lecture.3 conicsection(ellipse,parabola&hyperbola)

7/29/2019 Lecture.3 ConicSection(Ellipse,Parabola&Hyperbola)

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Engineering Curves

Conic Sections

Involute

Cycloid

Spiral

Helix

Conic section: The sections obtained by the intersection of a right circular cone by a plane in different positions

relative to the axis of the cone are called conics.

1. When the section plane is inclined to the axis and cuts all the generators on one side of the apex, the section

is an ellipse.

2. When the section plane is inclined to the axis and is parallel to one of the generators, the section is a

parabola.

3. When the section plane cuts both the parts of the double cone on one side of the axis, the section is a

hyperbola

The conic may be defined as the locus of a point moving in a plane in such a way that the ratio of its distances

from a fixed point and a fixed straight line is always constant. The fixed point is called the focus and the fixed line,the directrix.

The ratio between the distances of the point from the focus to the distance of the point from the directrix is called

eccentricity and is denoted by e. It is always less than 1 for ellipse, equal to 1 for parabola and greater than 1 forhyperbola.


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Section plane

parallel toAxis Hyperbola

Section plane parallel

toend eneratorParabola

Ellipse

Section plane through

generator

Conic Sections

Ellipse, Parabola and hyperbola are called conic sections because these curves appear on the surface of a conewhen it is cut by some typical cutting planes.


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Construction of an ellipseFoci method (Given major and minor axes)Given: Major and minor axes

1. Draw a line AB equal to the major axis and a line CD equal to the minor axis, bisecting each other at right angles at O.

2. With center C and radius equal to half AB (i.e. AO) draw arcs cutting AB at F1 and F2, the foci of the ellipse.

3. Mark a number of points 1, 2, 3 etc. on AB

4. With centers F1 and F2 and radius equal to A1, draw arcs on both sides of AB.

5. With same centers and radius equal to B1, draw arcs intersecting the previous arcs at four points marked P1.

6. Similarly, with radii A2 and B2, A3 and B3 etc. obtain more points.

7. Draw smooth curve through these points. This curve is the required ellipse.

A B

C

D

OF1 F21 2 3

P1

P1

P1

P1


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Concentric circles method (Given major and minor axes)

1. Draw the major axis AB and the minor axis CD intersecting each other at O.

2. With center O and diameters AB and CD respectively, draw two circles.

3. Divide the major axis circle into a number of equal divisions, say 12 and mark points 1, 2 etc. as shown.

4. Draw lines joining these points with the center O and cutting the minor axis circle at points 1, 2 etc.

5. Through point 1 on the major axis circle draw a line parallel to CD, the minor axis.

6. Through point 1 on the minor axis circle, draw a line parallel to AB, the major axis. The point P1, where

these two lines intersect is on the required ellipse.

7. Repeat the construction through all the points. Draw the ellipse through A, P1, P2..etc.

A B

C

D

1

2

2

1P1

P2

P2


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Oblong method or Rectangle Method

(Given major and minor axes)

1. Draw the two axes AB and CD intersecting each other at O.

2. Construct the oblong EFGH having its sides equal to the two axes.

3. Divide the semi major axis AO into a number of equal parts, say 3, and AE into the same number of equal parts,

numbering them from A as shown.

4. Draw lines joining 1, 2, and 3 with C.

5. From D draw lines through 1, 2 and 3 intersecting C1, C2 and C3 at points P1, P2 and P3 respectively.

6. Draw the curve through A, P1C. It will be one quarter of the ellipse.

7. Complete the curve by the same construction in each of the three remaining quadrants

AB

D

CE

F G

H

O1 2 3

1

2

3


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Construction of Parabola

Tangent method (Given base and the axis)

1. Draw the base AB and the axis EF.

2. Produce EF to O, so that EF = FO.

3. Join O with A and B. Divide lines OA and OB into the same number of equal parts, say 14.

4. Mark the division points in reverse order.

5. Draw lines joining 1 with 1, 2 with 2 etc. Draw a curve starting from A and tangent to lines 1-1, 2-2 etc. This

curve is the required parabola.

ABE

F

O

1

2

13

3

4

1

23

13


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Rectangle method (Given base and the axis)

1. Draw the base AB.

2. At its mid point E, draw the axis EF at right angles to AB.

3. Construct a rectangle ABCD, making side BC equal to EF.

4. Divide AE and AD into the same number of equal parts and name them accordingly.

5. Draw lines joining F with points 1, 2 and 3. Through 1, 2 and 3, draw perpendiculars to AB intersecting

F1, F2 and F3 at points P1, P2 and P3 respectively.

6. Draw a curve through A, P1, P2 etc. It will be a half parabola.


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Rectangle method (Given base and the axis)

1. Draw the base AB.

2. At its mid point E, draw the axis EF at right angles to AB.

3. Construct a rectangle ABCD, making side BC equal to EF.

4. Divide AE and AD into the same number of equal parts and name them accordingly.

5. Draw lines joining F with points 1, 2 and 3. Through 1, 2 and 3 draw perpendiculars to AB

intersecting F1, F2 and F3 at points P1, P2 and P3 respectively.

6. Draw a curve through A, P1, P2 etc. It will be a half parabola.

A BE

FD C

P1

P2

P3

lecture.3 conicsection(ellipse,parabola&hyperbola)

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