engineering graphics

1. M.KUMARASAMY COLLEGE OF
ENGINEERING
(AUTONOMOUS)
18MES101JT- ENGINEERING GRAPHICS
PRESENTED BY,
PRETHIGA S
SHREELAYA R
SINDHU G R
SINDHUMATHI R
VIKASINI M
VIVETHA R S
GUIDED BY,
Dr.K.RAJU.,M.E.,Ph.D.,
1

2. CONIC SECTIONS
AND
INVOLUTES

3. CONIC SECTION

4. DEFINITION OF CONIC SECTION
• Conic section, also called conic, in geometry, any curve produced by
the intersection of a plane and a right circular cone.
• Depending on the angle of the plane relative to the cone, the
intersection is a circle, an ellipse, a hyperbola, or a parabola.

5. ELLIPSE

6. DEFINITION OF ELLIPSE
• Ellipse is the locus of a point P which moves such that the ratio of its distance from the fixed
point F to its distance from a fixed line is a constant and is always less than 1.

7. CONSTRUCTION OF ELLIPSE
• Mark point V and F w.r.t the ratio of “e”
• Here, e=2/3 and the distance of focus
from directrix is 50mm
• Mark points where CV=20mm and
VF=30mm
7

8. • Draw a line at V equal to VF
• Join VE and extend it
• Mark points 1’,2’,....10’ on extended CE
and 1,2...10 on the axis
• With 11’ as radius and F as centre and
mark an arc on 11’
• Continue the same step for the next 10
lines
• When we join all the arcs marked, we get
an ellipse
8

9. APPLICATIONS
OF ELLIPSE
Electrons in the atom
move around the
nucleus in an
elliptical path of orbit
Property of Ellipse to
reflect sound and light
is used in pulverizing
kidney
stones(Lithotripsy)
An ellipse exhibits an
interesting acoustic
phenomenon.This
principle is used in
whispering galleries.
Your eye is an
horizontal ellipse!

11. PARABOLA

12. DEFINITION OF PARABOLA
• The locus of a point that is equidistant from a given point (focus) and a given line (directrix) is
called the Parabola.
• A parabola is a section of a right circular cone formed by cutting the cone by a plane parallel to the
generator of the cone.
• The standard equation of a regular parabola is y2 = 4ax.

13. CONSTRUCTION OF PARABOLA
• In the question, the distance between
directrix and focus is given 50mm.
• Let us mark the point F with the distance
AF=50mm
• Point to note is that eccentricity of
parabola is always 1
• Mark the point V of distance 25mm
from directrix and focus.
• Here AV=VF=25mm
13

14. • Draw a vertical line of length 25mm
from V and mark the point as E
• Join the point A in directrix and E
• Extend the line AE
• Mark the points 1,2,3… with the
distance of 10mm between them in AB
14

15. Join the points to the line AE and extend it
downwards.
Mark the points as 1’,2’,3’… in the line AE.
In the first line 11’, take 11’ as radius and fix
point F as centre and draw an arc in the line
11’
Follow the same for the other 5 lines.
15

16. • After drawing the curves join all the
curves as shown in the figure.
• Follow the same steps and cut the arc
downwards.
• Join the arcs.
• We obtain parabola after joining all
the arcs
16

17. APPLICATIONS
OF
PARABOLA
Satellite dishes use
parabolas to help reflect
signals that are
subsequently sent to a
receiver.
In the realm of
architecture and
engineering, parabolas are
used.
The shape of car
headlights, mirrors in
reflecting telescopes, TV
and radio antenna etc are
parabolic.
The solar power sector is
increasingly benefiting from
the use of parabolic reflectors
to concentrate light.

19. HYPERBOLA

20. DEFINITION OF HYPERBOLA
• The hyperbola is plane curve generated by a point moving so that the difference of it’s distance
from two fixed points, called “Focuses or Foci” is a constant.
• A hyperbola is a conic section created by intersecting a right circular cone with a plane at an
angle such that both halves of the cone are crossed in analytic geometry.
• This intersection yields two unbounded curves that are mirror reflections of one another
.

21. 21
• Draw the directrix as a vertical line and the axis as a horizontal line
• Mark the vertex V at a distance 33 mm from the directrix
• Mark the focus F on the axis at a distance 50 mm from the vertex
• Draw a vertical line from V and mark C on it such that VC = VF, then draw a
line from A passing through C for convenient length.
• Draw a vertical line at any distance and mark 1 on the axis and I’ on the
inclined line AC.
• Use the length 1-l’ as radius , focus F as centre , draw an arc to cut the line 1-
l’ at M1 and N1
• Repeat this procedure by drawing vertical lines 2-2’,3-3’,etc., and get
M2,N2,M3,N3,etc.
• Join these points by drawing a smooth curve to get the hyperbola.
• Mark the point P on the curve at 40 mm from the directrix and join focus F
and P then draw a line from F at 90° to the line FP to get B on the directrix .
• Draw a line from B passing through P which is the tangent to the hyperbola .
• Draw another line through P , perpendicular to the tangent which is normal to
the curve.
.
CONSTRUCTION OF HYPERBOLA

22. APPLICATIONS
OF
HYPERBOLA
For all nuclear cooling towers and several
coal-fired power facilities, the hyperboloid
is the design standard.
Hyperbolic shadows are cast on
a wall by a home lamp.
It is mostly used in the design of
channels.
When two stones are tossed into a pool of calm water
at the same time, ripples form in concentric circles. The
hyperbola is a curve formed when these circles overlap
in points..

23. INVOLUTE

24. DEFINITION OF INVOLUTE
• Attach a string to a point on a curve.
• Make the string a tangent to the curve at the point of attachment.
• Then wind the string up, keeping it always taut. The locus of points traced out by the
end of the string is called the involute of the original curve.
• The original curve is called the evolute of its involute.

25. INVOLUTE
OF
CIRCLE

26. DEFINITION OF INVOLUTE OF CIRCLE
• The involute of a circle is the path traced out by a point on a straight line that rolls
around a circle. It was studied by Huygens when he was considering clocks without
pendulums that might be used on ships at sea.
• It is similar to the Archimedes spiral.

27. 27
CONSTRUCTION OF INVOLUTE OF CIRCLE
• Draw a circle of diameter 50mm .
• Mark P on the circle and draw a line PQ of length equal to (n
D) circumference of the circle.
• Divide PQ and the circle into 12 equal parts and mark the
number’
• Draw tangents at points 1,2,3…..11 and mark P1 P2
P3….P12 on them such that 1P1=P1, 2P2=P2,3P3=P3.
• Draw involute curve through the point P1,P2,P3….P11, Q
• Mark the point G on the involute curve.
• Join G with O and mark the point C as the mid point of GO.
Keeping C as centre, GC as radius , draw a semicircle which
cuts the small circle at M.
• Join MG’. MG is the normal.
• Draw a perpendicular line (ST) to the normal through the
point G.
• This is the tangent to the involute.

28. APPLICATIONS
OF
INVOLUTE OF
CIRCLE
Gear industries – To make
teeth for two revolving
machines and gears.
Scroll compressing and Gas
Compressing – These are made in
this shape to reduce noise and to
make them efficient.
Picture as follows.

29. INVOLUTE OF
TRIANGLE

30. DEFINITION OF INVOLUTE OF TRIANGLE
Involute, of a curve C, a curve that intersects all the tangents of the curve C at
right angles.

31. 31
CONSTRUCTION OF INVOLUTE OF TRIANGLE
• Given triangle ABC, extend the sides of the triangle to any
convenient length.
• Using CA as a radius and C as a center, strike arc AD
terminating at the intersection of the extension BD
• With BD as a radius and B as a center, strike arc DE.
• With AE as a radius and A as a center , strike arc EF.
• Repeat this procedure until you reach a figure of the
desired size.
• To draw tangent and normal mark any point D on the
involute.
• Join the point D and C and the line DC is normal.
• Draw a line perpendicular to the line DC and name the line
as TT’ which acts as a tangent

32. INVOLUTE OF
HEXAGON

33. 33
CONSTRUCTION OF INVOLUTE OF HEXAGON
• Draw a regular hexagon ABCDEF of side 15 mm.
• Produce the lines BA , CB , DC , ED , FE and AF.
• With B as centre and BA as radius, draw an arc to intersect
CB induced at P1.
• With C as centre and CP1 as radius, draw an arc to
intersect DC produced at P2.
• With D as centre and DP2 as radius, draw an arc to
intersect ED produced at P3.
• With E as centre and EP3 as radius, draw an arc to
intersect FE produced at P4.
• With F as centre and FP4 as radius, draw an arc to
intersect AF produced at P5.
• With A as centre and AP5 as radius , draw an arc to
intersect BA produced at P6.
• Draw a smooth curve passing through P1,P2….and P6

34. CONCLUSION
Conic sections are in many objects that we use in our everyday life
they are also on objects we ride in and sometimes the buildings we
go in. Many things are shaped in parabolas, circles, ellipses, and
hyperbolas and because of they way things are shaped it gives us
the opportunity to use the items correctly because they are made
correctly in the right shape. An involute (also known as an
evolvent) is a form of curve in mathematics that is dependent on
another shape or curve. The location of a point on a taut string as it
is either unwrapped from or wrapped around a curve is called an
involute of a curve.