CHAPTER – 2

1. Engineering drawingis a language of all persons involved in engineering activity. Engineering ideas are recorded by preparing drawings and execution of work is also carried out on the basis of drawings. Communication in engineering field is done by drawings. It is called as a “Language of Engineers”.

2. CHAPTER – 2 ENGINEERING CURVES

3. USES OF ENGINEERING CURVES • Useful by their nature & characteristics. • Laws of nature represented on graph. • Useful in engineering in understanding laws, manufacturing of various items, designing mechanisms analysis of forces, construction of bridges, dams, water tanks etc.

4. CLASSIFICATION OF ENGG. CURVES 1. CONICS 2. CYCLOIDAL CURVES 3. INVOLUTE 4. SPIRAL 5. HELIX 6. SINE & COSINE

5. Vertex/Apex 90º Base What is Cone ? • It is a surface generated by moving a Straight line keeping one of its end fixed & other end makes a closed curve. • The fixed point is known as vertex or apex. • The closed curve is known as base. • If the base/closed curve is a circle, we get a cone. • If the base/closed curve is a polygon, we get a pyramid.

6. Vertex/Apex Cone Axis Generator Base • The line joins apex to the center of base is called axis. • If axes is perpendicular to base, it is called as rightcircular cone. • If axis of cone is not perpendicular to base, it is called as oblique cone. • The line joins vertex/ apex to the circumference of a cone is known as generator. 90º

7. CONICS • Definition :-The section obtained by the intersection of a right circular cone by a cutting plane in different position relative to the axis of the cone are called CONICS.

8. CONICS A - TRIANGLE B - CIRCLE C - ELLIPSE D – PARABOLA E - HYPERBOLA

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

10. Sec Plane Circle CIRCLE • When the cutting plane is perpendicular to the axis or parallel to the base in a right cone we get circle the section.

11. θ α ELLIPSE Definition :- • When the cutting plane is inclined to the axis but not parallel to generator or the inclination of the cutting plane(α) is greater than the semi cone angle(θ), we get an ellipse as the section. α > θ

12. θ α PARABOLA • When the cutting plane is inclined to the axis and parallel to one of the generators of the cone or the inclination of the plane(α) is equal to semi cone angle(θ), we get a parabola as the section. α = θ

13. HYPERBOLA α = 0 θ θ Definition :- • When the cutting plane is parallel to the axis or the inclination of the plane with cone axis(α) is less than semi cone angle(θ), we get a hyperbola as the section. α < θ

14. Conic Curve Directrix Focus CONICS • Definition :- The locus of point moves in a plane such a way that the ratio of its distance from fixed point (focus) to a fixed Straight line (Directrix) is always constant. M P F C V • Fixed straight line is called as directrix. • Fixed point is called as focus.

15. Conic Curve M P Directrix Axis F C V Focus Vertex • The line passing through focus & perpendicular to directrix is called as axis. • The intersection of conic curve with axis is called as vertex.

16. Conic Curve Directrix F C V Focus Axis Vertex M P N Q Distance of a point from focus Ratio = Distance of a point from directrix = Eccentricity = PF/PM = QF/QN = VF/VC = e

17. Ellipse Axis Vertex Focus ELLIPSE • Ellipse is the locus of a point which moves in a plane so that the ratio of its distance from a fixed point (focus) and a fixed straight line (Directrix) is a constant and less than one. P M Directrix F C V Eccentricity=PF/PM = QF/QN < 1. N Q

18. C P D O B A F2 F1 Q ELLIPSE • Ellipse is the locus of a point, which moves in a plane so that the sum of its distance from two fixed points, called focal points or foci, is a constant. The sum of distances is equal to the major axis of the ellipse.

19. P C B A D CF1 +CF2 = AB O but CF1 = CF2 F2 F1 hence, CF1=1/2AB Q PF1 + PF2 = QF1 + QF2 = CF1 +CF2 = constant = F1A + F1B = F2A + F2B But F1A = F2B F1A + F1B = F2B + F1B = AB = Major Axis

20. C C A A B B D D F1 F2 Major Axis = 100 mm Minor Axis = 60 mm O CF1 = ½ AB = AO F1 F2 Major Axis = 100 mm F1F2 = 60 mm O CF1 = ½ AB = AO

21. Uses :- • Shape of a man-hole. • Shape of tank in a tanker. • Flanges of pipes, glands and stuffing boxes. • Shape used in bridges and arches. • Monuments. • Path of earth around the sun. • Shape of trays etc.

22. Parabola M Directrix Axis Vertex Focus PARABOLA Definition :- • The parabola is the locus of a point, which moves in a plane so that its distance from a fixed point (focus) and a fixed straight line (directrix) are always equal. • Ratio (known as eccentricity) of its distances from focus to that of directrix is constant and equal to one (1). P F C V Eccentricity = PF/PM = QF/QN = 1. Q N

23. Uses :- Home • Motor car head lamp reflector. • Sound reflector and detector. • Bridges and arches construction • Shape of cooling towers. • Path of particle thrown at any angle with earth, etc.

24. HYPERBOLA Hyperbola P M Axis Directrix Vertex F C V Focus N Q • It is the locus of a point which moves in a plane so that the ratio of its distances from a fixed point (focus) and a fixed straight line (directrix) is constant and grater than one. Eccentricity = PF/PM = QF/QN > 1.

25. Uses :- • Nature of graph of Boyle’s law • Shape of overhead water tanks • Shape of cooling towers etc.

26. METHODS FOR DRAWING ELLIPSE 1. Arc of Circle’s Method 2. Concentric Circle Method 3. Loop Method 4. Oblong Method 5. Ellipse in Parallelogram 6. Trammel Method 7. Parallel Ellipse 8. Directrix Focus Method

27. ARC OF CIRCLE’S METHOD C B A   D P4 P4 P3 P3 P2 P2 P1 P1 Rad =B1 R =A1 F1 F2 O 1 2 3 4 `R=A2 R=B2 P1’ P1’ Tangent P2’ P2’ Normal P3’ P3’ 90° P4’ P4’

28. Axis A B Minor CONCENTRIC CIRCLE METHOD 10 11 9 P10 C 8 12 P11 P9 10 N 11 9 P12 P8 8 T 12 P7 Major Axis Q P1 1 7 O F1 7 1 F2 6 P6 P2` 2 5 3 P3 4 P5 2 6 P4 D e = AF1/AQ 3 5 CF1=CF2=1/2 AB 4

29. C P4’ E 4 4’ P3 P3’ 3 3’ P2 P2’ S 2 2’ P1 P1’ Ø 1 Ø 1’ F1 F2 B 1’ 3 3’ 2’ 1 2 4 P1 P1’’ P2 P2’’ P P3’’ P3 F P4 P4’’ D OBLONG METHOD P4 Normal Directrix Minor Axis R=AB/2 Major Axis P0 0 0’ 4’ A Tangent

30. C A B 60° D ELLIPSE IN PARALLELOGRAM 0 H P1 0 Q1 P0 P2 1 1 Q2 K 2 P3 2 Q3 3 Q4 P4 3 4 Q5 4 5 P5 Minor Axis 5 P6 Q6 O 5 6 2 3 1 0 4 1 2 5 3 4 6 Major Axis S4 R4 G J S3 R3 S2 R2 S1 I R1

31. D1 V1F1 QV1 2 = = 3 R1V1 R1V1 R=1a N D1 N ELLIPSE – DIRECTRIX FOCUS METHOD g f  < 45º e Ellipse d c Eccentricity = 2/3 P7 P6 P5 b P4 P3 a P2 P1 Q R=6f` Directrix  1 2 7 3 4 5 6 R1 V1 F1 Dist. Between directrix & focus = 50 mm 90° P1’ T Tangent P2’ 1 part = 50/(2+3)=10 mm P3’ P4’ P5’ P6’ P7’ V1F1 = 2 part = 20 mm S T V1R1 = 3 part = 30 mm Normal

32. PROBLEM :- The distance between two coplanar fixed points is 100 mm. Trace the complete path of a point G moving in the same plane in such a way that the sum of the distance from the fixed points is always 140 mm. Name the curve & find its eccentricity.

33. ARC OF CIRCLE’S METHOD G AF1 e = AE F1 B F2 A 100   90° 140 G’ G4 G4 G3 • e G3 G2 directrix G2 R=70 G1 G1 R=70 R=B1 R =A1 O E 90° 1 2 3 4 `R=A2 R=B2 G1’ G1’ Tangent G2’ G2’ Normal G3’ G3’ G4’ G4’ GF1 + GF2 = MAJOR AXIS = 140

34. PROBLEM :-3 Two points A & B are 100 mm apart. A point C is 75 mm from A and 45 mm from B. Draw an ellipse passing through points A, B, and C so that AB is a major axis.

35. 8 7 8 7 6 6 2 2 A B 5 3 4 3 5 4 D P8 1 C E P7 1 P1 45 75 P6 100 O P2 P5 P3 P4

36. PROBLEM :-5 ABCD is a rectangle of 100mm x 60mm. Draw an ellipse passing through all the four corners A, B, C and D of the rectangle considering mid – points of the smaller sides as focal points. Use “Concentric circles” method and find its eccentricity.

37. 4 1 4 1 50 Q P F1 F2 100 2 3 A B 3 2 R I1 I4 D C O I2 I3 S

38. PROBLEM :-1 Three points A, B & P while lying along a horizontal line in order have AB = 60 mm and AP = 80 mm, while A & B are fixed points and P starts moving such a way that AP + BP remains always constant and when they form isosceles triangle, AP = BP = 50 mm. Draw the path traced out by the point P from the commencement of its motion back to its initial position and name the path of P.

39. 2 2 1 1 A B P 60 1 2 2 1 80 M P2 Q2 Q1 P1 R = 50 O Q R1 S1 R2 S2 N

40. PROBLEM :-2 Draw an ellipse passing through 60º corner Q of a 30º - 60º set square having smallest side PQ vertical & 40 mm long while the foci of the ellipse coincide with corners P & R of the set square. Use “OBLONG METHOD”. Find its eccentricity.

41. 3 60º   2 1 3’’ ? 2’’ 1’’ ? C TANGENT directrix NORMAL O3 O3’ Q R=AB/2 3 MINOR AXIS O2 O2’ 2 ELLIPSE 40mm O1 O1’ 89mm 1 F1 F2 MAJOR AXIS 30º 80mm S B A P R 1’ 2’ 3’ D ECCENTRICITY = AP / AS MAJOR AXIS = PQ+QR = 129mm

42. PROBLEM :-4 Two points A & B are 100 mm apart. A point C is 75 mm from A and 45 mm from B. Draw an ellipse passing through points A, B, and C so that AB is not a major axis.

43. 75 45 A 100 B ELLIPSE C 0 0 H P1 P2 Q1 P0 1 1 P3 2 Q2 2 Q3 K 3 3 P4 Q4 4 4 P5 Q5 5 5 6 Q6 O P6 6 6 4 2 1 1 2 3 4 6 3 0 5 5 J G I D

44. PROBLEM :- Draw an ellipse passing through A & B of an equilateral triangle of ABC of 50 mm edges with side AB as vertical and the corner C coincides with the focus of an ellipse. Assume eccentricity of the curve as 2/3. Draw tangent & normal at point A.

45. PROBLEM :- Draw an ellipse passing through all the four corners A, B, C & D of a rhombus having diagonals AC=110mm and BD=70mm. Use “Arcs of circles” Method and find its eccentricity.

46. METHODS FOR DRAWING PARABOLA 1. Rectangle Method 2. Parabola in Parallelogram 3. Tangent Method 4. Directrix Focus Method

47. P1 P1 P2 P2 P3 P3 PARABOLA P4 P4 P5 P5 A B PARABOLA –RECTANGLE METHOD D C V 0 0 1 1 2 2 3 3 4 4 5 5 P6 P6 6 6 0 5 4 3 2 1 1 2 3 4 5

48. P’ P’ 2 1 P’ 3 P’ 4 P’ 5 P’ P 6 6 B 0 30° X A PARABOLA – IN PARALLELOGRAM C 0 1’ 2’ V 3’ P1 4’ P2 5’ P3 D 0 6’ 5’ 1 P4 4’ 3’ 2 2’ 1’ 3 P5 1 4 2 3 5 4 5 6

49. PARABOLA TANGENT METHOD V   O B A 0 10 9 1 8 2 7 3 4 6 5 5 4 6 F 3 7 2 8 1 9 10 0

50. PARABOLA DIRECTRIX FOCUS METHOD D N N D P4 P3 P2 R4 PF R3 R2 P1 RF R1 AXIS R V F 1 2 3 4 90° 90° T P1’ PF’ S P2’ P3’ DIRECTRIX P4’ T