PROJECTIONS OF PLANES Definition of Plane A plane

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PROJECTIONS OF PLANES Definition of Plane: A plane surface or figure has only two

PROJECTIONS OF PLANES Definition of Plane: A plane surface or figure has only two dimensions i. e length and breadth and has no or negligible thickness. A plane surface may be of any shape e. g. circular or polygon, regular or irregular etc. . A plane surface may be represented by; (a)Three Points not on a Straight line (b)Two intersecting lines (c)A line and a point not lying on that line (d)Two parallel lines

(1) Three Points not on a Straight line b’ . . . a’ c’

(1) Three Points not on a Straight line b’ . . . a’ c’ X Y . b a. . c

(2) Two intersecting lines a’ c’ d’ b’ X Y a d c b

(2) Two intersecting lines a’ c’ d’ b’ X Y a d c b

(3)A line and a point not lying on that line a’ . p’ b’

(3)A line and a point not lying on that line a’ . p’ b’ Y X p. b a

(4) Two parallel lines b’ a’ d’ c’ X Y d c b a

(4) Two parallel lines b’ a’ d’ c’ X Y d c b a

ORIENTATION OF PLANES IN SPACE Orientation of planes in space may be of any

ORIENTATION OF PLANES IN SPACE Orientation of planes in space may be of any type as stated below; Class A: Plane parallel to (or contained by)one reference plane and hence perpendicular to both the other planes; (a) Plane parallel to (or contained by)P. P. and hence perpendicular to both H. P. & V. P. (b) Plane parallel to ( or contained by) V. P. and hence perpendicular to both H. P. & P. P. (c) Plane parallel to (or contained by) H. P. and hence perpendicular to both V. P. & P. P.

ORIENTATION OF PLANES IN SPACE Class B: Plane perpendicular to one reference plane and

ORIENTATION OF PLANES IN SPACE Class B: Plane perpendicular to one reference plane and inclined to other two planes; (a) Plane perpendicular to V. P. and inclined to H. P. by & also inclined to P. P. (b) Plane perpendicular to H. P. and inclined to V. P. by & also inclined to P. P. (c) Plane perpendicular to P. P. and inclined to H. P. by & to V. P. by

ORIENTATION OF PLANES IN SPACE Class C: Planes inclined to all the reference planes

ORIENTATION OF PLANES IN SPACE Class C: Planes inclined to all the reference planes ( i. e. H. P. , V. P. and P. P. ); - Plane inclined to H. P. by , inclined to V. P. by and also inclined to P. P - Such planes are also called oblique planes

Class A(1): Plane parallel to P. P. and hence perpendicular to both H. P.

Class A(1): Plane parallel to P. P. and hence perpendicular to both H. P. & V. P. Y . P. V ’ b , a’ A D ’ d , ’ c X a, d a” P. P d” YB c” C b, c a’, b’ . b” . P. H X V. P. c’, d’ X a, d a” d” Y b, c H. P. P. b” c”

Class A(2): Plane parallel to V. P. and hence perpendicular to both H. P.

Class A(2): Plane parallel to V. P. and hence perpendicular to both H. P. & P. P. A, a’ Y B, d’ D, d’ F. V. . P V. b’ a’ C, c’ X c’ Y Y d’ X a, d T. V. b, c A D d , a B C c , b P. . H X

Class A(3): Plane parallel to H. P. and hence perpendicular to both V. P.

Class A(3): Plane parallel to H. P. and hence perpendicular to both V. P. & P. P. Y . P. V d’ c’, , b’ a’ A X D a’, b’ c’, d’ F. V. C Y B d Y X a d b c c. P. H a b X T. V.

Class B(1): Plane perpendicular to V. P. and inclined to H. P. by &

Class B(1): Plane perpendicular to V. P. and inclined to H. P. by & also inclined to P. P. Y . P. V b’, c’ B . a’, d’ F. V. X Y a d b T. V. c Y a’, d’ A X a b D C c . P. H d = X

Exercise 01: A regular hexagonal plate ABCDEF, 30 mm side, is resting on H.

Exercise 01: A regular hexagonal plate ABCDEF, 30 mm side, is resting on H. P. on one of the sides/edges with surface of the plate making 45º with H. P. and perpendicular to V. P. Draw the projections of the plate.

Data Given: (1) Hexa. Plate Size = 30 mm (2) = 45° . P.

Data Given: (1) Hexa. Plate Size = 30 mm (2) = 45° . P. V d’, e’ E c’, f’ F a’, b’ ’ e ’ d A, a ’ , f E, e f ’ c F ’ a’b X A, a B, b Y D e f C d B, b D, d C, c c P. . H

Data Given: (1) Hexa. Plate Size = 30 mm (2) = 45° d 1’,

Data Given: (1) Hexa. Plate Size = 30 mm (2) = 45° d 1’, e 1’ c 1’, f 1’ a’, b’ X c’, f’ f Y f 1 e a 1 e 1 d b 1 d 1 30 a d’, e’ a 1’, b 1’ b Scale: 1: 1 c c

Class B (2): Plane perpendicular to H. P. and inclined to V. P. by

Class B (2): Plane perpendicular to H. P. and inclined to V. P. by Φ & also inclined to P. P. Y . P. . V b’ b’ a’ d’ a’ c’ Ac’ F. V. X . a, d Y T. V. d’ X Y C D a, d B b , c . P H. b, c = X

Exercise 02: A square plate PQRS, edge 25 mm size, is in space with

Exercise 02: A square plate PQRS, edge 25 mm size, is in space with one of its corners on V. P. Surface of the plate makes 60º with V. P. and it is perpendicular to H. P. Draw its projections.

Data Given: - Square Plate Size=25 mm s’ - Φ=60º . P. V r’

Data Given: - Square Plate Size=25 mm s’ - Φ=60º . P. V r’ S s’ , S P, p’ ’ r , R Y q’ P ’ p q’ , Q q, s r p R Φ Q q, s r P. . H p X X

Data Given: (1) Square Plate Size=25 mm s’ s 1’ S (2) Φ=60º P

Data Given: (1) Square Plate Size=25 mm s’ s 1’ S (2) Φ=60º P r’ p ’ 1 p’ X r 1’ R q 1’ q’ Q p q, s r p 1 Φ q 1, s 1 r 1 Y

Class B (3): Plane perpendicular to P. P. and inclined to H. P. by

Class B (3): Plane perpendicular to P. P. and inclined to H. P. by & to V. P. by Φ Y P. V. b’ Y A d’ Z a . B a’ X a”, b” P. P c”, d” C D d P. . c. H X

Exercise 03: A rectangular plate PQRS 30 mm X 60 mm size, is in

Exercise 03: A rectangular plate PQRS 30 mm X 60 mm size, is in space with shorter edge parallel to H. P. and 20 mm above it. Plate PQRS is perpendicular to V. P. and inclined to H. P. by such an angle so that its plan becomes square. Draw the projections.

Data Given: c 1’d 1’ - Size of rectangle=30 mm. X 60 mm T.

Data Given: c 1’d 1’ - Size of rectangle=30 mm. X 60 mm T. L. -Distance of plane from H. P. =20 mm (plane is parallel to H. P. ) - Size of Plan=square of 30 mm X 30 mm c’, d’ a 1’, b 1’ a’, b’ 20 θ D, d A, a a 1 d 1 b 1 c 1 30 X B , b 60 C, c 30 Y

Exercise 04: A regular pentagonal plate of 30 mm sides, has one of its

Exercise 04: A regular pentagonal plate of 30 mm sides, has one of its corners on H. P. The plane of the pentagon is inclined at 45º to the H. P. The side of the pentagon, which is opposite to the corner, which is on H. P. is inclined at 30º to the V. P. Draw the projections of the plate.

Data Given - Size of Pentagonal Plate =30 mm - = 45° - =

Data Given - Size of Pentagonal Plate =30 mm - = 45° - = 30° d 2’ T. L. c 1’, d 1’ b’, e’ e c’, d’ a 1’ d 30 X a’ a b 1’, e 1’ θ e 1 b 2’ Φ d 1 c b 1 Y a 2’ d 2 c 2 e 2 a 1 b e 2’ c 2’ a 2 Scale = 1: 1 b 2

Exercise 05: Draw the projections of a circle of 75 mm diameter resting on

Exercise 05: Draw the projections of a circle of 75 mm diameter resting on the H. P. on a point A of the circumference. Plane is inclined to the H. P. such that the plan of it is an ellipse of minor axis 30 mm. The plan of the diameter, thorough the point A, is making an angle of 30º with the V. P. Measure the angle of the plane with the H. P.

Data Given: 75 a’ b’, h’ c’, g’ d’, f’ e’ g X f

Data Given: 75 a’ b’, h’ c’, g’ d’, f’ e’ g X f h c 1’, g 1 b 1’, h 1’ ’ a 1’ θ g 1 h 1 f 1 e a b c T. L. (1) Circle Dia. =75 mm (2) =30° e 1’ (3) Minor axis plan d 1’, f 1’ length=30 mm d Scale: 1: 1 a 1 e 1 b 1 d 1 c 1 30 e 2’ f 2 ’ d 2’ g 2’ h 2’ g 2 a 2’ h 2 c 2’ b 2’ f 2 e 2 a 2 d 2 b 2 c 2 Y

Exercise 06: A regular hexagonal plate 40 mm side is resting on one of

Exercise 06: A regular hexagonal plate 40 mm side is resting on one of its edges on H. P. The plane makes an angle of 30º to H. P. and its longest diagonal makes an angle of 45 º with V. P. Draw its projections.

Data given : - (1) Regular hexagonal plate size = 40 mm (2) Θ

Data given : - (1) Regular hexagonal plate size = 40 mm (2) Θ =30º (3) = = 45º Scale = 1: 1 e 2’ d 1’e 1’ c’f’ f a c 2’ b 2’ e Y 2 d 2 40 X a’b’ L. f 2’ T ’ c ’f ’ 1 a 2’ d’e’ a 1’b θ 1 1 f 2 a e 1 d 2’ d b c d 1 b 1 c 1 a 2 b 2 c 2

Exercise 07: A square plate of side 50 mm is held on a corner

Exercise 07: A square plate of side 50 mm is held on a corner on H. P. with a diagonal horizontal and inclined at 40º to V. P. The plate is seen as a rhombus in plan with one of its diagonals measuring 40 mm. Draw its projections and determine the angle it makes with H. P.

Data Given: (1)Plate =50 mm sq. (2)Plate Diagonal =parallel to H. P. & at

Data Given: (1)Plate =50 mm sq. (2)Plate Diagonal =parallel to H. P. & at 40° ( ) to V. P. L b’d’ d c’ T. S. square c a 1’ b 1’d 1’ a d 1 b c 1 a 1 40 b 1 d 2’ b 2’ a 2’ Y = d 2 c 2 50 X a’ R= L (3)Plate is seen as rhombus with one diagonal=40 mm c 2’ c 1’ a 2 b 2

Exercise 08: A regular pentagon ABCDE, of 25 mm sides, has its side AB

Exercise 08: A regular pentagon ABCDE, of 25 mm sides, has its side AB in the V. P. and inclined at an angle of 45º to the H. P. The corner A is 20 mm above H. P. and the corner D is 25 mm in front of V. P. Draw the projections of the plane and find its inclination with the V. P.

Data Given: (1) Regular pentagonal plate size=25 mm (2) =45° (3) A= 20 mm

Data Given: (1) Regular pentagonal plate size=25 mm (2) =45° (3) A= 20 mm above XY line (4) D= 25 mm below XY line a’ a, b c’ c, e d Scale=1: 1 d 2’ c 2’ d 1’ e 2’ b 1’ c 1’ a 1, b 1 c 1, e 1 Locus of d b 2’ a 2 25 25 d’ b’ X a 1’ e 1’ d 1 e 2 a 2’ d 2 20 e’ -: Answer: = 31° b 2 Y c 2

Exercise 09: A 30º- 60º set square has its shortest side 40 mm long

Exercise 09: A 30º- 60º set square has its shortest side 40 mm long and is in the H. P. The top view of the set square is an isosceles triangle and the hypotenuse of the set square is inclined at an angle of 45º with the V. P. Draw the projections of the set square and find its inclination with the H. P.

Data: - Answer: - (1) θ = 55º (1) T. L. (AB)=40 mm (2)

Data: - Answer: - (1) θ = 55º (1) T. L. (AB)=40 mm (2) Φ=45º R=T. L. c 1’ . 30º b 2’ a 2’ Y a 2 Φβ T. b 1 40 c 1 b 2 L. . Scale: -1: 1 c θ P. L b T. L 40 60º c’ a 1’, b 1’ a 1 L. P. 40 a’, b’ X a Locus of c’ c 2 Locus of c

Exercise 10: ABCD is a rhombus of diagonals AC=120 mm and BD=60 mm. Its

Exercise 10: ABCD is a rhombus of diagonals AC=120 mm and BD=60 mm. Its corner A is in the H. P. and the Plane is inclined to the H. P. such that the plan appears to be a square. The plan of diagonal AC makes an angle of 30º to the V. P. Draw the projections of the plane and find its inclination with H. P.

Data: (1) Size of rhombus = 120 mm X 60 mm (2) =30º c

Data: (1) Size of rhombus = 120 mm X 60 mm (2) =30º c 1’ c 2’ Answer: - T. L . (1) θ = 60º b 1’d 1’ a’ X b’, d’ c’ a 1’ 60 Scale: -1: 1 a 2’ c 1 c a 1 120 b b 2’ Y d 1 d a d 2’ b 60 1 d 2 a 2 30º c 2 b 2

Exercise 11: A regular hexagonal plate 40 mm side is resting on one of

Exercise 11: A regular hexagonal plate 40 mm side is resting on one of its corners in H. P. The diagonal through that corner is inclined at 30º to H. P. and (a)the plan of that diagonal (b) is inclined to V. P. by 30º and (b)diagonal is inclined at 30º. Draw its projections.

Data Given: (1) Hexagonal plate size=40 mm (2) =30° (3) =30° [for case (a)]

Data Given: (1) Hexagonal plate size=40 mm (2) =30° (3) =30° [for case (a)] (4) =30° [for case (b)] d 2’ e ’ e 2’ 2 c 2’ T. c ’e ’ c 2’ b 1’f 1’ 1 1 f 2’ f ’ a’ b’f’ c’e’ b ’ d’ 2 X b 2’ 1 2 e a ’ 2 e 2 2 2 f f 1 e d 2 d 1 f 2 d 2 a d 1 a β Φ c 2 a c 2 2 a 2 b b 2 c 1 b 2 40 c Locus of d L. d 1’ Scale=1: 1 Y

A square plate having size 40 mm, is resting on V. P. with one

A square plate having size 40 mm, is resting on V. P. with one of its corner. The plane is inclined to V. P. by 40°. The diagonal passing through the point which is on V. P. , is inclined at 30° to the H. P. Draw its projection.

DATA : (1) Ф =40° (2) β =30° (3) Size of square =40 mm

DATA : (1) Ф =40° (2) β =30° (3) Size of square =40 mm C 3’ C 2’ L. . P. T. L LOCUS OF C 2’ d 2’ Ф Scale : - 1: 1

PROJECTIONS OF PLANES

PROJECTIONS OF PLANES