ORTHOGRAPHIC PROJECTIONS OF POINTS LINES PLANES AND SOLIDS

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ORTHOGRAPHIC PROJECTIONS OF POINTS, LINES, PLANES, AND SOLIDS. TO DRAW PROJECTIONS OF ANY OBJECT,

ORTHOGRAPHIC PROJECTIONS OF POINTS, LINES, PLANES, AND SOLIDS. TO DRAW PROJECTIONS OF ANY OBJECT, ONE MUST HAVE FOLLOWING INFORMATION A) OBJECT { WITH IT’S DESCRIPTION, WELL DEFINED. } B) OBSERVER { ALWAYS OBSERVING PERPENDICULAR TO RESP. REF. PLANE}. C) LOCATION OF OBJECT, { MEANS IT’S POSITION WITH REFFERENCE TO H. P. & V. P. } TERMS ‘ABOVE’ & ‘BELOW’ WITH RESPECTIVE TO H. P. AND TERMS ‘INFRONT’ & ‘BEHIND’ WITH RESPECTIVE TO V. P FORM 4 QUADRANTS. OBJECTS CAN BE PLACED IN ANY ONE OF THESE 4 QUADRANTS. IT IS INTERESTING TO LEARN THE EFFECT ON THE POSITIONS OF VIEWS ( FV, TV ) OF THE OBJECT WITH RESP. TO X-Y LINE, WHEN PLACED IN DIFFERENT QUADRANTS. STUDY ILLUSTRATIONS GIVEN ON HEXT PAGES AND NOTE THE RESULTS. TO MAKE IT EASY HERE A POINT A IS TAKEN AS AN OBJECT. BECAUSE IT’S ALL VIEWS ARE JUST POINTS.

NOTATIONS FOLLOWING NOTATIONS SHOULD BE FOLLOWED WHILE NAMEING DIFFERENT VIEWS IN ORTHOGRAPHIC PROJECTIONS. OBJECT

NOTATIONS FOLLOWING NOTATIONS SHOULD BE FOLLOWED WHILE NAMEING DIFFERENT VIEWS IN ORTHOGRAPHIC PROJECTIONS. OBJECT POINT A LINE AB IT’S TOP VIEW a ab IT’S FRONT VIEW a’ a’ b’ IT’S SIDE VIEW a” a” b” SAME SYSTEM OF NOTATIONS SHOULD BE FOLLOWED INCASE NUMBERS, LIKE 1, 2, 3 – ARE USED.

VP 2 nd Quad. 1 ST Quad. Y Observer HP X Y X 3

VP 2 nd Quad. 1 ST Quad. Y Observer HP X Y X 3 rd Quad. 4 th Quad. THIS QUADRANT PATTERN, IF OBSERVED ALONG X-Y LINE ( IN RED ARROW DIRECTION) WILL EXACTLY APPEAR AS SHOWN ON RIGHT SIDE AND HENCE, IT IS FURTHER USED TO UNDERSTAND ILLUSTRATION PROPERLLY.

POINT A IN Point A is ND Placed In 2 QUADRANT different A quadrants

POINT A IN Point A is ND Placed In 2 QUADRANT different A quadrants and it’s Fv & Tv are brought in same plane for Observer to see HP clearly. Fv is visible as it is a view on VP. But as Tv is is a view on Hp, it is rotated downward 900, In clockwise direction. The In front part of Hp comes below xy line and the part behind Vp HP comes above. Observe and note the process. A POINT A IN RD 3 QUADRANT POINT A IN 1 ST QUADRANT VP a’ A a HP OBSERVER a’ a a’ VP OBSERVER VP A POINT A IN 4 TH QUADRANT

Basic concepts for drawing projection of point FV & TV of a point always

Basic concepts for drawing projection of point FV & TV of a point always lie in the same vertical line FV of a point ‘P’ is represented by p’. It shows position of the point with respect to HP. If the point lies above HP, p’ lies above the XY line. If the point lies in the HP, p’ lies on the XY line. If the point lies below the HP, p’ lies below the XY line. TV of a point ‘P’ is represented by p. It shows position of the point with respect to VP. If the point lies in front of VP, p lies below the XY line. If the point lies in the VP, p lies on the XY line. If the point lies behind the VP, p lies above the XY line.

PROJECTIONS OF A POINT IN FIRST QUADRANT. POINT A ABOVE HP & INFRONT OF

PROJECTIONS OF A POINT IN FIRST QUADRANT. POINT A ABOVE HP & INFRONT OF VP For Tv PICTORIAL PRESENTATION a’ A Y X POINT A IN HP & INFRONT OF VP POINT A ABOVE HP & IN VP a’ A For Y For F v Fv Y a’ a X X a For Tv PICTORIAL PRESENTATION A a For ORTHOGRAPHIC PRESENTATIONS OF ALL ABOVE CASES. Fv above xy, Tv below xy. VP Fv above xy, Tv on xy. Fv on xy, Tv below xy. VP VP a’ X a’ Y X a Y a’ X a a HP HP HP Y Fv

PROJECTIONS OF STRAIGHT LINES. INFORMATION REGARDING A LINE means IT’S LENGTH, POSITION OF IT’S

PROJECTIONS OF STRAIGHT LINES. INFORMATION REGARDING A LINE means IT’S LENGTH, POSITION OF IT’S ENDS WITH HP & VP IT’S INCLINATIONS WITH HP & VP WILL BE GIVEN. AIM: - TO DRAW IT’S PROJECTIONS - MEANS FV & TV. SIMPLE CASES OF THE LINE 1. A VERTICAL LINE ( LINE PERPENDICULAR TO HP & // TO VP) 2. LINE PARALLEL TO BOTH HP & VP. 3. LINE INCLINED TO HP & PARALLEL TO VP. 4. LINE INCLINED TO VP & PARALLEL TO HP. 5. LINE INCLINED TO BOTH HP & VP. STUDY ILLUSTRATIONS GIVEN ON NEXT PAGE SHOWING CLEARLY THE NATURE OF FV & TV OF LINES LISTED ABOVE AND NOTE RESULTS.

For Tv (Pictorial Presentation) Note: Fv is a vertical line Showing True Length &

For Tv (Pictorial Presentation) Note: Fv is a vertical line Showing True Length & Tv is a point. a’ . V. P 1. A Line perpendicular to Hp & // to Vp A FV b’ Y a’ Fv b’ X Fo B Orthographic Pattern V. P. Y r. F v TV a b Tv a b X H. P. Orthographic Pattern (Pictorial Presentation) 2. . P. V A Line // to Hp & // to Vp . F. V V. P. Note: Fv & Tv both are // to xy & both show T. L. For Tv b’ B a’ Fv b’ a’ A X Y Fo Y r. F b X a T. v a V. H. P. Tv b

. V. P 3. b’ . a’ b’ F. V B F. V. A

. V. P 3. b’ . a’ b’ F. V B F. V. A Line inclined to Hp and parallel to Vp V. P. Fv inclined to xy Tv parallel to xy. Y X (Pictorial presentation) a’ A . Y a b T. V X a H. P. Orthographic Projections 4. A Line inclined to Vp and parallel to Hp . V. P a’ V. P. Tv inclined to xy Fv parallel to xy. . b’ F. V A a’ Ø B (Pictorial presentation) Ø T. V. b’ X Y a a Fv Ø Tv b H. P. b

For Tv 5. b’ . V. P For Fv A Y For F v

For Tv 5. b’ . V. P For Fv A Y For F v A X T. V. a’ On removal of object i. e. Line AB Fv as a image on Vp. Tv as a image on Hp, a B . F. V Y a’ X . B b’ V. P (Pictorial presentation) . F. V A Line inclined to both Hp and Vp b a T. V. b V. P. b’ FV a’ X Y Orthographic Projections Fv is seen on Vp clearly. To see Tv clearly, HP is rotated 900 downwards, a Hence it comes below xy. Note These Facts: Both Fv & Tv are inclined to xy. (No view is parallel to xy) Both Fv & Tv are reduced lengths. (No view shows True Length) TV H. P. b

Note the procedure When Fv & Tv known, How to find True Length. (Views

Note the procedure When Fv & Tv known, How to find True Length. (Views are rotated to determine True Length & it’s inclinations with Hp & Vp). Orthographic Projections Means Fv & Tv of Line AB are shown below, with their apparent Inclinations & V. P. b’ b’ FV a’ Fv TL a’ Y a’ X Y a TV H. P. b’ b 1’ FV X a Note the procedure When True Length is known, How to locate FV & TV. (Component a’b 2’ of TL is drawn which is further rotated to determine FV) TV b 1 Here TV (ab) is not // to XY line Hence it’s corresponding FV a’ b’ is not showing True Length & True Inclination with Hp. H. P. Y a b 1 Ø Tv b In this sketch, TV is rotated and made // to XY line. Hence it’s corresponding FV a’ b 1’ Is showing True Length & True Inclination with Hp. b 2 ’ X TV b TL H. P. TL b b 2 Here a’b 1’ is component of TL ab 1 gives length of FV. Hence it is brought Up to Locus of a’ and further rotated to get point b’. a’ b’ will be Fv. Similarly drawing component of other TL(a’b 1‘) TV can be drawn.

Projection of straight line Line inclined to both HP & VP Type-I Given projections

Projection of straight line Line inclined to both HP & VP Type-I Given projections (FV & TV) of the line. To find True length & true inclination of the line with HP (θ) and with VP(Φ). PROBLEM End A of a line AB is 20 mm above HP & 20 mm in front of VP while its end B is 55 mm above HP and 75 mm in front of VP. The distance between end projectors of the line is 50 mm. Draw projections of the line and find its true length and true inclination with the principal planes. Also mark its traces.

b 1 ’ b’ θ: True inclination of the line with HP = 24º

b 1 ’ b’ θ: True inclination of the line with HP = 24º TL 55 a’ X HT VT’ h’ 20 θ α b 2 ’ α : Inclination of FV of the line with HP/XY Y v 50 20 Φ β b 1 a β : Inclination of TV of the line with VP/XY 75 TL b b Ø: True inclination of the line with VP = 41º 2

Type –II Line inclined to both HP & VP Given (i) T. L. ,

Type –II Line inclined to both HP & VP Given (i) T. L. , θ and Ø, (ii) T. L. , F. V. , T. V. to draw projections, find α, β, H. T. and V. T. PROBLEM A line AB, 70 mm long, has its end A 20 mm above HP and 20 mm in front of VP. It is inclined at 30° to HP and 45°to VP. Draw its projections and mark its traces.

b’ b 1 ’ 70 a’ 15 X 30° b 2 ’ HT VT’

b’ b 1 ’ 70 a’ 15 X 30° b 2 ’ HT VT’ h’ Y v 20 b 1 a 45° 70 b b 2

Q 10. 11 The top view of a 75 mm long line AB measures

Q 10. 11 The top view of a 75 mm long line AB measures 65 mm, while its front view measures 50 mm. Its one end A is in HP and 12 mm in front of VP. Draw the projections of AB and determine its inclination with HP and VP To draw FV &TV of the line AB Given, TL=75 mm, TV=65 mm, FV=50 mm To find θ & Ø b 1 ’ b’ Ans. θ=31º 50 A is in HP & 12 mm→VP 75 Ans. Ø=49º a’ X Y 12 31º a Hint: Draw ab 1=65 mm // to XY. Because when TV is // to XY, FV gives TL. b 1 65 49º 75 65 b b 2

Q 10. 12 A line AB, 65 mm long has its end A 20

Q 10. 12 A line AB, 65 mm long has its end A 20 mm above H. P. and 25 mm in front of VP. The end B is 40 mm above H. P. and 65 mm in front of V. P. Draw the projections of AB and show its inclination with H. P. and V. P. Given, To draw FV &TV of the line AB A is 20 mm ↑ HP & 25 mm →V. P. B is 40 mm ↑ & 65 mm → V. P. To find θ & Ø b 1 ’ b’ 65 a’ b 2 ’ Hint 2: Draw locus of b’ 40 mm above XY & locus of b 65 mm below XY 40 TL=65 mm Hint 1: Mark a’ 20 mm above H. P & a 25 mm below XY 20 18º X 25 Y 38º Ans. θ=18º b 1 65 a 65 b b 2 Ans. Ø=38º

Q 10. 13: The projectors of the ends of a line AB are 5

Q 10. 13: The projectors of the ends of a line AB are 5 cm apart. The end A is 2 cm above the H. P and 3 cm in front of V. P. The end B is 1 cm below H. P. and 4 cm behind the V. P. Determine the true length and traces of AB, and its inclination with the two planes Given, To find, True Length, θ, Ø, H. T. and V. T. A 0 B 0=50 mm A is 20 mm ↑ HP & 30 mm →V. P. B is 10 mm ↓ & 40 mm ← V. P. b b 2 20 HT X b 2’ 91 VT’ 40 a’ v h’ 10 50 30 b’ a 50º Y 20º Ans. θ=20º b 1 Ans. Ø=50º

Q 10. 14: A line AB, 90 mm long, is inclined at 45 to

Q 10. 14: A line AB, 90 mm long, is inclined at 45 to the H. P. and its top view makes an angle of 60 with the V. P. The end A is in the H. P. and 12 mm in front of V. P. Draw its front view and find its true inclination with the V. P. b’ Given, b 1 ’ T. L. =90 mm, θ=45º, β=60º A is in the H. P. & 12 mm→V. P. To find/draw, F. V. , T. V. & Ø 90 Ans. Ø = 38º a’ 12 X a Y 45º 60º 38º b 1 90 b b 2

Q 10. 16: The end A of a line AB is 25 mm behind

Q 10. 16: The end A of a line AB is 25 mm behind the V. P. and is below the H. P. The end B is 12 mm in front of the VP and is above the HP The distance between the projectors is 65 mm. The line is inclined at 40 to the HP and its HT is 20 mm behind the VP. Draw the projections of the line and determine its true length and the VT Given, To find/draw, A 0 B 0=65 mm A is 25 mm ←V. P. & is ↓H. P. is 12 mm →V. P. & is above HP = 40º F. V. , T. L. , VT’ B θ b’ b 1 ’ b 2 ’ Ans. TL= a’b 2’=123 mm VT’ b 2 b 1 HT 20 X h’ a’ v 40º 12 25 a b 65 Y

10. 17: A line AB, 90 mm long, is inclined at 30 to the

10. 17: A line AB, 90 mm long, is inclined at 30 to the HP. Its end A is 12 mm above the HP and 20 mm in front of the VP. Its FV measures 65 mm. Draw the TV of AB and determine its inclination with the VP 65 b’ b 1 ’ 90 a’ 12 Y 20 X 30° b 1 44° a Ans: Ø = 44º 90 b b 2

Q 10. 23: Two lines AB & AC make an angle of 120 between

Q 10. 23: Two lines AB & AC make an angle of 120 between them in their FV & TV. AB is parallel to both the HP & VP. Determine the real angle between AB & AC. C c’ c 2’ c 1’ Ans. 112º 112° b’ 120° a’ X Y a b c 2 120° c c 1

Q 8: A line AB 65 mm long has its end A in the

Q 8: A line AB 65 mm long has its end A in the H. P. & 15 mm in front of the V. P. The end B is in the third quadrant. The line is inclined at 30 to the H. P. and at 60 to the V. P. Draw its projections. VP b 2 b 65 b 15 b 2 ’ 30º 15 X a’ 60º 65 a’ Y b 1 ’ 30º 60º a a b’ X Y b” b’ a” HP

Q 10. 19 A line AB, inclined at 40º to the V. P. has

Q 10. 19 A line AB, inclined at 40º to the V. P. has its end 50 mm and 20 mm above the H. P. the length of its front view is 65 mm and its V. T. is 10 mm above the H. P. determine. the true length of AB its inclination with the H. P. and its H. T. Given, To find, Ø = 40º, A is 20 mm↑HP, B is 50 mm ↑ HP, FV=65 mm, VT is TL, θ & HT ↑ HP b 1’ b’ 85 50 a’ 21º VT’ HT 10 b 2’ X v 40º b 1 a h’ 20 10 mm Y Ans, TL = 85 mm, = 21º & HT is 17 mm behind VP b 2 θ

Q 10. 19 A line AB, inclined at 40º to the V. P. has

Q 10. 19 A line AB, inclined at 40º to the V. P. has its end 50 mm and 20 mm above the H. P. the length of its front view is 65 mm and its V. T. is 10 mm above the H. P. determine. the true length of AB its inclination with the H. P. and its H. T. B 1’ Given, To find, Ø = 40º, A is 20 mm↑HP, B is 50 mm ↑ HP, FV=65 mm, VT is 10 mm ↑ HP TL, θ & HT Step 1: For solving the problem by trapezoidal method, draw a line at 40º(Ø) from VT’. Then draw perpendiculars from a’ and b’ on this line. Step 2: Then draw projectors from a’ and b’ and mark the distance of b’B 1’ on the projector of b’ below XY. Similarly mark the distance a’A 1’ on the projector of a’ below XY b’ A 1’ 65 50 a’ 40º v h’ 10 X 20 VT’ 21º HT Y a Ans: A 1’B 1’=TL=85 mm Ans: HT is 17 mm behind VP Ans: θ = 21º b

Q 6. The top view of a 75 mm long line CD measures 50

Q 6. The top view of a 75 mm long line CD measures 50 mm. C is 50 mm in front of the VP & 15 mm below the HP. D is 15 mm in front of the VP & is above the HP. Draw the FV of CD & find its inclinations with the HP and the VP. Show also its traces. Given, To draw, TL = 75 mm, FV =50 mm, C is 15 mm ↓ HP & 50 mm → VP, D is 15 mm → VP FV & to find θ & Ø VT’ d’ Hint 1: Cut an arc of 50 mm from c on locus of D to get the TV of the line Hint 2: Make TV (cd), // to XY so that FV will give TL d 1’ 75 h’ X To mark HT & VT Y 15 v 50 c’ θ=48º HT 50 d 2 d Locus of D 75 c Ø=28º d 1 Ans: θ=48º Ans: Ø=28º

Q 10. 10 A line PQ 100 mm long is inclined at 30º to

Q 10. 10 A line PQ 100 mm long is inclined at 30º to the H. P. and at 45º to the V. P. Its mid point is in the V. P. and 20 mm above the H. P. Draw its projections, if its end P is in the third quadrant and Q is in the first quadrant. Given, To draw, TL = 100, θ = 30º, Mid point M is 20 mm↑HP & in the VP End P in third quadrant & End Q in first quadrant FV & TV q’ p 2 X p 1’ p 50 m’ 30º 50 p’ 50 q 2’ 20 p 2’ q 1’ 45º m q 1 50 q q 2 Y

Problem 3: The front view of a 125 mm long line PQ measures 75

Problem 3: The front view of a 125 mm long line PQ measures 75 mm while its top view measures 100 mm. Its end Q and the mid point M are in the first quadrant. M being 20 mm from both the planes. Draw the projections of line PQ.

For Tv A Line inclined to both Hp and Vp 5. b’ . V.

For Tv A Line inclined to both Hp and Vp 5. b’ . V. P Y For Fv A Y For F v A X T. V. a’ On removal of object i. e. Line AB Fv as a image on Vp. Tv as a image on Hp, a B . F. V . B a’ X . V. P (Pictorial presentation) b’ b a T. V. b V. P. B b’ FV a’ A X Y Orthographic Projections Fv is seen on Vp clearly. To see Tv clearly, HP is rotated 900 downwards, a Hence it comes below xy. Note These Facts: Both Fv & Tv are inclined to xy. (No view is parallel to xy) Both Fv & Tv are reduced lengths. (No view shows True Length) TV H. P. b