
Lecture 8
Projection of Planes
1

Example.1
A rectangle ABCD of size 30 mm x 20mm is parallel to
the HP and has a shorter side AB perpendicular to the
VP. Draw its projections
3

Example.2 : Draw the projection of a circle of 5 cm diameter,
having its plane vertical and inclined at 30 to the VP. Its center is
3 cm above the HP and 2 cm in front of the VP.
10

Let us first assume that the plane is perpendicular to HP and parallel to VP.
So in top view a straight line and in front view the circle.
Divide the circle into 12 equal parts and name them 1´, 2´, 3´, 4´ etc.
Project these points to top view. 11

Rotate the top view by 30 to the XY line (because it is inclined to VP at 30)
Project the points to front view to get the finalfront view. 12

Example: A rectangle ABCD of size 30 mm x 20 mm is inclined to
the HP at 30°. Its shorter side AB is parallel to the HP and inclined
at 45°to the VP. Draw the projections of the rectangle.
13

Example 4. A Pentagonal plane lamina of edge 25 mm is resting on HP
with one of its corners touching it such that the plane surface makes
an angle of 60 with HP. Two of the edges containing the corner on
which the lamina rests make equal inclinations with HP. When the
edge opposite to this corner makes an angle of 45 with VP and nearer
to the observer, draw the TV and FV of the lamina.
14

A. The lamina is resting on HP.
B. The lamina rests with corner C on HP such that lamina is inclined at
60 to
HP. Edge ae is perpendicular to XY line.
C. Lamina is rotated such that edge AE is inclined at 45 to VP and
redrawn.

Engineering Drawing
Lecture 9
Projections on Auxiliary Planes

Need for 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
The auxiliary view method may be applied
▪ To find the true length of a line.
▪ To project a line which is inclined to both HP and VP as a
point.
▪ To project a plane surface or a lamina as a line.

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.
If the auxiliary plane selected is perpendicular to HP and inclined to
VP, the views of the object projected on the auxiliary plane is called
auxiliary front view and the auxiliary plane is called auxiliary
vertical plane and denoted as AVP.
If the auxiliary plane is perpendicular to VP and inclined to HP, the
view of the object projected on the auxiliary plane is called auxiliary
top view and the auxiliary plane is called auxiliary inclined plane
and denoted as AIP.

Auxiliary Vertical Plane
AVP is placed in the first
quadrant with its surface
perpendicular to HP and
inclined at  to VP.
The object is to be placed
in the space in between
HP, VP and AVP. The AVP
intersects HP along the
X1Y1 line.
The direction of sight to
project the auxiliary front
view will be normal to 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

Auxiliary Inclined Plane
AIP is placed in the first quadrant
with its surface perpendicular to
VP and inclined at  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.
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-plane with that
of VP.

Projection of Points on Auxiliary Planes
Projection on AVP
Point Pis situated in the first quadrant at a height m above HP.An auxiliary vertical
planeAVPissetupperpendiculartoHPandinclinedattoVP.ThepointPis
projected on VP, HP andAVP.
p'istheprojectiononVP,pisthe projectiononHPandP1'isthe projectiononAVP.
Since point is at a height m above HP, both p' and p1’ are at a height m above the
XY and X1Y1lines, respectively

HPisrotatedby90degreetobringit
in plane of VP.
After the HP lies in-plane with VP,
the AVP is rotated about the X1Y1,
line, so that it becomes in-plane with
that of both HP and VP.

Orthographic
projections
Draw the XY line and mark
p and p', the top and front
views of the point P.
Since AVP is inclined at 
to VP, draw the X1Y1 line
inclined at  to the XYline
at any convenient distance
from p.
Since point P is at a height m above HP, theauxiliaryfrontviewp1'willalsobeat
aheight m above the X1Y1 line.
Therefore, mark P1
’ by measuring o1p1’=op’ = m on the projector drawn fromp
perpendicular to the X1Y1 line.

Projection on AIP
PointPissituatedinfirstquadrantatadistancenfromVP. Anauxiliary plane AIPis
setupperpendiculartoVPandinclinedattoHP.ThepointPisprojectedonVP,
HP and AIP.
p'istheprojectiononVP,pisthe projectiononHPandP1 istheprojection onAIP.
Since the point is at a distance n from VP, bothpandp1 areatadistancenabove
the XY and X1Y1 lines,respectively

HP is rotated by 90 degree
about XY line to bring it in
plane with VP.
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 andVP.
p and p’ lie on a vertical
projector perpendicular to the
XY line, and p’ and p1 lie on a
projector perpendicular to the
X1Y1 line which it self is
inclined at  to XYline.

Orthographic
projections
Draw the XY line and mark
p and p', the top and front
views of the point P.
SinceAIPisinclined at to
HP, draw the X1Y1 line
inclined at  to the XYline
at any convenient distance
from p’.
Since point P is at a distance n infront of VP, theauxiliarytopviewp1 willalsobe
atadistance n from the X1Y1 line.
Therefore, mark P1 by measuring o1p1=op = n on the projector drawn from p‘

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 X1Y1 line inclined at 
(the inclination of AVP with
VP) to the XY line.
• Draw X1Y1 line inclined at 
(the inclination of AIP with HP)
to XY line.
• Draw the projectors through
the top views of the points
perpendicular to the X1Y1
line.
• Draw the projectors through the
front views of the points
• The auxiliary front view of a
point is obtained by stepping
off a distance from the X1Y1
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 X1Y1 line equal to
to the distance of the top view
of the given point from the XY
line

Projection of lines on auxiliary planes
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
auxiliary plane will be perpendicular to VP and inclined at 
to HP, i.e., AIP, and the other will be perpendicular to HP
and inclined at  (true inclination) or  (apparent
inclination) to VP.

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 infront 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 infront of VP.
Assume that the line is
parallel to both HP and VP
and draw its top and front
views.
Since the line is to be
inclined at 300 to HP, set
up an AIP inclined at 300
to HP and perpendicular
to VP.

Draw X1Y1 line inclined at
300 to XY line at any
convenient distance from it.
To project an auxiliary top
view on AIP, draw
projections from a1’ and b1’
perpendicular to X1Y1 line,
and on them step off
1a1=3a and 2b1=4b from
the X1Y1 line.
Connect ab which will be
the auxiliary top view.
Since the top view of the line appears inclined to VP at 600, draw the
X2Y2 line inclined at 600 to the auxiliary top view ab at any convenient
distance from it. Draw the projections from a and b perpendicular to
X2Y2 and on them step off 5a’ = 3a1
’ and 6b’=4b1
’. Connect a’b’ which
will be the auxiliary front view.

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
Let A be one of the extremities
of the line AB at distance 60 mm
infront of VP and 45 mm above
HP.
Mark a1 and a1’ the top and the
front views of the extremity A.
Initially the line is assumed to
be parallel to HP and VP.
a1b1 and a1’b1’ 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 , 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 X1Y1 line inclined at  = 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  to HP and parallel
to VP.
Since the line is inclined at true
inclination  to VP, to project the
auxiliary front view an AVP
inclined at  to VP should be
setup.

To draw the Auxiliary F.V. on AVP
Already the line is inclined at  to
AIP and parallel to VP. If the line is
to be inclined at  to VP, an AVP
inclined at  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  to
ab.
The apparent inclination  of ab with the X2Y2 line should be found out. To
find , through a draw ab2 equal to 60 mm, the true length of AB inclined
at  = 450 to ab.

Through b2, draw the
locus of B parallel to
X1Y1 line. With center a
and radius ab strike an
arc to intersect the locus
of B at b3. Connect ab3
and measure its
inclination  with ab. Now
draw the X2Y2 line
inclined at  to ab. Mark
AVP and AIP on either
side of X2Y2. 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 shouldbe
shown in their point views.
If the point views of the lines are required, then first they have to be shownin
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
Projections of a pair of parallel lines
AB and PQ are shown. 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.
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

Draw the X1Y1 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 X1Y1 line.
Measure 5a1’=1a1’ along the
projector drawn through a from
the X1Y1 line, and 6b1’=2b’ along
the projector drawn through b
from the X1Y1 line.
Connect a1'b1' which will be equal
to the true length of the line AB.
Similarly by measuring 7p1' = 3p'
and 8q1' = 4q' obtain p1'q1' the
true length view of the line PQ.

The line AB and PQ are 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 X2Y2 perpendicular
to a1’b1' and p1'q1' at any
convenient distance from them.
Produce a1'b1' and p1'q1'.
Measure a5 = b6 = 9a1 along
a1'b1' produced from X2Y2.
Similarly obtain the point,view
p1(q1) by measuring p1(10)=p7
= q8.
Connect p1a1 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 andCD.
Determine the shortest
distance EF between the line
segments
First an Auxiliary A1B1 is made showing the true length of AB.
A second auxiliary view showing the point view ofAB is projected.
For this draw the reference line normal to A1B1 and draw the
projectors C2 D2 (of C1 and D1).
The shortest distance F2E2 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.

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