Engineering-curves 1 3v512r

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CONIC SECTIONS Ellipse, Parabola and Hyperbola are called conic sections Because These curves appear on the surface of a cone When it is cut by some typical cutting planes. OBSERVE ILLUSTRATIONS GIVEN BELOW..

Section Plane Through Generators

Pa rab ola

Ellipse

Section Plane Parallel to end generator.

Section Plane Parallel to Axis.

Hyperbola

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ENGINEERING CURVES PART -1 {Conic Sections} ELLIPSE

PARABOLA

HYPERBOLA

1.General Method 1.General Method 1.General Method (Directrix – focus Method) (Directrix – focus (Directrix – focus Method) 2. Arcs of Circle Method 2.Rectangle Method Method) 3. Oblong Method 2.Rectangular 3. Method of Tangents ( Rectangle Method)(Triangle Method) Hyperbola 4. Concentric Circle Method (coordinates given) 3. Rectangular Hyperbola (P-V diagram Methods of Drawing Tangents & Normals Equation given) To These Curves. 3

COMMON DEFINITION OF ELLIPSE, PARABOLA & HYPERBOLA: These are the loci of points moving in a plane such that the ratio of it’s distances from a fixed point And a fixed line (directrix) always remains constant. The Ratio is called ECCENTRICITY. (E)

A) For Ellipse E<1 B) For Parabola E=1 C) For Hyperbola E>1

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PROBLEM 1:- Point F is 60 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 }

DIRECTRIX-FOCUS METHOD

A

ELLIPSE

DIRECTRIX

STEPS: 1.Draw a vertical line AB and point F 60 mm from it. 2.Divide 60 mm distance in 5 parts. 3.Name 2nd part from F as V. It is 24mm and 36mm from F and AB line resp. 4.Draw perpendicular line at V with a distance of VF. ( VG = VF). 5.Consider few vertical lines 1-11 , 2-21 , 3-31…… 6.With center F and radius 1-11 draw arc on the 1st vertical line and also with the same center and radius 2-21 draw arc on the 2nd vertical line and repeat steps. 7. all these Points with V in smooth curve.

ELLIPSE

(vertex) V

F ( focus)

B 5

ELLIPSE TANGENT & NORMAL TO DRAW TANGENT & NORMAL TO THE CURVE FROM A GIVEN POINT ( Q )

A

DIRECTRIX

Problem 14:

ELLIPSE

STEPS: 1. point Q to F. 2.Construct 900 angle with This line at point F 3.Extend the line to meet directrix At T 4. this point to Q and extend. This is tangent to ellipse from Q 5.To this tangent draw perpendicular Line from Q. It is normal to curve.

T

(vertex) V

F ( focus) 900

N

N

Q

B

T

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PROBLEM 9: Point F is 60 mm from a vertical straight line AB. Draw locus of point P, moving in a plane such that it always remains equidistant from point F and line AB. STEPS: 1.Draw a vertical line AB and point F 60 mm from it. 2.Divide CF (60 mm) distance in 2 parts. 3.Name midpoint as V. 4.Draw perpendicular line at V with a distance of VF. ( VG = VF). 5.Consider few vertical lines 1-11 , 2-21 , 3-31…… 6.With center F and radius 1-11 draw arc on the 1st vertical line and also with the same center and radius 2-21 draw arc on the 2nd vertical line and repeat steps. 7. all these Points with V in smooth curve.

PARABOLA

DIRECTRIX-FOCUS METHOD

PARABOLA

A

(VERTEX) V 1 2 3 4

F ( focus)

It will be the locus of P equidistance from line AB and fixed point F.

B 7

PARABOLA TANGENT & NORMAL TO DRAW TANGENT & NORMAL TO THE CURVE FROM A GIVEN POINT ( Q )

T

PARABOLA

A

1. POINT Q TO F. 2.CONSTRUCT 900 ANGLE WITH THIS LINE AT POINT F 3.EXTEND THE LINE TO MEET DIRECTRIX AT T 4. THIS POINT TO Q AND EXTEND. THIS IS TANGENT TO THE CURVE FROM Q 5.TO THIS TANGENT DRAW PERPENDICULAR LINE FROM Q. IT IS NORMAL TO CURVE.

VERTEX V

900

F ( focus) N

Q

B

N

T 8

PROBLEM 12:- POINT F IS 60 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 3/2 DRAW LOCUS OF POINT P. { ECCENTRICITY = 3/2 }

HYPERBOLA DIRECTRIX FOCUS METHOD

A

STEPS: 1.Draw a vertical line AB and point F 60 mm from it. 2.Divide 60 mm distance in 5 parts. 3.Name 2nd part from F as V. It is 36mm and 24mm from F and AB line resp. 4.Draw perpendicular line at V with a distance of VF. ( VG = VF). 5.Consider few vertical lines 1-11 , 2-21 , 331…… 6.With center F and radius 1-11 draw arc on the 1st vertical line and also with the same center and radius 2-21 draw arc on the 2nd vertical line and repeat steps. 7. all these Points with V in smooth curve.

(vertex)

V

F ( focus)

B

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HYPERBOLA TANGENT & NORMAL A

TO DRAW TANGENT & NORMAL TO THE CURVE FROM A GIVEN POINT ( Q ) T 1. POINT Q TO F. 2.CONSTRUCT 900 ANGLE WITH THIS LINE AT POINT F 3.EXTEND THE LINE TO MEET DIRECTRIX AT T 4. THIS POINT TO Q AND EXTEND. THIS IS TANGENT TO CURVE FROM Q 5.TO THIS TANGENT DRAW PERPENDICULAR LINE FROM Q. IT IS NORMAL TO CURVE.

(vertex)

F ( focus)

V

900 N

N

Q

B T

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SECOND DEFINATION OF AN ELLIPSE:It is a locus of a point moving in a plane such that the SUM of it’s distances from TWO fixed points always remains constant. {And this sum equals to the length of major axis.} These TWO fixed points are FOCUS 1 & FOCUS 2

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PROBLEM 4. Major axis AB & minor axis CD are 150mm and 100mm long respectively. Draw ellipse by “arcs of circles Method”. Draw a tangent to a ellipse at a point on it 25mm below the major axis.

BY ARCS OF CIRCLE METHOD As per the definition Ellipse is locus of point P moving in a plane such that the SUM of it’s distances from two fixed points (F1 & F2) remains constant and equals to the length of major axis AB.(Note A .1+ B .1=A . 2 + B. 2 = AB)

STEPS: 1.Draw both axes as usual. 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 can be located below AB. 7. all points by smooth curve to get an ellipse/

ELLIPSE

p4

p3

C

p2 p1

A

F1

1

2

3

4

B

O

F2

D 12

TO DRAW TANGENT & NORMAL TO THE CURVE FROM A GIVEN POINT ( Q ) 1. Point Q To F1 & F2 2. Bisect Angle F1Q F2 . The Angle Bisector Is Normal 3. A Perpendicular Line Drawn To It Is Tangent To The Curve.

p4

p3

ELLIPSE TANGENT & NORMAL

C

p2 p1

1

2

3

4

B F2

AL

F1

O

NOR M

A

Q

TAN GE NT

D 13

Steps: 1 Draw a rectangle taking major and minor axes as sides. 2. In this rectangle draw both axes as perpendicular bisectors of each other.. 3. 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) 4. Name those as shown.. 5. Now 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. 6. Then extend C-1 line upto D-1 and mark that point. Similarly extend C-2, C-3, C-4 lines up to D-2, D-3, & D-4 lines. 7. Mark all these points properly and all along with ends A and D in smooth possible curve. Do similar construction in right side part. along with lower half of the rectangle. all points in smooth curve. It is required ellipse.

ELLIPSE

BY RECTANGLE METHOD

Problem 2 Draw ellipse by Rectangle method. Take major axis 150 mm and minor axis 90 mm long.

D

4

4

3

3

2

2

1

1

A

1

2

3

3

4

C

2

B

1

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Problem 3:- Draw ellipse by Oblong method. Draw a parallelogram of 100 mm and 70 mm long sides with included angle of 750.Inscribe Ellipse in it. STEPS ARE SIMILAR TO THE PREVIOUS CASE (RECTANGLE METHOD) ONLY IN PLACE OF RECTANGLE, HERE IS A PARALLELOGRAM.

ELLIPSE

BY OBLONG METHOD

D 4

4

3

3 2

2 1

1

A

1

2

3

C

4

3

2

1

B

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ELLIPSE

Problem 1 :Draw ellipse by concentric circle method. Take major axis 100 mm and minor axis 70 mm long.

Steps: 1. Draw both axes as perpendicular bisectors of each other & name their ends as shown. 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. 5.From all points of inner circle draw horizontal lines to intersect those vertical lines. 6. Mark all intersecting points properly as those are the points on ellipse. 7. all these points along with the ends of both axes in smooth possible curve. It is required ellipse.

BY CONCENTRIC CIRCLE METHOD

3 2

4 C

1 2

3

5 4

1

5

A

B 10 10

6 9

8 D

9

7

6

7 8 16

Problems 1.A fixed point is 75mm from a straight line. Draw the locus of a point P moving such a way that its distance from fixed straight line is i) twice its distance from the fixed point ii) equal to its distance from the fixed point. Name the curves. 2.Two fixed points F1 and F2 are 100mm apart. Trace the complete path of a point P moving (in the same plane as that of F1 and F2) in such a way that the sum of distance F 1 and F2 is always the same and equal to 125mm. Name the curve. Draw the another curve parallel to 25mm away from this curve. 3.The sum of the distance of point P from two fixed points is 120mm and the distance between the fixed points is 80mm draw the locus of the Point P. 4.Construct an ellipse when the major axis is 120mm and the distance between foci is 108mm. Determine the length of the minor axis. 5.The major axis of an ellipse is 100mm and the minor axis is 55mm. Find the foci and construct the ellipse by Intersecting Arcs method. [EEE JAN 2015] 6.A Plot of a ground is in the shape of a rectangle 110mm X 50mm. inscribe an elliptical lawn in it. Take a suitable scale. [ECE JAN 2015] 7.Construct an ellipse of 120mm major axis and 80mm minor axis using concentric circles method? [CSEAUG 2014] 8.The major axis of an ellipse is 150mm long and the minor axis is 100mm long. Find the foci and draw the ellipse by arcs of circle method. Draw a tangent to the ellipse at a point on it 25mm above the major axis. [ECE JAN 2015] 9.The foci of an ellipse are 90mm apart and the minor axis is 72mm long. Determine the length of the major axis. Construct the ellipse. [ECE FEB 2014]. 17 10.The major and minor axes of an elliptical fish pond are 10m and 6m respectively. Draw the