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PROJECTIONS OF STRAIGHT LINESDefinition of Straight line:A straight line is the shortest distancebetween two points.-Top views of two end points of a straight line, when joined, give the top view of the straight line.-Front views of the two end points of a straight line, when joined, give the front view of the straight line.-Both the above projections are straight lines.
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Orientation of Straight Line in Space- A line in space may be parallel, perpendicular or inclined to either the H.P. or V.P. or both.- It may be in one or both the reference Planes.- Line ends may be in different Quadrants.- Position of Straight Line in space can be fixed by various combinations of data like distance of its end points from reference planes, inclinations of the line with the reference planes, distance between end projectors of the line etc.
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Notatioans used for Straight LineTrue length of the line:Denoted by Capital letters. e.g. AB=100 mm,means that true length of the line is 100 mm.Front View Length:Denoted by small letters. e.g. a’b’=70 mm,means that Front View Length is 70 mm.Top View Length:Denoted by small letters. e.g. ab=80 mm,means that Top View Length is 80 mm.Inclination of True Length of Line with H.P.:It is denoted by θ. e.g. Inclination of the linewith H.P. (or Ground) is given as 30º meansthat θ = 30º.
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Inclination of True Length of Line with V.P.:It is denoted by Φ. e.g. Inclination of the linewith V.P. is given as 40º means that Φ = 40º.Inclination of Front View Length with XY :It is denoted by α. e.g. Inclination of the FrontView of the line with XY is given as 50º meansthat α = 50º.Inclination of Top View Length with XY :It is denoted by β. e.g. Inclination of the TopView of the line with XY is given as 30º meansthat β = 30º.End Projector Distance:It is the distance between two projectorspassing through end points of F.V. & T.V.measured parallel to XY line.
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Line in Different Positions with respect to H.P. & V.P.CLASS A: Line perpendicular to (or in) one reference plane & hence parallel to both the other planes(1) Line perpendicular to P.P. & (hence) parallel to both H.P. & V.P.(2) Line perpendicular to V.P. & (hence) parallel to both H.P. & P.P.(3) Line perpendicular to H.P. & (hence) parallel to both V.P. & P.P.
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Line in Different Positions with respect to H.P. & V.P.CLASS B: Line parallel to (or in) one reference plane & inclined to other two planes(1) Line parallel to ( or in) V.P. & inclined to H.P. by .(2) Line parallel to ( or in) H.P. & inclined to V.P. by .(3) Line parallel to ( or in) P.P. & inclined to H.P. by & V.P. by .
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Line in Different Positions with respect to H.P. & V.P.CLASS C: Line inclined to all three reference planes ( Oblique lines )Line inclined to H.P. by , to V.P. by and alsoinclined to profile plane.
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Class A(1) : Line perpendicular to P.P. & hence parallel to both the other planes Y . V.P b’ a” P.P. a’ . b” B Y A X b H. P. a z x
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Class A(1) : Line perpendicular to P.P. & hence parallel to both the other planes . V.P b a Y X b’ P. a’ H.
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. P.P a”, b” . P. V. b’ Y1 a’ Y X b a H.P .Class A(1) : Line perpendicular to P.P. & hence parallel to both the other planes
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a’ b’ a”, b” . F.V. L.H.S.V. X Y Y1 a T.V. bClass A(1) : Line perpendicular to P.P. & hence parallel to both the other planes
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Exercise 1 :- A Line AB, 50mm long is perpendicular tothe profile plane. The end A is 20mm below H.P. , 30mmbehind V.P. & 10mm to the left of P.P. Draw theprojections of straight line AB (i.e. Front View & TopView). Data given :- Y1 50 Profile (1) T.L. = 50mm b. .a Plane (2)Point A 20 below H.P. T.V.=T.L. 30mm Behind V.P. 30 (3) Line is perpendicular 10 to P.P.X Y (4) Line is 10mm left of 20 b’. . a’ P.P. F.V.=T.L. Y1 Scale :- 1:1
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Y V.P . a’,. b’ A Y B a X H. P. b XClass A(2):Line perpendicular to V.P. & (hence) parallel to both the other Planes (i.e. H.P. & P.P.)
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Class A(2):Line perpendicular to V.P. & (hence) parallel to both the other Planes . V.P a’, b’ . Y X a P. H. b
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Class A(2):Line perpendicular to V.P. & (hence) parallel to both the other Planes V.P. a’, b’ . F.V. X Y a T.V. b H.P.
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Exercise 2 :- A Line ABC, 80mm long is perpendicularto V.P & 50mm below H.P. Point B, 20mm from A ison V.P. A is in 4th quadrant. Draw the projections ofline ABC. .c Data given :- (1) T.L. = 80mm (2) AB = 20, BC = 60 60 (3) Point B is in V.P. X .b Y (4) Line is 50mm below H.P. 20 .a 50 - Point A is in 4th quadrant .a’,b’,c’- Line is perpendicular to V.P. Scale :- 1:1
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Y . V.P b’ B Y a’ A X . H. P. a,b XClass A(3):Line perpendicular to H.P. & (hence) parallel to both the other Planes
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Class A(3):Line perpendicular to H.P. & (hence) parallel to both the other Planes . V.P b’ Y a’ X . a,b P. H.
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Class A(3):Line perpendicular to H.P. & (hence) parallel to both the other Planes V.P. a’ b’ X Y a,. b H.P.
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Exercise 3:- A Line AB, 50mm long is perpendicularto H.P. & it is below H.P. Point A is on H.P. & 30mmbehind V.P. Draw the projections of the line AB. Data given :- . a,b (1) T.L. = 50mm 30 On H.P. (2) Point A 30mm BehindX . Y a’ V.P. (3) Line is perpendicular F.V.=T.L. to H.P. 50 .b’ Scale :- 1:1
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Class B(1): Line contained by ( or parallel to) V.P. & inclined to H.P. by Y . V.P B b’ A θ b Y a’ a X H.P . X
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Class B(1): Line contained by ( or parallel to) V.P. & inclined to H.P. by . V.P b’ a’ θ b Y a X . H.P
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Class B(1): Line contained by ( or parallel to) V.P. & inclined to H.P. by V.P. b’ a’ θ a b X Y V.P.
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Exercise 4 :- A Line AB, 75mm long, is in V.P. Itmakes an angle of 30º with the H.P. Point A is20mm above H.P. Draw the projections of line AB. b’ Data given :- T. L. .= (1)T.L. = 75mm F.V (2) = 30º . a’ ==30º (3)Point A = 20mm 20 X a. b Y above H.P. T.V. - Line AB is in V.P.Scale :- 1:1
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Class B(2) : Line parallel to (or contained by) H.P. & inclined to V.P. by Y V.P. V.P . b’ a’ b’ a’ A ø B Y X Y β a .P . X b H a β=φ X b H.P.
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Exercise 5 :- A Line AB, 120mm long, is parallel to H.P.and inclined to V.P. by 50º. Point B is 10mm above H.P.and 40mm on in front of V.P. Point A is behind V.P. Drawthe projection of line AB. Data given :- . a (1) T.L. = 120mm . a’ F.V. . b’ (2) = 50ºX 10 Y Point B (3) 10 above H.P. P. = 40mm in Front of L. V.P. = 40 T. L - Line is parallel to H.P. . - Point A is behind V.P. b Scale :- 1:1
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Class B(3): Line parallel to (or contained by) P.P., inclined to H.P. by & to V.P. by Y V.P. a” P.P . a’ A Y b” b’ X a B H.P. b X Z
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Class B(3): Line parallel to (or contained by) P.P., inclined to H.P. by & to V.P. by V.P. a’ a” P.P. b’ b” X Y a b H.P.
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Exercise 6 :- The distance between the end projectors ofline MN is zero. Point M is 40 mm below H.P. & 25 mmbehind V.P. Point N is 15 mm below H.P. & 65 mm behindV.P. Draw its projections and find the angle of the linewith H.P. and V.P. Also find the true length of the line. Data given :- .n (1) Point M 40 below H.P. 25 mm behind T.V. V.P. . m 15 below H.P.65 (2) Point N 65 mm behind 25 45 Y V.P.X (3) End projector dist. = 0 .n’ .n” 15 40 F.V. T.L. .m’ .m”
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Class C:Line inclined to H.P. by & V.P. by ( i.e. Line inclined to both the planes) Y . V.P b’ B Y a’ A . X a b H.P X
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Class C:Line inclined to H.P. by & V.P. by ( i.e. Line inclined to both the planes) . V.P b’ Y a’ X a b H.P .
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Class C:Line inclined to H.P. by & V.P. by ( i.e. Line inclined to both the planes) V.P. b’ a’ X Y a b H.P.
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Exercise 7:- A Line AB, 90 mm long, is inclined to H.P. by 30° and inclined to V.P. by 45º. The line is in first quadrant with Point A 15 mm above H.P. and 25 mm in front of V.P. Draw the projection of line AB. Locus of b’ b’ b1 ’ Data Given :- F.V. .L.= 90 (1) T.L.=90 mm T (2) Θ =30° a’ . θ α b2 ’ (3) Φ =45°25 15 15 above H.P.X Y (4) Point A 25 mm in Front of a .Φ b1 V.P. Answers :- β (1) F.V.= 64 mm T.V. (2) T.V = 78 mm b2 Locus of b (3) = 45° (4) = 55° b
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Exercise 8 :- The distance between the end projectors of astraight line AB is 80mm. Point A is 10mm above H.P. and30mm in front of V.P. Point B is 40mm above H.P. and 50mmbehind V.P. Draw the projections and find the inclination ofstraight line AB with H.P & V.P. and the true length of theline. Data given :- (1) E.P.D. = 80mm 10 above H.P. Locus of b (2) Point A 30mm in front of . V.P. Locus of b’ b b1 . (3) Point B 40 above H.P. b’ 50mm behind . b2’ V.P. F.V . 50 a’ b1’ 40 Answers :- 30 10 V. Y (1) = 15ºX T. a . 80 b2 (2) = (3) T.L. = 43º 117mm
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Exersice 9 : A room is 5m X 4.5m X 4 m high. Determineby method of projections of straight lines, distancebetween diagonally(solid) opposite corners of the room. Locus of b’ b’ Data Given :- b2’ (1) Length of the room=L=5mH (2) Breadth of the room=B=4.5m θ (3) Height of the room=H=4m a’X a Y b2 Answer :- (1) Diagonal distance betweenB opposite corners of the room a’b2’= 7.826m b L
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Exercise 10 :- Two unequal legs AB and AC, hinged at Amake an angle of 135º between them in their elevation andplan. Leg AB is perpendicular to the Profile Plane.Determine the real angle between them. Data Given :- (1) = 135º Locus of c’ C (2) = 135º c’ - AB is perpendicular to P.P. . c3’c2’ T.L L. - Legs are unequal (AB > AC) T. B,b’ A,a’ Answer :- T.L. - The real angle between twoX Y unequal legs = ∠BAC=125º a c3 b c2 c Locus of c
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Exercise 11 :- Two Mangoes on a tree, planted near thecompound wall of a bunglow, are 1m and 1.25m above theground and 0.5m & 0.75m from a 15cm thick compound wallbut on the opposite sides of it. The distance between Mangoesmeasured along the ground and parallel to the wall is 1m.Determine the real distance between centres of two mangoes. p q2 Data Given :- (1) E.P.D. = 1m 0.5m T. 15cmPoint P 1m above ground V (2) . 0.5 behind wallX Locus of q’ Y 1.25m above ground q’ q2’ (3) Point Q F.V. 0.75m 0.75m in front of p’ wall (4) Wall thickness = 15cm q Scale :- 1:20 1.25m Answer :- 1m Locus of q - the real distance between centres of two mangoes =G 1m L p’q2’= 1.63m
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Exercise 12 : The F.V. of a line MN, 90 mm long, measures65 mm. Point M is in V.P. and 20 mm below H.P. Point N isin the first quadrant. Draw the projections and findinclinations of line with H.P. and V.P. Data Given: Locus of n’ 20mm below H.P. n’ n1’ Point M (1) In V.P. (2) T.L.= 90 mm F.V. (3) F.V.= 65 mm X m . Φ n1 Y (4) α = 45° (5) Point N is in 20 α θ . m’ n2’ first Quadrant Answers: T.V. (1) Θ = 31° (2) Φ = 44° n n2 Locus of n Scale:- 1:1
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TRACES OF A LINEDefinition: When a line is inclined to a plane, it will meet that plane, produced if necessary. The point where the line or line produced meets the plane is called trace.Horizontal Trace: The point of intersection of the inclined line with the H.P. is called Horizontal Trace or simply H.T.Vertical Trace: The point of intersection of the inclined line with the V.P. is called Vertical Trace or simply V.T.
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Example to illustrate V.P. the concept of traces b’ F.V . Y B a’ A .a h v b H.P. T.V.X H.T. . V.T. V.P.
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IMPORTANT POINTS REGARDING TRACES OF A LINE - If a line is inclined to both H.P. & V.P. then its Front view, h’ and V.T. must be on the same straight line. e.g. if front view of a line AB is a’b’, then h,a’,b’ and V.T. must be on a same straight line. - If a line is inclined to both H.P. & V.P. then its Top view, v and H.T. must be on the same straight line. e.g. if Top View of a line AB is ab, then v, a, b and H.T. must be on a same straight line.
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IMPORTANT POINTS REGARDING TRACES OF A LINE(1) If a line is parallel to any of the plane, it has no trace upon that plane. P.e.g. If the line is parallel to V.T. V. horizontal plane then a’,. b’ that line will not meet A H.P and hence there Y will be no H.T. and B only V.T. a P. H. b
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IMPORTANT POINTS REGARDING TRACES OF A LINE(1) If a line is parallel to any of the plane, it has no trace upon that plane.e.g. If the line is parallel to horizontal plane then V.T. b’ that line will not meet . a’ A ø B H.P and hence there will be no H.T. and only V.T. β a b
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IMPORTANT POINTS REGARDING TRACES OF A LINEe.g. If the line is parallel to . V.P Vertical Plane then b’ that line will not meet V.P and hence there B will be no V.T. and Y a’ only H.T. A P. X H.T. . H. a,b
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IMPORTANT POINTS REGARDING TRACES OF A LINE e.g. If the line is parallel to Vertical Plane then that line will not meet V.P and hence there will be no V.T. and only H.T. b’ T. L. .= F.V a’ = =30º 20 h’ a b H.T. T.V.
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Exercise 13 : A line AB, 80 mm long is seen as a straight line of length 55mm in its front view and of length 65 mm in its top view. Its end A is 10 mm above H.P. & is in first quadrant where as end B is 25 mm behind V.P. and is in Second Quadrant. Draw its projections and find out its inclinations with H.P. & V.P. and also locate its traces.
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Data Given: (1) T.L.=80 mm (2) F.V. = 55mm (3) T.V. = 65mm 10 mm above H.P.(4) End A ??? I.F.O V.P. Locus of b’ ??? above H.P.(5) End B b1 ’ 25mm behind V.P. b’ V.T. F.V. . b b2 Locus of b Answers: (1) = 36° a’ 25 b2’ 10 (2) = 46° X h v Y(3) H.T.=46mm T.V. I.F.O. V.P. a (4) V.T.=36mm H.T. . 65 b1 above H.P.
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Exercise 14 : The end projectors distance of a line MN is zero. Its end M is 25mm below H.P. & 40mm behind V.P. where as end N is 10 above H.P. & 55mm in front of V.P. Draw its projections and find out its inclinations with H.P. & V.P. and also locate its traces.
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Data Given: 25 below H.P. (1) End M 40 behind V.P. 10 above H.P. Z (2) End N 55mm I.F.O.V.P. . m (3) E.P.D. = 0 Answers: .n’ .n” (1) = 20°40 .H.T. (2) = 70° 25 10X .v . V.T. Y (3) T.L.= 101mm .hm’ m” . (4) H.T.=27mm55 I.F.O. V.P. .n (5) V.T.=12mm below H.P. Z (6) + = 90°
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Exercise 15 : A divider instrument of a compass boxhaving two equal arms AB & BC hinged at B is keptin H.P. on its needle point A & C with the linejoining A & C is perpendicular to V.P. It is seen infront view as a straight line 100mm long inclined at30° to H.P. while it is seen in top view as an angle abc with <abc = 60°. Draw its front view and top viewand find;(1) The height of point B above H.P.(2) The apart distance between the needle points A&C(3)The lengths of arms AB & BC with real between them
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Data Given: Locus of b’ b’(1) F.V.=100 mm f F.V.(2) = 30° H(3) <abc =2 = 60° a’,c’ X b1 ’ Y(4) ac is perpendicular to V.P. a,A Answers: AB dist.(1) AB = 112mm 2 60° b(2) 2 = 53° b1, B(3) AC = 100 mm Locus of b’ c,C
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Exercise 16 : The end A of a straight line AB,120mm long, is 50 mm behind V.P. & 35 mmbelow H.P. The line is inclined to H.P. by 30°& has a point C on it in both the referenceplanes. Draw the projections of the line andfind out its inclinations with V.P. Also locateits traces.
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Data Given: 35 below H.P. (1) End A 50 behind V.P. a. c1 b1 (2) T.L.=120 mm Locus of b’ (3) = 30° T.V. of AC b’ b1’ (4) C on AB in 50 both ref. PlanesX C,c’,c . H.T. & V.T. c1’ Y Answers: F.V. of AC (1) = 45° 35 (2) H.T.= in . a’ b b2 b2’ V.P.& H.P. (3) V.T.= in V.P. Locus of b’ & H.P.