Projection of lines
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The document outlines the graphical techniques for projecting straight lines using orthographic projections with respect to horizontal (hp) and vertical planes (vp). It details various cases such as lines that are vertical, parallel, or inclined to hp and vp, along with the procedures to determine true length and inclinations. Several problem-solving examples are included to illustrate the drawing of projections based on given parameters and to find angles with hp and vp.
Projection of lines
- 1. 12/18/13 Hareesha N G, DSCE, Bengaluru 1
- 2. 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.
- 3. 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 Fo r B Orthographic Pattern V.P. a’ Fv b’ X Y Fv TV a b Tv a b X (Pictorial Presentation) 2. . V.P A Line // to Hp & // to Vp . F.V H.P. Orthographic Pattern 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 rF v b X a Y a V. T. H.P. Tv b
- 4. .P . V 3. b’ F. V a’ V.P. b’ . F.V B . A Line inclined to Hp and parallel to Vp Fv inclined to xy Tv parallel to xy. θ Y X θ (Pictorial presentation) A . T.V X θ a’ Y a b T.V. b a H.P. Orthographic Projections 4. A Line inclined to Vp and parallel to Hp .P. V a’ Tv inclined to xy Fv parallel to xy. a’ . b’ F.V A V.P. Ø B (Pictorial presentation) Ø T.V. b’ X Y a a Fv Ø Tv b H.P. b
- 5. For Tv For Tv 5. b’ α Y a’ For Fv A X a On removal of object i.e. Line AB Fv as a image on Vp. Tv as a image on Hp, β Y For Fv a’ A X T.V. B F.V . B α b’ . V.P (Pictorial presentation) F.V . . V.P A Line inclined to both Hp and Vp b a β T.V. b V.P. b’ FV a’ α X Y Orthographic Projections Fv is seen on Vp clearly. To see Tv clearly, HP is rotated 900 downwards, a 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) β TV Hence it comes below xy. H.P. b
- 6. 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 α&β V.P. V.P. V.P. b’ b’ FV a’ α a’ Y β Fv TL a’ θ X Y a TV H.P. θ β TV b2 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. α In this sketch, TV is rotated and made // to XY line. Hence it’s corresponding FV a’ b1’ Is showing True Length & True Inclination with Hp. 1’ Y a 1 Ø β Tv b TL X TV b b’ b1’ b 1’ FV X a Note the procedure When True Length is known, How to locate Fv & Tv. (Component a-1 of TL is drawn which is further rotated to determine Fv) H.P. TL b b1 Here a -1 is component of TL ab1 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’ b1‘) Tv can be drawn.
- 7. 1) The most important diagram showing graphical relations True Length ( TL) – a’ b1’ & a b among all important parameters of this topic. 2) Angle of TL with Hp Study and memorize it as a CIRCUIT DIAGRAM 3) Angle of TL with Vp – Ø And use in solving various problems. 4) Angle of FV with xy – θ α V.P. β 5) Angle of TV with xy – Distance between End Projectors. 6) LTV (length of FV) – Component (a-1) b 1’ b’ Important TEN parameters to be remembered with Notations used here onward 7) LFV (length of TV) – Component (a’-1’) 8) Position of A- Distances of a & a’ from xy Fv a’ 9) Position of B- Distances of b & b’ from xy TL 10) Distance between End Projectors α θ 1’ LTV X Y LFV a Ø θ& α Ø&β 1 NOTE this Construct with a’ Construct with a b’ & b1’ on same locus. β b & b1 on same locus. Tv H.P. TL b b1 Also Remember True Length is never rotated. It’s horizontal component is drawn & it is further rotated to locate view. Views are always rotated, made horizontal & further extended to locate TL, θ & Ø
- 8. GROUP (A) PROBLEM 1) GENERAL CASES OF THE LINE INCLINED TO BOTH HP & VP ( based on 10 parameters). Line AB is 75 mm long and it is 300 & 400 Inclined to Hp & Vp respectively. End A is 12mm above Hp and 10 mm in front of Vp. Draw projections. Line is in 1st quadrant. b’ FV SOLUTION STEPS: 1) Draw xy line and one projector. 2) Locate a’ 12mm above xy line & a 10mm below xy line. 3) Take 300 angle from a’ & 400 from a and mark TL I.e. 75mm on both lines. Name those points b1’ and b1 respectively. 4) Join both points with a’ and a resp. 5) Draw horizontal lines (Locus) from both points. 6) Draw horizontal component of TL a b1 from point b1 and name it 1. ( the length a-1 gives length of Fv as we have seen already.) 7) Extend it up to locus of a’ and rotating a’ as center locate b’ as shown. Join a’ b’ as Fv. 8) From b’ drop a projector down ward & get point b. Join a & b I.e. Tv. b’1 TL θ a’ X Y a LFV Ø TV 1 TL b b1
- 9. PROBLEM 2: Line AB 75mm long makes 450 inclination with Vp while it’s Fv makes 550. End A is 10 mm above Hp and 15 mm in front of Vp.If line is in 1st quadrant draw it’s projections and find it’s inclination with Hp. b’ b’1 LOCUS OF TL 550 a’ y X 45 0 LFV 1 Φ a TL TV 1.Draw x-y line. 2.Draw one projector for a’ & a 3.Locate a’ 10mm above x-y & Tv a 15 mm below xy. 4.Draw a line 450 inclined to xy from point a and cut TL 75 mm on it and name that point b1 Draw locus from point b1 5.Take 550 angle from a’ for Fv above xy line. 6.Draw a vertical line from b1 up to locus of a and name it 1. It is horizontal component of TL & is LFV. 7.Continue it to locus of a’ and rotate upward up to the line of Fv and name it b’.This a’ b’ line is Fv. 8. Drop a projector from b’ on locus from point b1 and name intersecting point b. Line a b is Tv of line AB. 9.Draw locus from b’ and from a’ with TL distance cut point b1‘ 10.Join a’ b1’ as TL and measure it’s angle at a’. It will be true angle of line with HP. FV Solution Steps:- LOCUS OF b b b1
- 10. 1.Draw xy line and one projector. 2.Locate a’ 10 mm above xy and a 15 mm below xy line. 3.Draw locus from these points. 4.Draw Fv 500 to xy from a’ and mark b’ Cutting 55mm on it. 5.Similarly draw Tv 600 to xy from a & drawing projector from b’ Locate point b and join a b. 6.Then rotating views as shown, locate True Lengths ab1 & a’b1’ and their angles with Hp and Vp. b’ FV PROBLEM 3: Fv of line AB is 500 inclined to xy and measures 55 mm long while it’s Tv is 600 inclined to xy line. If end A is 10 mm above Hp and 15 mm in front of Vp, draw it’s projections,find TL, inclinations of line with Hp & Vp. SOLUTION STEPS: X a’ θ b’1 TL 500 y a Φ 600 TL b b1
- 11. a’ θ LTV b’1 TL 1’ X Y a LFV Φ 1 TL TV SOLUTION STEPS: 1.Draw xy line and one projector. 2.Locate a’ 10 mm above xy and a 15 mm below xy line. 3.Draw locus from these points. 4.Cut 60mm distance on locus of a’ & mark 1’ on it as it is LTV. 5.Similarly Similarly cut 50mm on locus of a and mark point 1 as it is LFV. 6.From 1’ draw a vertical line upward and from a’ taking TL ( 75mm ) in compass, mark b’1 point on it. Join a’ b’1 points. 7. Draw locus from b’1 8. With same steps below get b1 point and draw also locus from it. 9. Now rotating one of the components I.e. a-1 locate b’ and join a’ with it to get Fv. 10. Locate tv similarly and measure & Φ Angles θ FV PROBLEM 4 :Line AB is 75 mm long .It’s Fv and Tv measure 50 mm & 60 mm long respectively. End A is 10 mm above Hp and 15 mm in front of Vp. Draw projections of line AB if end B is in first quadrant.Find angle with Hp and Vp. b’ b b1
- 12. ROBLEM 5 :- T.V. of a 75 mm long Line CD, measures 50 mm. End C is in Hp and 50 mm in front of Vp. End D is 15 mm in front of Vp and it is above Hp. d’ Draw projections of CD and find angles with Hp and Vp. X c’ θ d1 d TL TV c LOCUS OF d’ & d’1 TL FV SOLUTION STEPS: 1.Draw xy line and one projector. 2.Locate c’ on xy and c 50mm below xy line. 3.Draw locus from these points. 4.Draw locus of d 15 mm below xy 5.Cut 50mm & 75 mm distances on locus of d from c and mark points d & d1 as these are Tv and line CD lengths resp.& join both with c. 6.From d1 draw a vertical line upward up to xy I.e. up to locus of c’ and draw an arc as shown. 7 Then draw one projector from d to meet this arc in d’ point & join c’ d’ 8. Draw locus of d’ and cut 75 mm on it from c’ as TL θ & Φ 9.Measure Angles d’1 Φ Y LOCUS OF d & d1