Conic Sections

• Sections of a correct circular cone obtained by cutting the cone in dissimilar ways
• Depending on the position of the cutting plane relative to the axis of the cone, three conic sections can exist obtained
– ellipse,
– parabola and
– hyperbola

An ellipse is obtained when a department plane A–A, inclined to the axis cuts all
the generators of the cone.

• A parabola is obtained when a section plane B–B, parallel to one of the
generators cuts the cone. Obviously, the department airplane will cutting the base of the
cone.

• A hyperbola is obtained when a section plane C–C, inclined to the axis cuts the
cone on 1 side of the centrality.

• A rectangular hyperbola is obtained when a section plane D–D, parallel to
the axis cuts the cone.
conic sections

conics section curves

Conic is defined as the locus of a bespeak moving in a airplane such that the ratio of its distance from a fixed indicate (F) to the stock-still directly line is always a constant. This ratio is called eccentricity.

Ellipse: eccentricity is always <1
Parabola: eccentricity is always=i
Hyperbola: eccentricity is >1
The fixed point is called the Focus
The fixed-line is called the Directrix
Centrality is the line passing through the focus and perpendicular to the directrix.
Vertex is a point at which the conic cuts its axis.

definition of focus,

Ellipse:

Human relationship betwixt Major axis, Small-scale axis and Foci

• If minor axis is given instead of the distance between the foci, then locate the foci F and F' by cut the arcs on major axis with C every bit a center and radius= ½ major axis = OA.

Ellipse definition

• If the major axis and minor axis are given, the 2 fixed points F1 and F2
can be located with the post-obit fact:

The sum of the distances of a bespeak on the ellipse from the ii foci is equal to the major axis
• The distance of any end of the pocket-sized axis from any focus is equal to the half of the
major axis.

Methods for Conics Department Construction:

ELLIPSE
1.Concentric Circle Method
two.Rectangle Method
3.Ellipsoidal Method
4.Arcs of Circumvolve Method
5.Rhombus Metho
6.Basic Locus Method (Directrix – focus)

HYPERBOLA
1.Rectangular Hyperbola (coordinates are given)
2 Rectangular Hyperbola (P-V diagram – Equation given)
3.Basic Locus Method (Directrix – focus)

PARABOLA
1.Rectangle Method
two Method of Tangents ( Triangle Method)
3.Basic Locus Method (Directrix – focus)

Concentric Circle Method:

Given Major and Minor Axis

Steps:

1. Describe both axes as perpendicular bisectors of each other & name their ends as shown.

2. Taking their intersecting signal as a heart, describe 2 concentric circles considering both every bit respective diameters.

iii. Split both circles into 12 equal parts & name equally shown.

four. From all points of the outer circle draw vertical lines down and upwards respectively.

five. From all points of the inner circle describe horizontal lines to intersect those vertical lines.

6. Mark all intersecting points properly every bit those are the points on the ellipse.

7. Bring together all these points forth with the ends of both axes in the smooth possible curve.

It is required ellipse.

Concentric Circle Method

Ellipse by Concentric Circle Method

Ellipse by Rectangle Method

Steps: ane Describe a rectangle taking major and minor axes as sides.

2. In this rectangle draw both axes as perpendicular bisectors of each other.

3. For structure, select the upper left part of the rectangle. Split vertical small side and horizontal long side into same number of equal parts. ( here divided into four parts)

4. Proper noun those as shown.

5. Now join all vertical points i,2,three,four, to the upper end of the minor axis. And all horizontal points i.eastward.ane,2,3,4 to the lower end of the minor axis.

6. And so extend C-1 line up to D-1 and mark that point. Similarly extend C-2, C-3, C-4 lines up to D-two, D-iii, & D-4 lines.

7. Mark all these points properly and join all along with ends A and D in the smooth possible bend. Do similar construction in the right side part.along with the lower half of the rectangle.Join all points in the smoothen curve.

Information technology is required ellipse.

Oblong Method

Ellipse past Rectangle Method

Parabola General Method:

Construction

Draw the axis AB and the directrix CD, at right to each other.

Mark the focus F on the centrality with given length for suppose AF=50 or twoscore etc..

Locate the vertex 5 on AB such that AV=VF= ½(AF)

Describe a line VE, perpendicular to AB such that VE=VF

Bring together A,E and extend , by construction VE/VA=VF/VA=1, the eccentricity.

Locate a number of points 1,2,3, etc . to the correct of V on the axis, which demand not be equidistant.

Through the points one,2,3 etc, draw lines perpendicular to the axis and to meet the line AE extended at 1',2',three', etc.

With the center F and radius 1-1', describe arcs intersecting the line through one at P1 and P1'. P1 and P1' are the points on the parabola, considering, the altitude of P1(P1') from Fis one-i'and from CD. Information technology is A-1 and

1-one'/A-1=VE/VA=VF/VA=1

Similarly locate the points P2,P2';P3,P3'; etc.. on either side of the axis.

Bring together the points by a smooth curve, forming the required parabola

Notation: OUR REQUIRED I ( PARABOLA) SHOULD Be DARK

TO Draw the Tangent and Normal:

To draw the tangent and normal to the parabola , locate the bespeak M. which at a given altitude from directrix

Then join M and F and draw a line through F, perpendicular to MF, coming together the directrix at T.

The line joining T and M and extended (T-T) is the tangent and line N-Due north , through M and perpendicular to TM is the normal to the curve.

parabola60

General Method (Directrix, Eccentricity Method)

RECTANGLE METHOD PARABOLA:

STEPS:
ane.Describe rectangle of above size and split it in two equal vertical parts
2.Consider left part for construction. Carve up height and length in equal number of parts and name those 1,2,three,4,5& 6
3.Join vertical 1,2,three,4,five & vi to the top center of rectangle
iv.Similarly depict upward vertical lines from horizontal1,2,three,4,5 And wherever these lines intersect previously drawn inclined lines in sequence Mark those points and further join in smoothen possible curve.
v.Repeat the construction on right side rectangle also.Join all in sequence.

This locus is Parabola.
Rect Parabola Rectangle Method of Parabola

METHOD OF TANGENTS OF PARABOLA:

Step-1 Draw a horizontal major axis of the length 140 mm and give the notations A & B as shown in the effigy. And mark a midpoint C on it.

Footstep-2 Draw a vertical axis, perpendicular to the horizontal axis & passing through the point C; of the length equal to the length of small-scale centrality, which is 100 mm and give the notations C & D as shown in the effigy. Like in the aforementioned way extend the line CD equally DE of the length equal to 100 mm, as shown in the effigy.

Step-three Then connect the bespeak E with the points A & B past direct inclined lines every bit shown in the figure. And divide these 2 lines AE & Be in to 10 equal divisions. At present give the notations on these points as 1,2,3 etc. but in contrary way as shown in the figure.

Step-v Connect these points 1-1, 2-2, 3-iii etc. past straight lines equally shown in the figure.

Step-6 Draw a smooth free hand medium dark curve starting from the point A and intersecting the lines one-1, 2-2, 3-3 etc. by tangent and catastrophe at the point B every bit shown in the effigy. This is the required parabola.

Pace-7 Give the dimensions by any one method of dimensions and requite the proper name of the components by leader lines wherever necessary.

Tangent Mehtod Parabola

Method of Tangent Parabola

Lecture 2 Engineering curves

Orthographic Projection