Engineering Drawing Unit – III III Projections of Straight Lines (Inclined to both the planes) 3.0 Line inclined to both HP and VP 3.1 Projections of a line inclined to HP and VP
When a line is inclined to both HP and VP, the projections are shorter than the true length and the inclination angles with XY are greater than the true inclinations of the line. The line makes an angle of θ with the HP and Φ with the VP. Point P is above HP and in front of the VP. Stage 1:
Stage 2:
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Above two stages are integrated as in fig.
1. A line CD, 80 mm long, is inclined at 30 0 to the HP and at 45 0 to the VP. It’s one end C 20 mm above the HP and 30 mm in front of the VP. Draw the projections of CD.
Draw the reference line XY.
Mark c' in the front view at a distance of 20 mm above XY and c in the top view at a distance of 30 mm below XY.
Draw the loci of c' and c in the front view and in the top view.
Draw c'd1' equal to 80 mm at an angle of 30 0 to the XY in the front view and project it to the top view cd 1. 2
Draw cd2 equal to 80 mm at an angle of 45 0 to the XY in the top view and project it to the front view cd2'.
Draw the loci of d' and d in the front view and in the top view.
With c as centre and radius equal to cd1, draw an arc cutting locus of d in d. Join c and d is the required final top view of the line CD.
With c' as centre and radius equal to cd 2', draw an arc cutting locus of d' in d'. Join c' and d' is the required final front view of the line CD.
Check that d and d' lie on the same projector.
2. A line AB, 65 mm long, has its end A 20 mm above the HP and 25 mm in front of the VP. The end B is 40 mm above the HP and 65 mm in front of the VP. Draw the projections of the AB and show its inclinations with the HP and the VP.
Mark a' 20 mm above XY and draw a line cd (locus of a') passing through a'.
Mark a 25 mm below XY and draw a line gh (locus of a) passing through a.
Draw a line ef, 40 mm above XY is the locus of b'.
Draw a line jk 65 mm below XY is the locus of b.
With a' as centre and radius equal to 65 mm draw an arc cutting ef in b'. Join a' with b'. ɵ is the inclination of a'b' with XY, is the inclination of AB with the HP. 3
Project b' to b on gh.
With a as centre and radius equal to ab draw an arc cutting jk in b 2. Join b 2 and a, is the required final top view of the line AB.
With a as centre and radius equal to 65 mm draw an arc cutting jk in b 1. Join a with b 1. Ø is the inclination of ab 1 with XY, is the inclination of AB with the VP.
Project b1 to b1' on cd.
With a' as centre and radius equal to a'b1' draw an arc cutting ef in b 2'. Join b 2' and a', is the required final front view of the line AB.
3.2 To find out true length of a line and its inclinations with reference planes, when projections of a line are given.
When projections of a line are given, its true length and inclinations with the planes are determined by the application of the following rule: When a line is parallel to a plane, its projection on that plane will show its true
length and the true inclination with the other plane. The line may be made parallel to a plane, and its true length obtained by any one of the following two methods:
Rotating line method
Trapezoidal plane method
' Example: The top view ab and the front view a b' of a line AB are given. To determine
its true length and the inclinations with the HP and VP
Rotating line method :
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Figure shows the front view a'b' and top view ab of the line AB.
In the top view, move b along an arc drawn with a as centre and radius equal to ab to b 1, so that ab1 is parallel to XY.
Draw a projector through b 1 to cut pq at b 1'.
Draw the line a'b1' which is the true length of AB.
The angle ɵ, which it makes with XY is the inclination of AB with the HP.
Similarly, with a' as centre and radius equal to a'b', draw an arc to cut cd at b2'.
Draw projector through b 2' to cut rs at b 2.
Draw the line ab2, which is the true length of AB. The angle Ø which it makes with XY is the inclination of AB with the VP.
Trapezoidal method :
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Figure shows the front view a'b' and top view ab of the line AB
Draw perpendiculars to a'b', viz. a'A 1 (equal to ao1) and b'B1 (equal to bo 2) and then joining A1 with B1. The line A 1B1 is the true length of AB and its inclination Ø with a'b' is the inclination of AB with the VP.
Similarly, draw perpendiculars to ab', viz. aA 2 (equal to a'o1) and bB2 (equal to b'o2) and then joining A 2 with B2. The line A 2B2 is the true length of AB and its inclination ɵ with ab is the inclination of AB with the HP.
3. The top view of a 75 mm long line AB measures 65 mm, while the length of its front view is 50 mm. It’s one end A is in the HP and 12 mm in front of the VP. Draw the projections of AB and determine its inclinations with the HP and the VP.
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4. A line AB, 90 mm long, is inclined at 45 0 to the HP and its top view makes an angle of 600 with the VP. The end A is in the HP and 12 mm in front of the VP. Draw its front view and find its true inclination with the VP.
5. A line AB is 75 mm long. A is 50 mm in front of VP and 15 mm above HP. B is 15 mm in front of VP and is above HP. Top view of AB is 50 mm long. Draw and measure the front view.
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6. The mid- point of a straight line AB is 60 mm above HP and 50 mm in front of VP. The line measures 80 mm long and inclined at an angle of 30 0 to the HP and 45 0 to VP. Draw its projections.
7. A line PQ 100 mm long is inclined at 30 0 to the HP and 45 0 to the VP. Its mid point is in the VP and 20 mm above HP. Draw its projections, if it s end P is in the third quadrant and Q in the first quadrant.
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8. Draw the projections of a straight line AB of 100 mm long when one of its ends is touching the VP and the other end touching HP. The angles of inclination with HP and VP are 400 and 500 respectively.
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3.3 Methods of determining Traces of a line
Figure shows c'd' and cd are the projections of the line CD inclined to both the planes. Its traces may be determined by any of the following two methods.
Method I:
Produce the front view c'd' to meet XY at a point h.
Through h, draw a projector to meet the top view cd produced at the H.T. of the line.
Similarly produce the top view cd to meet XY at appoint v.
Through v, draw a projector to meet the front view c'd' produced at the V.T. of the line.
Method II:
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Positions of Traces of a Line:
A line AB, inclined at 40 0 to t 9. A line AB inclined at 40 0 to the VP has its ends 50 mm and 20 mm above the HP. The length of its front view is 65 mm and its VT is 10 mm above the HP. Determine the true length of AB, its inclination with the HP and its HT
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Questions from Previous Papers :
1. A line AB, 60 mm long, has its end A in both the H.P and the V.P. It is inclined at 45 degrees to the H.P and 30 degrees to the V.P. Draw the projections of the s traight line. 2. A line PQ 75 mm long has its end P in the VP and end Q in the HP. The line is inclined at 300 to HP and 60 0 to VP. Draw its projections and traces. 3. A line AB 70 mm long has its end A 10 mm above H.P and 15 mm in front of V.P. Its front view and top view measures 50 mm and 60 mm respectively. Draw the projections of the line, determine its inclinations with H.P and V.P and also locate its traces. 4. The front view of a line AB measuring 125 mm long is 75 mm and its top view is 100 mm long. Its end B is 30 mm from both the planes. Draw the projections and find its inclinations and with the reference planes. 5. The front view of a line AB measures 70 mm and makes an angle of 45 degrees with xy. A is in the H.P. and the V.T. of the line is 15 mm below the H.P. The line is inclined at 30 degrees to the V.P. Draw the projections of AB and find its true length, inclination with the H.P. and its H.T. 6. A line AB, 70 mm long, has its end A in both the H.P. and the V.P. It is inclined at 30 degrees to the H.P. and at 60 degrees to the V.P. Draw the projecti ons of the straight line. 7. The straight line AB is inclined at 30 degrees to H.P., while its top view at 45 degrees to line xy. The end A is 20 mm in front of the V.P. and it is below the H.P. The end B is 75 mm behind the V.P. and it is above the H.P. Draw the projections of the line when its V.T. is 40 mm below. Find the true length of the portion of the straight line which is in the second quadrant and locate its H.T. 8. The end P of a line PQ is 25 mm behind the V.P. and is below the H.P. The end Q is 12 mm in front of the V.P. and is above the H.P. The distance between the projectors is 65 mm. The line is inclined at 40 degrees to the H.P. and its H.T. is 20 mm behind the V.P. Draw the projections of the line and determine its true length and the V.T. 9. The top view and the front view of the line GH, measures 53 mm and 65 mm respectively. The line is 75 mm long. Point G is on the ground and 40 mm behind VP. Draw the projections of the line GH and determine its inclinations with HP and VP. 10. The front view of the line AB, inclined at 30 0 to the VP measures 65 mm. Two ends of the line A and B are 50 mm and 20 mm below HP respectively. The VT of the line is 5 mm below HP. Draw the projections and determine the true length of AB, its inclinations with HP and its HT. Assume the line AB in 3rd quadrant. 12
11. The ends of a line PQ are on the same projector. The end P is 30 mm below the HP and 15 mm behind the VP. The end Q is 55 mm above the HP and 45 mm in front of the VP. Determine the true length and traces of PQ and its inclinations with the two planes. 12. Top view of a 75 mm long line CD measures 50mm. End C is in HP and 50mm in front of VP. End D is 15mm in front of VP and it is above HP. Draw projections of CD and find angles of inclination with HP and VP. 13. A line CD is 75mm long. Its front and top views measure 50mm & 60mm respectively. End A is 20mm above HP and 20mm in front of VP. Draw projections of line CD. Find angles of inclination of the line with HP and VP . 14. Line AB is 75mm long and it is inclined at 30 0 & 400 to HP and VP respectively. End A is 12mm above HP and 10mm in front of VP. Draw the projections and find their apparent inclinations to HP and VP
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