Download presentation

Published byBailee Martell Modified over 3 years ago

1
**Hyperbola Directrix e>1 O: center F1, F2: foci V1, V2: vertices**

PF2 – PF1 = V1F2 – V1F1 = V1V2 = 2a Product of shortest distances from P to the asymptotes is a constant. When the asymptotes are perpendicular it is a called a rectangular hyperbola. P Axis b F2 V2 O V1 F1 Hyperbola a/e a Asymptote ae

2
**DIRECTRIX-FOCUS METHOD**

Draw an ellipse, focus is 50 mm from the directrix and the eccentricity is 3/2 HYPERBOLA DIRECTRIX-FOCUS METHOD 2’ P2 A 1’ VE = VF1 P1 E DIRECTRIX F1-P1=F1-P1’ = 1-1’ F1-P1/(P1 to directrix AB) = 1-1’/C-1=VE/VC (similar triangles) =VF1/VC=2/3 THEREFORE P1 AND P1’ LIE ON THE HYPERBOLA (vertex) V F1 ( focus) C 1 2 F1-P2=F1-P2’= 2-2’ P2 AND P2’ ALSO LIE ON THE HYPERBOLA P1’ B P2’

3
**HYPERBOLA THROUGH A POINT OF KNOWN CO-ORDINATES**

Problem: Point P is 40 mm and 30 mm from horizontal and vertical axes respectively. Draw a Hyperbola through it. Solution Steps: 1) Extend horizontal line from P to right side. 2) Extend vertical line from P upward. 3) On horizontal line from P, mark some points taking any distance and name them 1, 2, 3 etc. 4) Join points to pole O. Let them cut part [P-B] at 1’,2’,3’ points. 5) From horizontal 1,2,3 draw vertical lines downwards and 6) From vertical 1’,2’,3’ points [from P-B] draw horizontal lines. 7) Vertical line from 1 and horizontal line from 1’ P1.Similarly mark P2, P3, P4 points. 8) Repeat the procedure by marking points 4, 5 on upward vertical line from P and joining all those to pole O. They cut the horizontal line from P at 4’ and 5’. Repeat earlier procedure to obtain points P4, P5. Join them by a smooth curve. 5 P5 4 P4 5’ 4’ P 1 2 3 40 mm P1 1’ P2 2’ P3 3’ 30 mm O B

4
**Hyperbola-rectangle method**

1’ 2’ 2 1 Base Height of hyperbola Axis height

5
**HYPERBOLA P-V DIAGRAM 1 2 3 4 5 6 7 8 9 10 + PRESSURE ( Kg/cm2) 1 2 3**

Problem: A sample of gas is expanded in a cylinder from 10 unit pressure to 1 unit pressure. Expansion follows law PV=Constant. If initial volume being 1 unit, draw the curve of expansion. Also Name the curve. 1 2 3 4 5 6 7 8 9 10 Form a table giving few more values of P & V P V = C + 10 5 4 2.5 2 1 = Now draw a Graph of Pressure against Volume. It is a PV Diagram and it is a Hyperbola. Take pressure on vertical axis and Volume on horizontal axis. PRESSURE ( Kg/cm2) 1 2 3 4 5 6 7 8 9 10 VOLUME:( M3 )

6
**TO DRAW TANGENT & NORMAL TO THE CURVE AT A GIVEN POINT ( Q )**

ELLIPSE TANGENT & NORMAL Problem : TO DRAW TANGENT & NORMAL TO THE CURVE AT A GIVEN POINT ( Q ) JOIN POINT Q TO F1 & F2 BISECT ANGLE F1Q F2 THE ANGLE BISECTOR IS NORMAL A PERPENDICULAR LINE DRAWN TO IT IS TANGENT TO THE CURVE. D F1 F2 A B C p1 p2 p3 p4 O NORMAL Q TANGENT

7
**TO DRAW TANGENT & NORMAL**

ELLIPSE TANGENT & NORMAL Problem: TO DRAW TANGENT & NORMAL TO THE CURVE AT A GIVEN POINT ( Q ) F ( focus) DIRECTRIX V ELLIPSE (vertex) A B 1.JOIN POINT Q TO F. 2.CONSTRUCT 900 ANGLE WITH THIS LINE AT POINT F 3.EXTEND THE LINE TO MEET DIRECTRIX AT T 4. JOIN 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 900 N Q N T

8
**TO DRAW TANGENT & NORMAL**

PARABOLA TANGENT & NORMAL Problem: TO DRAW TANGENT & NORMAL TO THE CURVE AT A GIVEN POINT ( Q ) A B PARABOLA VERTEX F ( focus) V T 1.JOIN POINT Q TO F. 2.CONSTRUCT 900 ANGLE WITH THIS LINE AT POINT F 3.EXTEND THE LINE TO MEET DIRECTRIX AT T 4. JOIN 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. 900 N Q N T

9
**TO DRAW TANGENT & NORMAL**

HYPERBOLA TANGENT & NORMAL Problem 16 TO DRAW TANGENT & NORMAL TO THE CURVE FROM A GIVEN POINT ( Q ) F ( focus) V (vertex) A B 1.JOIN POINT Q TO F. 2.CONSTRUCT 900 ANGLE WITH THIS LINE AT POINT F 3.EXTEND THE LINE TO MEET DIRECTRIX AT T 4. JOIN 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. T 900 N N Q T

10
**Concept of Principal lines of a plane**

All the points lie on a straight line representing the edge of the plane Point view C B TL A1 T Draw a line on the plane in one view parallel to the other plane. The corresponding projection in the other plane will give the true length. A T F A’ C’ Principal line B’

11
Principal lines: Lines on the boundary or within the surface, parallel to the principal planes of projection -They can be frontal lines (parallel to frontal plane) -Horizontal lines (parallel to top plane) Frontal line (parallel to frontal plane) b True length b l a f l a c T c T F c’ F c’ a’ l’ f’ a’ True length b’ l’ Horizontal line (parallel to top plane) b’

12
**To obtain the edge view of a plane**

c1 a1 b1 x1 y1 Edge view of the plane b True length l a c -Draw a principle line in one principle view and project the true length line in the other principle view -With the reference line perpendicular to the true length line, draw a primary auxiliary view of the plane, to obtain the edge view Distances: a1, b1, c1 from x1y1 = a’, b’, c’ from xy respectively T x y c’ F l’ a’ Horizontal line (parallel to top plane) b’

13
**Auxiliary view of TRUE SHAPE of a plane always gives an EDGE VIEW**

True shape is the auxiliary view obtained from the edge view c1 a1 b1 x1 y1 a c2 a2 b2 c4 a4 b4 Edge view of the plane a is the angle of the plane with the HP b True length l c3 a3 b3 a c Edge view of plane T True shape and dimensions of the plane y x F c’ l’ a’ b’ Horizontal line (parallel to top plane)

Similar presentations

OK

DEVELOPMENT OF SURFACES Part I Prof.T.JEYAPOOVAN Department of Mechanical Engineering Hindustan Institute of Technology and Science Chennai-603103, India.

DEVELOPMENT OF SURFACES Part I Prof.T.JEYAPOOVAN Department of Mechanical Engineering Hindustan Institute of Technology and Science Chennai-603103, India.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on product advertising ideas Free ppt on etiquettes Ppt on taj lands end Download ppt on motivational stories Ppt online open university Ppt on new zealand culture Ppt on food chains and webs Ppt on world diabetes day 2017 Ppt on forward rate agreement legs Free download ppt on unity in diversity