Conics_Ellipse1.ppt

7/27/2019 Conics_Ellipse1.ppt

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CHAPTER2

ENGINEERING

CURVES


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1. CONICS

2. CYCLOIDAL CURVES

3. INVOLUTE

4. SPIRAL5. HELIX

CLASSIFICATION OF ENGG. CURVES


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It is a surface generated by moving a

Straight line keeping one of its end fixed &other end makes a closed curve.

What is Cone ?

If the base/closedcurve is a polygon, we

get a pyramid.

If the base/closed curve

is a circle, we get a cone.

The closed curve isknown as base.

The fixed point is known as vertex or apex.

Vertex/Ape

90

Base


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If axis of cone is notperpendicular to base, it is

called as oblique cone.

The line joins vertex/apex to the

circumference of a cone

is known as generator.

If axes is perpendicular to base, it is called as

rightcircular cone.

Generator

Cone Axis

The line joins apex to the center of base is

called axis.

90

Base

Vertex/Apex


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Definition :-The section obtained by the

intersection of a right circular cone by acutting plane in different position relativeto the axis of the cone are calledCONICS.

CONICS


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B - CIRCLE

A - TRIANGLE

CONICS

C - ELLIPSE

D PARABOLA

E - HYPERBOLA


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When the cutting plane is perpendicular to

the axis or parallel to the base in a rightcone we get circle the section.

CIRCLE

Sec Plane

Circle


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Definition :-When the cutting plane is inclined to theaxis but not parallel to generator or theinclination of the cutting plane() is greaterthan the semi cone angle(), we get anellipse as the section.

ELLIPSE

>


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When the cutting plane is inclined to the axisand parallel to one of the generators of the

cone or the inclination of the plane() is equalto semi cone angle(), we get a parabola asthe section.

PARABOLA

=


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When the cutting plane is parallel to the

axis or the inclination of the plane withcone axis() is less than semi coneangle(), we get a hyperbola as the

section.

HYPERBOLADefinition :-

< = 0


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When the cutting plane contains theapex, we get a triangle as thesection.

TRIANGLE


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CONICSDefinition :- The locus of point moves in aplane such a way that the ratio of itsdistance from fixed point (focus) to a fixedStraight line (Directrix) is always constant.

Fixed point is called as focus.

Fixed straight line is called as directrix.

M

C FV

P

Focus

Conic Curve

Directrix


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The line passing through focus &

perpendicular to directrix is called as axis.

The intersection of conic curve with axis is

called as vertex.

AxisM

C FV

P

Focus

Conic Curve

Directrix

Vertex


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N Q

Ratio =Distance of a point from focus

Distance of a point from directrix

= Eccentricity= PF/PM = QF/QN = VF/VC= e

M P

F

Axis

CV

Focus

Conic Curve

Directrix

Vertex


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Vertex

Ellipse is the locus of a point which moves ina plane so that the ratio of its distance

from a fixed point (focus) and a fixedstraight line (Directrix) is a constant andless than one.

ELLIPSE

M

N Q

P

C

F

V

Axis

Focus

EllipseDirectrix

Eccentricity=PF/PM

= QF/QN

< 1.


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METHODS FOR DRAWING ELLIPSE

(1) DirectrixFocus Method

(2) Arcs Of Circles Method

(3) Concentric Circle Method

(4) Oblong Method Or

Rectangle Method
http://localhost/var/www/apps/conversion/tmp/gdb02/New%20Folder/ELLIPSERECTANGULAR1.ppt


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P6

P5 P7P6

P1

P1

T

T

V1

P5

P4

P4

P3P2

F1

D1

D

1

R1

ba

c d

ef

g

Q

P7P3P2

D

irectrix

90

1 2 3 4 5 6 7

Eccentricity = 2/3

3R1V1

QV1=

R1V1

V1F1=

2

Ellipse

ELLIPSEDIRECTRIX FOCUS METHOD

Dist. Between directrix

& focus = 50 mm

1 part = 50/(2+3)=10 mmV1F1 = 2 part = 20 mm

V1R1 = 3 part = 30 mm

< 45

S


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Ellipse is the locus of a point, which moves in aplane so that the sum of its distance from twofixed points, called focal points or foci, is aconstant. The sum of distances is equal to themajor axis of the ellipse.

ELLIPSE

F1A B

P

F2O

Q

C

D

P


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F1A B

D

P

F2O

PF1 + PF2 = QF1 + QF2 = CF1 +CF2 = constant

= Major Axis

Q

= F1A + F1B = F2A + F2B

But F1A = F2B

F1A + F1B = F2B + F1B = AB

CF1 +CF2 = AB

but CF1 = CF2

hence, CF1=1/2AB

C


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F1 F2OA B

C

D

Major Axis = 100 mm

Minor Axis = 60 mm

CF1 = AB = AO

F1 F2

OA B

C

D

Major Axis = 100 mm

F1F2 = 60 mm

CF1 = AB = AO

ARC OF CIRCLES


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P2

1 2 3 4A B

C

D

P1

P3

P2

P4 P4P3

P2

P1

P1

F2

P3

P4 P4P3

P2

P1

90

F1 O

ARC OF CIRCLE S

METHOD

ARC OF CIRCLES


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ARC OF CIRCLE S

METHOD

G2

1 2 3 4A B

G

G

G1

G3G

2

G4 G4G3

G2

G1

G1

G3 G4 G4G3

G2G1

F2F1 O

90

90

dire

ctrix

100

140

GF1 + GF2 = MAJOR AXIS = 140

E

eAF1

AEe =

CO C C


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A BMajor Axis 7

8

910

11

9

8

7

6

543

2

1

12

11

P6

P5P4

P3

P2`

P1

P12

P11 P10

P9

P8

P7

6

543

2

1

12 C10

O

CONCENTRIC

CIRCLE

METHOD

F2F1

D

CF1=CF2=1/2 AB

T

N

Q

e = AF1/AQ


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01

2

3

4

1 2 3 4 10

234

1

2

3

4

A B

C

D

Major Axis

MinorAxis

F1 F2

Directrix

E

F

S

P

P1

P2

P3

P4

P1

P2

P3

P4

P0

P1

P2

P3

P4P4

P3

P2

P1

OBLONG METHOD


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BA

P4

P0

D

C

60

6

543

21

0

5 4 3 2 1 0 1 2 3 4 5 6

5

3210P1P2

P3

Q1Q2

Q3Q4

Q5P6 Q6O

4

ELLIPSE IN PARALLELOGRAM

R4

R3R2

R1S1

S2

S3

S4

P5

G

H

I

K

J


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Uses Of Ellipse :-

Shape of a man-hole.

Flanges of pipes, glands and stuffing boxes.

Shape of tank in a tanker.

Shape used in bridges and arches.

Monuments.

Path of earth around the sun.

Shape of trays etc.

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