conics presentation-r.surya kiran ,assistant professor ,gitam university
TRANSCRIPT
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Contents
Introduction
Ellipse
Parabola Hyperbola
Cycloids
Involutes
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Definition
CONIC is defined as
the locus of the point
moving in a plane in
such a way that theratio of its distance
from a point to the
fixed straight line is
always constant.
This constant is known
as ECCENTRICITY.
APEX
AXIS
GENERATOR
BASE
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Section Plane
Parallel to Axis.Hyperbola
Section Plane Parallel
to one - end generator.
Section Plane
Through Generators
Ellipse
Conic Sections
Ellipse, Parabola and Hyperbola are called conic
sections
These curves appear on the surface of a cone
when it is cut by some typical cutting planes.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Engineering Applications
Ellipse:
Arches, bridges, dam, gears, man-holes, flanges
of the pipes, stuffing boxes.
Parabola:
Headlights, solar connectors, reflectors.
Hyperbola:
Cooling towers, hydraulic channels, transmitters,
receivers.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Ellipse
The locus of a point moving in a plane in such a
way that the ratio of its distance from a point to
the fixed straight line is always constant.
Distance of the moving point from the
directrix
Distance of the moving point from the
focusE =
Eccentricity (E) less than 1 gives
ELLIPSE
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Concentric circles method
1
2
3
4
5
6
7
8
9
10
BA
D
C
1
23
4
5
6
78
9
10
Q1. Construct an ellipse
when its major axis
is equal to 100 mm
and the minor axis isequal 65mm.
65
100
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Rectangular method
1
2
3
1
2
3
A B
Q2. A plot of ground is in the shape of a rectangle
110mm*50mm. Inscribe an elliptical lawn in it.
C
D
O
50
110
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Parallelogram or Oblong
methodQ2. A plate is in the shape of 100mm and 70mm long sides
with included angle of 75o. Draw an elliptical mark on the plate.
1
2
3
A B
100
1
2
3
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Rectangular + Concentric circles
method
Q3. The major and minor axis of an elliptical
fish pond is 100mm and 65mm
respectively. Draw one half of the ellipse by
rectangular method and the other half by
the concentric circles method.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Arcs of circles method
Q4. Construct an ellipse when a pair of conjugate
axes is equal to 110mm and 50mm
respectively. The angle between the conjugate
axes is 700
. Determine the lengths of the majorand the minor axes.
[conjugate axes-these are the two lines
intersecting each other at the centre of an
ellipse when they are parallel to the tangents
drawn at their extremities.]
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
A Bf1 f2
C
D
1 2 3 4 5
110
50
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Eccentricity method
Q5. Construct an ellipse when the distance
between the focus and the directrix is 30mm
and the eccentricity is . Draw the tangent
and the normal at any point on the curve
using the directrix.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
30
D
D
O
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Parabola
The locus of a point moving in a plane in such a
way that the ratio of its distance from a point to
the fixed straight line is always constant.
Distance of the moving point from the
directrix
Distance of the moving point from the
focusE =
Eccentricity (E) equal to1 gives
PARABOLA
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Eccentricity method
Q. Construct a parabola when the distance
between the focus and the directrix is
50mm. Draw a tangent and the normal at
any point P on your curve.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
A
B
V
(VERTEX)
F( focus)O 1 2 3 4
50
OV =
VF
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Rectangular method
Q2. A cricket ball is thrown from the ground
level reaches the wicket keepers gloves.Maximum height reached by the ball
is 5 cm. The ball travels a horizontal
distance of 10 cm from the point of projection.
Trace the path of the ball.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
1
2
3
4
5
6
1 2 3 4 5 6
1
2
3
4
5
6
5 4 3 2 1
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Parallelogram method
Q3. Construct a parabolic arch within a
parallelogram of sides 110mm*50mm.One
of the included angles between the sides is700.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Tangent method
Q4. A fountain jet discharges water from the
ground level at an inclination of 500to the
ground. The jet travels a horizontaldistance of 90 mm from the point of
discharge and falls on the ground. Trace
the path of the jet.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Hyperbola
The locus of a point moving in a plane in such a
way that the ratio of its distance from a point to
the fixed straight line is always constant.
Distance of the moving point from the
directrix
Distance of the moving point from the
focusE =
Eccentricity (E) greater than1 gives
HYPERBOLA
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Methods(Hyperbola)
Eccentricity method
Ordinate, Abscissa &Transverse
method
Asymptotes method.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Eccentricity method
Q. Construct a hyperbola when the distance
between the focus and the directrix is 40mm. The
eccentricity is 4/3. Draw a tangent and normalat any point on the hyperbola.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
F ( focus)(vertex)
A
B
V
O di t Ab i & T
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Ordinate ,Abscissa & Transverse
axis
Q. Draw a hyperbola when its double ordinate is
90mm, abscissa is 35mm and half the
transverse axis is 45mm.
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O
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Asymptotes method
Q. Construct a rectangular hyperbola when a
point P on it is at a distance of 15mm and
20mm respectively from the two asymptotes.
[when the angle between the asymptotes is
900,the curve is called as rectangular
hyperbola.]
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
P
O
20 mm
15 mm
1
2
3
12 1 2 3
1
2
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Definition
A curve is a line traced out by a moving point by constantlychanging its directions
Plane curves:
A plane curve is a curve traced by a point which movesin a constantly changing directions on a plane
surface.Eg: Cycloidal curves ,involutes ,spirals.
Space curves:
A space curve is a curve traced by a point which movesin a constantly changing directions in space.
Eg. Helixes.
Cycloidal curves: -cycloids, epicycloids, hypocycloids,
Helixes:- cylindrical, conical.
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CYCLOIDS
A Cycloid is a curve traced by a point on the
circumference of a circle which rolls along a
straight line without slipping.
Eg.watches,dial gauges.As the rolling circle is assumed to roll
clockwise numbering of the division points on
it should be in counter clockwise direction.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
1 2 3 4 5 6 7 80
7
6
5
4
3
2
1
8
C0 C1 C2 C3 C4 C5 C6 C7 C8
(pD)
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Epicycloids
Epicycloids is a curve generated by a point on
the circumference of a circle which rolls
without slipping on the outside of another
circle. The fixed circle on the outside of which the
generating circle rolls is called as the Base
circle/directing circle.
Eg. Cycloidal teeth gears ,the faces are of
epicycloidal profile.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Q. Draw an epicycloids of rolling circle
50mm,which rolls outside another circle of
150mm dia for one revolution. Draw a tangent
and normal at any point on the curve. [(r/R)*360]
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
rR
3600=
O
C2
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
HYPOCYCLOIDS
It is a curve traced by a point on the
circumference of a circle which rolls without
slipping on the inside of another circle.
[(r/R)*360]
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
INVOlUTES
It is a single curve line traced out by an end of
a string when unwound itself from a straight
line or a circle or a polygon , the string being
always kept taut.Eg. Involute of a circle is the profile of the gear
teeth wheel.
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
Q. A coir is unwound from a drum of dia 30mm.
Draw the locus of the free end of the coir for
unwinding through an angle of 3600.
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(pD)
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R. SURYA KIRAN, Asst. Professor & A. RAVINDRA, Tech. Asst., GIT, GITAM
INVOLUTE OF A SQUARE
Q. Draw the involute of square of side 20mm.
Draw a tangent and a normal at any point M.
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