Engineering GraphicsActive Learning Assignment

Engineering Curves (conics)

Mechanical BranchL division

Group members:

Vrushang Sangani 140120119193

Chintan Sathvara 140120119195

Introduction

Ellipse

Parabola

Hyperbola

The intersection of a cone by cutting plane gives us various curves of

intersections known as conics. Different shapes by intersections are

achieved by taking different positions of cutting planes, with respect to

cone axis or base or generator.

The conics are

1. Triangle

2. Circle

3. Ellipse

4. Parabola

5. Hyperbola

This is shown in figure,

a. When the cutting plane contains the apex, we get a triangle as the section.

b. When the cutting plane is perpendicular to the axis or parallel to the base

in a right cone we get circle as the section.

c. When the cutting plane is inclined to the axis but not parallel to generator

we get an ellipse as the section.

d. When the cutting plane is inclined to the axis and parallel to one of the

generators of the cone we get a parabola as the section.

e. When the cutting plane is parallel to the axis we get a hyperbola as the

section.

Definition:

Ellipse is the locus of a point which moves in a plane so thatthe ratio of its distances from a fixed point (focus) and afixed straight line (directrix) is a constant and less than one.

Uses:

1. Shape of a man-hole

2. Shape of tank in tanker

3. Flanges of pipes, glands and stuffing boxes

4. Shape used in bridges and arches

5. Monuments

6. Path of planets around the sun

7. Shape of trays etc.

Methods of drawing ellipse:

Arc of a circle method

Concentric circle method

Loop method

Oblong method

Ellipse in parallelogram

Trammel method

Directrix-focus method

1.Draw both axes. Name the ends & intersecting point.

2.Taking AO distance i.e. half major axis, from C, mark F1 & F2 On AB. (focus 1 and 2)

3.On line F1- O taking any distance, mark points 1,2,3, & 4

4.Taking F1 center, with distance A-1 draw an arc above AB and taking F2 center, with B-1 distance cut this arc. Name the point p1

5.Repeat this step with same centers but taking now A-2 & B-2 distances for drawing arcs. Name the point p2

6.Similarly get all other P points. With same steps positions of P D can be located below AB. 7. Join all points by smooth curve to get an ellipse.

1.Draw both the major & minor axes as perpendicular bisectors of each other.

2.Taking their intersecting point as a center, draw two concentric circles considering both as respective diameters.

3. Divide both circles in 12 equal parts & name as shown.

4.From all points of outer circle draw vertical lines downwards and upwards respectively. From all points of inner circle draw horizontal lines to intersect those vertical lines.

5. Join all these intersecting lines along with the ends of both axes in smooth possible curve. It is required ellipse.

1.Take a loop of thread having L peripheral length, Where L=length of major axis + distance between two foci

2.Fix two points at foci F1 and F2, arrange the loop of thread around pins as shown.

3.Insert pencil point inside the loop and move it, keeping the thread of loop always tight, then the pencil will draw an ellipse.

1.Draw a rectangle taking major and minor axes as sides. In this rectangle draw both axes as perpendicular bisectors of each other.

2.For construction, select upper left part of rectangle. Divide vertical small side and horizontal long side into same number of equal parts.( here divided in four parts)

3.Now join all vertical points 1,2,3,4, to the upper end of minor axis. And all horizontal points i.e.1,2,3,4 to the lower end of minor axis.

4.Then extend D-1 line upto C-1 and mark that point. Similarly extend D-2, D-3, D-4 lines up to C-2, C-3, & C-4 lines.

5.Mark all these points properly and join all along with ends A and C in smooth possible curve. Do similar construction in right side part along with lower half of the rectangle. Join all points in smooth curve. It is required ellipse.

Draw a parallelogram first of given two sides at given angle and follow the same procedure as followed in the oblong method.

1.First draw two axes, major AB and minor CD, bisecting at right angle at point O.

2.Take trammel of hard board or hard paper and mark on it EP equal to half major axis and FP equal to half minor axis, as shown in figure. Trammel is now ready for use.

3.Now arrange the trammel in all possible ways keeping the point E of the trammel on minor axis CD and the point F of the trammel on the major axis AB and mark points against P of the trammel on the paper.

4.Join the points, marked against the point P in different positions of the trammel, by a smooth curve to get an ellipse.

1. Draw the directrix and axis as shown.2. Mark F on axis such that CF 1= 70 mm.3. Divide CF into 3 + 4 = 7 equal parts and mark V at the fourth

division from C. Now, e = FV/ CV = 3/4.4. At V, erect a perpendicular VB = VF. Join CB.5. Through F, draw a line at 45° to meet CB produced at D. Through D,

drop a perpendicular DV’ on CC’. Mark O at the midpoint of V– V’.6. Mark a few points, 1, 2, 3, … on V– V’ and erect perpendiculars

though themmeeting CD at 1’, 2’, 3’…. Also erect a perpendicular through O.

7. With F as a centre and radius = 1–1’, cut two arcs on the perpendicular through 1 to locate P1 and P1¢. Similarly, with F as a centre and radii = 2–2’, 3–3’, etc., cut arcs on the corresponding perpendiculars to locate P/2 and P/2’, P/3 and P/3’, etc. Also, cut similar arcs on the perpendicular through O to locate V1 and V1’.

Definition:

The parabola is the locus of a point, which moves in a plane sothat its distances from a fixed point (focus) and a fixed straightline (directrix) are always equal.

Uses:

1. Motor car head lamp reflector

2. Sound reflector and detector

3. Bridges and arches construction

4. Shape of cooling towers

5. Path of particle thrown at any angle with earth, etc.

Methods of drawing parabola:

Rectangle method

Parabola in parallelogram

Tangent method

Directrix-focus method

1.Draw rectangle of the size given and divide it in two equal vertical parts.

2.Consider left part for construction. Divide height and length in equal number of parts and name those 1,2,3,4,5& 6.

3.Join vertical 1,2,3,4,5 & 6 to the top center of rectangle.

4.Similarly draw upward vertical lines from horizontal1,2,3,4,5 and wherever these lines intersect previously drawn inclined lines in sequence, mark those points and further join in smooth possible curve.

5.Repeat the construction on right side rectangle also. Join all in sequence. This locus is Parabola.

This method is similar to the rectangle method. Here, instead of drawing a rectangle we have to draw a parallelogram.

1. Construct triangle as per the given dimensions.

2. Divide it’s both sides into same no. of equal parts.

3. Name the parts in ascending and descending manner, as shown.

4. Join 1-1, 2-2,3-3 and so on.

5. Draw the curve as shown i.e. tangent to all these lines. The above all lines being tangents to the curve, it is called method of tangents.

1. Draw directrix AB and axis CC’ as shown.

2. Mark F on CC’ such that CF = 60 mm.

3. Mark V at the midpoint of CF. Therefore, e = VF/VC = 1.

4. At V, erect a perpendicular VB = VF. Join CB.

5. Mark a few points, say, 1, 2, 3, … on VC’ and erect perpendiculars through them meeting CB produced at 1’, 2’, 3’, …

6. With F as a centre and radius = 1–1’, cut two arcs on the perpendicular through 1 to locate P1 and P1’. Similarly, with F as a centre and radii = 2–2’, 3–3’, etc., cut arcs on the corresponding perpendiculars to locate P2 and P2’, P3 and P3’,etc.

7. Draw a smooth curve passing through V, P1, P2, P3 … P3

Definition:

It is the locus of a point which moves in a plane so that the ratioof its distances from a fixed point (focus) and a fixed straight line(directrix) is a constant and greater than one.

Uses:

1. Nature of graph of Boyle’s law

2. Shape of overhead water tanks

3. Shape of cooling towers etc.

Methods of drawing hyperbola:

Rectangular hyperbola

Oblique hyperbola

Foci and vertices method

Directrix-focus method

1. Draw horizontal and vertical axes first and then locate point P at 40 and 30 mm from lines

2. On horizontal and vertical lines from P, mark some points taking any distance and name them after P-1, 2,3,4 etc.

3. Extend horizontal line from P to right side and extend vertical line from P upward.

4. Join 1-2-3-4 points to O. Let them cut horizontal and vertical also at 1,2,3,4 points.

5. From horizontal 1,2,3 draw vertical lines downwards and

6. From vertical 1,2,3 points draw horizontal lines.

7. Line from 1 horizontal and line from 1 vertical will meet at P1.Similarly mark P2, P3 points.

8.Repeat the procedure by marking four points on upward vertical line from P and joining all those to O. Name these points and join them by smooth curve.

Foci and vertices method is quite similar to arc of circle method.

1.First draw transverse axis and mark on it two foci F1 and F2 and two vertices V1 and V2.

2.On the axis mark points 1,2,3……n beyond F2 at nearly equal distances.

3.Now with F1 and F2 centers and radius equal to (V1,V11), (V2,V12),……(V5,V15) draw arcs on both sides to intersect at points P1,P2,P3,P4….

4.Join P1, P2,P3,P4…. On two sides by a smooth curve to get two hyperbolas.

Construction similar to ellipse and parabola in directrix-focus method.