Engineering Graphics and Engineering Curves

Engineering Graphics I
Unit - 4
Engineering Curves
Hope Foundation’s
International Institute of Information Technology, I²IT, P-14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411 057
Toll Free - 1800 233 4499 Website - www.isquareit.edu.in ; Email - info@isquareit.edu.in
By
Rakhi Wagh
As per Guidelines of Savitribai Phule Pune
University (SPPU) First Year Syllabus (2015)

As per Guidelines of Savitribai Phule Pune University (SPPU) First Year Syllabus.
ENGINEERING CURVES
Part- I {Conic Sections}
ELLIPSE
1.Rectangle Method
2.Basic Locus Method
(Directrix – focus)
HYPERBOLA
PARABOLA
1.Rectangle Method
Hope Foundation’s International Institute of Information Technology, I²IT, P-14 Rajiv Gandhi Infotech Park,
Hinjawadi, Pune - 411 057 Toll Free - 1800 233 4499 Website - www.isquareit.edu.in ;
Email - info@isquareit.edu.in

CONIC SECTIONS
ELLIPSE, PARABOLA AND HYPERBOLA ARE CALLED CONIC
SECTIONS
BECAUSE
THESE CURVES APPEAR ON THE SURFACE OF A CONE
WHEN IT IS CUT BY SOME TYPICAL CUTTING PLANES.
Section Plane
Through Generators
Ellipse
Section Plane Parallel
to end generator.
Section Plane
Parallel to Axis.
Hyperbola
OBSERVE
ILLUSTRATIONS
GIVEN BELOW.

1
2
3
4
1
2
3
4
A B
C
D
Problem :
Draw ellipse by Rectangle method.
Take major axis 100 mm and minor axis 70 mm long.
Steps:
1 Draw a rectangle taking major
and minor axes as sides.
2. In this rectangle draw both
axes as perpendicular bisectors of
each other..
3. For construction, select upper
left part of rectangle. Divide
vertical small side and horizontal
long side into same number of
equal parts.( here divided in four
parts)
4. Name those as shown..
5. Now join all vertical points
1,2,3,4, to the upper end of minor
axis. And all horizontal points
i.e.1,2,3,4 to the lower end of
minor axis.
6. Then extend C-1 line upto D-1
and mark that point. Similarly
extend C-2, C-3, C-4 lines up to
D-2, D-3, & D-4 lines.
7. Mark all these points properly
and join all along with ends A
and D in smooth possible curve.
Do similar construction in right
side part.along with lower half of
the rectangle.Join all points in
smooth curve.
It is required ellipse.
ELLIPSE
BY RECTANGLE METHOD
Hope Foundation’s International Institute of Information Technology, I²IT,
P-14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411 057 Toll Free -
1800 233 4499 Website - www.isquareit.edu.in ;

ELLIPSE
DIRECTRIX-FOCUS METHOD
PROBLEM :- POINT F IS 50 MM FROM A LINE AB.A POINT P IS MOVING IN A PLANE
SUCH THAT THE RATIO OF IT’S DISTANCES FROM F AND LINE AB REMAINS CONSTANT
AND EQUALS TO 2/3 DRAW LOCUS OF POINT P. { ECCENTRICITY = 2/3 }
F ( focus)
V
ELLIPSE
(vertex)
A
B
STEPS:
1 .Draw a vertical line AB and point F
50 mm from it.
2 .Divide 50 mm distance in 5 parts.
3 .Name 2nd part from F as V. It is 20mm
and 30mm from F and AB line resp.
It is first point giving ratio of it’s
distances from F and AB 2/3 i.e 20/30
4 Form more points giving same ratio such
as 30/45, 40/60, 50/75 etc.
5.Taking 45,60 and 75mm distances from
line AB, draw three vertical lines to the
right side of it.
6. Now with 30, 40 and 50mm distances in
compass cut these lines above and below,
with F as center.
7. Join these points through V in smooth
curve.
This is required locus of P.It is an ELLIPSE.
45mm
Hope Foundation’s International Institute of Information Technology, I²IT, P-14 Rajiv Gandhi Infotech
Park, Hinjawadi, Pune - 411 057 Toll Free - 1800 233 4499 Website - www.isquareit.edu.in ;

1
2
3
4
5
6
1 2 3 4 5 6
1
2
3
4
5
6
5 4 3 2 1
PARABOLA
RECTANGLE METHOD
PROBLEM : A BALL THROWN IN AIR ATTAINS 100 M HIEGHT
AND COVERS HORIZONTAL DISTANCE 150 M ON GROUND.
Draw the path of the ball (projectile)-
STEPS:
1.Draw rectangle of above size and
divide it in two equal vertical parts
2.Consider left part for construction.
Divide height and length in equal
number of parts and name those
1,2,3,4,5& 6
3.Join vertical 1,2,3,4,5 & 6 to the
top center of rectangle
4.Similarly draw upward vertical
lines from horizontal1,2,3,4,5
And wherever these lines intersect
previously drawn inclined lines in
sequence Mark those points and
further join in smooth possible curve.
5.Repeat the construction on right side
rectangle also.Join all in sequence.
This locus is Parabola.
.

A
B
V
PARABOLA
(VERTEX)
F
( focus)
1 2 3 4
PARABOLA
DIRECTRIX-FOCUS
METHOD
SOLUTION STEPS:
1.Locate center of line, perpendicular to
AB from point F. This will be initial
point P and also the vertex.
2.Mark 5 mm distance to its right side,
name those points 1,2,3,4 and from
those
draw lines parallel to AB.
3.Mark 5 mm distance to its left of P and
name it 1.
4.Take O-1 distance as radius and F as
center draw an arc
cutting first parallel line to AB. Name
upper point P1 and lower point P2.
(FP1=O1)
5.Similarly repeat this process by taking
again 5mm to right and left and locate
P3P4.
6.Join all these points in smooth curve.
It will be the locus of P equidistance
from line AB and fixed point F.
PROBLEM : Point F is 50 mm from a vertical straight line AB.
Draw locus of point P, moving in a plane such that
it always remains equidistant from point F and line AB.
O
P1
P2
Hope Foundation’s International Institute of Information
Technology, I²IT, P-14 Rajiv Gandhi Infotech Park, Hinjawadi,
Pune - 411 057 Toll Free - 1800 233 4499 Website -
www.isquareit.edu.in ; Email - info@isquareit.edu.in

P
O
40 mm
30 mm
1
2
3
12 1 2 3
1
2
HYPERBOLA
THROUGH A POINT
OF KNOWN CO-ORDINATES
Solution Steps:
1) Extend horizontal
line from P to right side.
2) Extend vertical line
from P upward.
3) On horizontal line
from P, mark some points
taking any distance and
name them after P-1,
2,3,4 etc.
4) Join 1-2-3-4 points
to pole O. Let them cut
part [P-B] also at 1,2,3,4
points.
5) From horizontal
1,2,3,4 draw vertical
lines downwards and
6) From vertical 1,2,3,4
points [from P-B] draw
horizontal lines.
7) Line from 1
horizontal and line from
1 vertical will meet at
P1.Similarly mark P2, P3,
P4 points.
8) Repeat the procedure
by marking four points
on upward vertical line
from P and joining all
those to pole O. Name
this points P6, P7, P8 etc.
and join them by smooth
curve.
Problem : Point P is 40 mm and 30 mm from horizontal
and vertical axes respectively.Draw Hyperbola through it.
Hope Foundation’s International Institute of Information Technology, I²IT,
P-14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411 057
Toll Free - 1800 233 4499 Website - www.isquareit.edu.in ;

INVOLUTE CYCLOID SPIRAL HELIX
ENGINEERING CURVES
Part-II
1. Involute of a circle 1. General Cycloid 1. Spiral of
One Convolution.
1. On Cylinder

CYCLOID:
IT IS A LOCUS OF A POINT ON THE
PERIPHERY OF A CIRCLE WHICH
ROLLS ON A STRAIGHT LINE PATH.
INVOLUTE:
IT IS A LOCUS OF A FREE END OF A STRING
WHEN IT IS WOUND ROUND A CIRCULAR POLE
SPIRAL:
IT IS A CURVE GENERATED BY A POINT
WHICH REVOLVES AROUND A FIXED POINT
AND AT THE SAME MOVES TOWARDS IT.
HELIX:
IT IS A CURVE GENERATED BY A POINT WHICH
MOVES AROUND THE SURFACE OF A RIGHT CIRCULAR
CYLINDER / CONE AND AT THE SAME TIME ADVANCES IN AXIAL DIRECTION
AT A SPEED BEARING A CONSTANT RATIO TO THE SPPED OF ROTATION.
( for problems refer topic Development of surfaces)
DEFINITIONS

INVOLUTE OF A CIRCLE
Problem : Draw Involute of a circle.
String length is equal to the circumference of circle.
1 2 3 4 5 6 7 8
P
P8
1
2
3
4
5
6
7
8
P3
P4
4 to p
P5
P7
P6
P2
P1

D
A
Solution Steps:
1) Point or end P of string AP is
exactly D distance away from A.
Means if this string is wound round
the circle, it will completely cover
given circle. B will meet A after
winding.
2) Divide D (AP) distance into 8
number of equal parts.
3) Divide circle also into 8 number
of equal parts.
4) Name after A, 1, 2, 3, 4, etc. up
to 8 on D line AP as well as on
circle (in anticlockwise direction).
5) To radius C-1, C-2, C-3 up to C-8
draw tangents (from 1,2,3,4,etc to
circle).
6) Take distance 1 to P in compass
and mark it on tangent from point 1
on circle (means one division less
than distance AP).
7) Name this point P1
8) Take 2-B distance in compass
and mark it on the tangent from
point 2. Name it point P2.
9) Similarly take 3 to P, 4 to P, 5 to
P up to 7 to P distance in compass
and mark on respective tangents
and locate P3, P4, P5 up to P8 (i.e.
A) points and join them in smooth
curve it is an INVOLUTE of a given
circle.
Hope Foundation’s International Institute of Information Technology, I²IT, P-
14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411 057 Toll Free - 1800
233 4499 Website - www.isquareit.edu.in ; Email - info@isquareit.edu.in

P
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12
p1
p2
p3
p4
p5
p6
p7
p8
D
CYCLOID
PROBLEM : DRAW LOCUS OF A POINT ON THE PERIPHERY OF A CIRCLE
WHICH ROLLS ON STRAIGHT LINE PATH. Take Circle diameter as 50 mm
Solution Steps:
1) From center C draw a horizontal line equal to D distance.
2) Divide D distance into 12 number of equal parts and name them C1, C2, C3__ etc.
3) Divide the circle also into 12 number of equal parts and in clock wise direction, after P name 1, 2, 3 up to 12.
4) From all these points on circle draw horizontal lines. (parallel to locus of C)
5) With a fixed distance C-P in compass, C1 as center, mark a point on horizontal line from 1. Name it P.
6) Repeat this procedure from C2, C3, C4 upto C12 as centers. Mark points P2, P3, P4, P5 up to P8 on the
horizontal lines drawn from 1,2, 3, 4, 5, 6, 7 respectively.
7) Join all these points by curve. It is Cycloid.
p9
p10
p11
p121
2
3
5
4
6
7
8
9
10
11
12

7 6 5 4 3 2 1
P
1
2
3
4
5
6
7
P2
P6
P1
P3
P5
P7
P4 O
SPIRAL
Problem : Draw a spiral of one convolution. Take distance PO 40 mm.
Solution Steps
1. With PO radius draw a circle
and divide it in EIGHT parts.
Name those 1,2,3,4, etc. up to 8
2 .Similarly divided line PO also in
EIGHT parts and name those
1,2,3,-- as shown.
3. Take o-1 distance from op line
and draw an arc up to O1 radius
vector. Name the point P1
4. Similarly mark points P2, P3, P4
up to P8
And join those in a smooth curve.
It is a SPIRAL of one convolution.
IMPORTANT APPROACH FOR CONSTRUCTION!
FIND TOTAL ANGULAR AND TOTAL LINEAR DISPLACEMENT
AND DIVIDE BOTH IN TO SAME NUMBER OF EQUAL PARTS.
Hope Foundation’s International Institute of Information Technology,
I²IT, P-14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411 057
Toll Free - 1800 233 4499 Website - www.isquareit.edu.in ;

1
2
3
4
5
6
7
8
P
P1
P
P2
P3
P4
P5
P6
P7
P8
1
2
3
4
5
6
7
HELIX
(UPON A CYLINDER)
PROBLEM: Draw a helix of one convolution, upon a cylinder.
Given 80 mm pitch and 50 mm diameter of a cylinder.
(The axial advance during one complete revolution is called
The pitch of the helix)
SOLUTION:
Draw projections of a cylinder.
Divide circle and axis in to same no. of equal parts. ( 8 )
Name those as shown.
Mark initial position of point ‘P’
Mark various positions of P as shown in animation.
Join all points by smooth possible curve.
Make upper half dotted, as it is going behind the solid
and hence will not be seen from front side.

1. N. D. Bhatt , Engineering Drawing , 50th Edition, Charotar Publications
House.
2. M.B. Shah, B.C. Rana, Engineering Drawing, 2nd Edition, Pearson
Publications.
3. K.C. John, Engineering Graphics for Degree, (2009), PHI Learning Private
Limited.
References

International Institute of Information Technology, I²IT, P-14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411057
Toll Free - 1800 233 4499 Website - www.isquareit.edu.in ; Email - info@isquareit.edu.in
THANK YOU
For further details please contact
Rakhi Wagh
Department of Applied Science & Engineering
Hope Foundation’s
International Institute of Information Technology,
I²IT
P-14, Rajiv Gandhi Infotech Park, MIDC Phase 1,
Hinjawadi, Pune – 411 057
www.isquareit.edu.in
Phone : +91 20 22933441 / 2 / 3
rakhiw@isquareit.edu.in | info@isquareit.edu.in

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