W ELCOME Engineering Graphics - Lect 2 1

O VERVIEW OF P LANE C URVES Regular Polygons up to hexagon. Conic Section Involute Cycloid Archimedian Spiral. 2

Conic section Ellipse Parabola Hyperbola O VERVIEW 3

M ETHODS TO D RAW C ONIC S ECTION Directrix Focus Method Ellipse Parabola Hyperbola Ellipse Concentric Circles Method Rectangle or oblong Method Arc of circle method Parabola Rectangle or oblong method Tangent Method Hyperbola Directrix Focus Method 4

C ONIC S ECTION When a right circular cone is cut by section plane in different positions with respect to the axis of the cone, the sections obtained are called as conics. 5

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C ONIC S ECTION When the section plane is inclined to the axis and cuts all the generators on one side of the vertex we get an ellipse as section. When the section plane is inclined to the axis and is parallel to the one of the generators we get parabola as section. When the section plane is inclined the axis and makes a smaller angle than that of made by the generator hyperbola as section. 7

D EFINITION It is a path generated by a point in one plane in such way that ratio of its distance from fixed point (Focus) to its distance from fixed line (Directrix) remains constant. This ratio is called as Eccentricity (e) If Ratio (e) < 1 it is Ellipse IF Ratio (e) =1 it is Parabola If Ratio (e) >1 it is Hyperbola The line at right angle to the directrix and passing through a focus is called axis. The point at which the conic curve cuts its axis is called as the vertex. 8

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D IRECTX F OCUS METHOD To construct an ellipse when the distance of the focus from directrix is equal to 60mm and eccentricity = 2/3 by using directrix focus method. 13

Draw a vertical line which represents directrix, At any point O, a right angle draw an axis OO 2 and mark focus F on the axis so that OF=60mm 14

Now divide OF into five equal parts Mark vertex V on the 3rd part of from O so that VF/VO=2/3=e (VF=60x2/5=24mm,VO=60x3/5=36mm) 15

Through V, draw a perpendicular VV´ to the axis, which is equal to the V 1 F. 16

Draw a line joining O and V’, and extend it. 17

Mark points 1,2,3…on the axis, and draw perpendicular lines to meet OA when produced at 1´ 18

With F as a centre and radius equal to 11´, Draw arcs to cut the perpendicular through 1 on the both sides of the axis. 19

Repeat the same procedure for other points, e.g. for point 2, take radius equal to 22’ and F as a centre, draw arks to cut the perpendicular through 2 on the both sides of the axis. 20

Finally, draw a smooth curve through all these points. Then this curve is required ellipse. 21

ELLIPSE DIRECTRIX-FOCUS METHOD PROBLEM 6:- POINT F IS 50 MM FROM A LINE AB.A POINT P IS MOVING IN A PLANE SUCH THAT THE RATIO OF IT’S DISTANCES FROM F AND LINE AB REMAINS CONSTANT AND EQUALS TO 2/3 DRAW LOCUS OF POINT P. { ECCENTRICITY = 2/3 } F ( focus) DIRECTRIX V ELLIPSE (vertex) A B STEPS: 1.Draw a vertical line AB and point F 50 mm from it. 2.Divide 50 mm distance in 5 parts. 3.Name 2 nd part from F as V. It is 20mm and 30mm from F and AB line resp. It is first point giving ratio of it’s distances from F and AB 2/3 i.e 20/30 4 Form more points giving same ratio such as 30/45, 40/60, 50/75 etc. 5.Taking 45,60 and 75mm distances from line AB, draw three vertical lines to the right side of it. 6. Now with 30, 40 and 50mm distances in compass cut these lines above and below, with F as center. 7. Join these points through V in smooth curve. This is required locus of P.It is an ELLIPSE. 30mm 45mm 23