Projection Of Lines Problems With Solution || Line Incline To Both HP and VP

Problem:1- The front view (FV) and top view (TV) of a line measures 55mm and 65mm respectively. Their end projectors are 40mm apart and one end of the line is 10mm above HP and in the VP. Draw the projections and find true length and true inclinations of the line.

Solution:

Given:

 1- length of the front view (FV)--55mm. 

 2- length of the Top View (TV)--65mm.

 3- distance between the projectors--40mm.

 4- one end of the line is 10mm above HP--         front view.

 5- and in the VP (top view).            

Steps:

1- from the given question we get five information means we can proceed to the further steps to draw the projection. 

2- draw XY line and mention HP and VP. Draw two projectors at a distance of 40mm from each other on the XY line on which both the ends of the line is going to be lie. 

3- let us name the given line as AB.

4- take a point a'(front view) 10mm above XY line on the projector as the line is 10mm above HP as shown in the figure.

5- take another point a(top view) on the XY line as one end of the line is on the VP.

6- as the ends of the front view and the top view lies on the same projector , hence from the point a' draw an arc of length 55mm (front view length is 55mm) which cuts the other projector at point b' as shown in the figure. Join the points a'b' (front view of the given line).

7- similarly from the point a draw an arc of length 65mm (top view length is 65mm) below the XY line which cuts the other projector at point b as shown in the figure. Join the points ab (top view of the given line).

8- now through the point a', b' and b draw locus and name the locus as locus a', locus b' and locus b respectively as shown in the figure below.

9-  project the point b' towards the locus b and this could be done by drawing an arc by taking radius a'b' which cuts the locus a' at point b1' as shown in the figure.

10- from the point b1' project it straight towards the locus b and cuts it at point b1. Join the points ab1 this is the true length of the line in top view as shown in the figure.

11- according to rule the true length in the top view and the true length in the front view are equal, therefore by taking a' as centre and the radius is equal to the length of the line ab1 [true length in top view] cut the locus b' at point b2' and join the points a'b2' and this is the true length of the line in front view.

12- mention the dimensions and measure the true length of the line and also the true inclination of the line with the XY line.

13- The true length of the line is 77mm. The true inclination in top view is 43degree and in front view the true inclination is 30degree.

Problem:2- A line AB is 70mm long is incline at an angle of 30degree to HP. it's end A is 10mm above HP and 15mm infront of VP. The front view of the line is  50mm. Draw the projection of the line AB.

Solution:

Given:

 1- length of the front view (FV)--50mm. 

 2- true length of the line--70mm.  

 3- true inclination of the line with HP is 30 degree.

 4- end A is 10mm above HP-front view(FV).

 5- and 15mm infront of VP (top view).     

Steps:  

1- from the given question we get five information means we can proceed to the further steps to draw the projection. 

2- draw XY line and mention HP and VP. Draw a projector on which the point A is going to be lie.

3- take a point a'(front view) on the projector and  10mm above XY line as the end A is 10 mm above HP.

4- take another point a(top view) on the same projector in which the point a' lies and 15mm below the XY line as the end A is 15mm infront of VP.

5- Now through the point a and a' draw  locus and name these locus as locus a and locus a' respectively.

6- from the point a' draw a line of length 70mm(true length) and  it incline to the locus a' at an angle of 30degree. Name the other end of the line as b1'. Through the point b1' draw a locus and name the locus as locus b1'.

7- now from the point a' draw draw an arc of length 50mm towards the locus b1' and it cuts the locus at point b'. Now join a'b' and this line is the front view of the given line.

8- take another point a(top view) 15mm below the XY line as the line end A is 15mm infront of VP. Through this point draw an locus and name this locus as locus a.

9- we know that if we project the front view of the line in downward or below the XY line we will get the true length of the line in top view of the line.So project the point b' in downward direction for this take a'b' as radius and a' as centre draw an arc which cuts the locus a' at point b2'.

10- from the point b2' draw a vertical line in downward direction.

11- as we know the true length of the line in top view and in the front view remains the same hence from the point a  cut the projected line of point b' by taking 70mm long line and name this cutting point as point b2. Now join the point ab2 and this will be the true length of the line in top view.

12- now through the point b2 draw a locus and name it as locus b2.

13- we also know that the end points of the front view and top view lines are lies on the same projector hence from the point b' draw a vertical line which cuts the locus b2 at point b. Join the points ab and this ab line is the top view of the line AB.

14- mention all the dimensions.

Problem:3- A line PQ 80mm length has it's end P 30mm above HP and 25mm infront of VP. Line makes an angle of 30degree to HP and 45degree to VP. Draw the projection of the line.

Solution:

Given:

 1- true length of the line--80mm(TL).  

 2- true inclination of the line with HP is 30 degree(theta).

 3- true inclination of the line with VP is 45 degree(phi)

 4- end P is 30mm above HP-front view(FV).

 5- and 25mm infront of VP (top view).     

Steps: 

1- from the given question we get five information means we can proceed to the further steps to draw the projection. 

2- draw XY line and mention HP and VP. Draw a projector on which the point P is going to be lie.

3- take a point p'(front view) on the projector and 30mm above the XY line as the end P of the line is 30mm above HP. Through the point p' draw a locus and name the locus as locus p'.

4- take another point p(top view) on the same projector on which the point p' lies and 25mm below the XY line as the point P is 25mm infront of VP. Through the point p draw a locus and name the locus as locus p.

5- from the point p' draw a line of length 80mm (TL) and it's incline to the locus p' at an angle of 30degree ( theta) and name the other end of the line as q1'. Now from the point q1' draw a locus and name the locus as locus q1' . As we know if the line is incline to HP means we have to show the inclination in VP.

6- from the point p draw a line of length 80mm (TL) and it's incline to the locus p at an angle of 45degree ( beta) and name the other end of the line as q2. Now from the point q2 draw a locus and name the locus as locus q2. As we know if the line is incline to VP means we have to show the inclination in HP.

7- we know that if we project the end point of the true length in front view in downward direction i.e, towards HP we will get the apparent length of the line in top view.

8- project the point q1'  towards downward as shown in the figure which cuts the locus q2 at point q. Join pq this is the apparent length of the line in top view.

9- similarly project the point q2 in upward direction as shown in the figure which cuts the locus q1' at point q'. Join the points p'q' this is the apparent length of the line in front view.

10- mension all the dimension.

Problem:4- Top view(TV) of 80mm long line AB measures 55mm while the length of its front view(FV) is 70mm. It's one  end is 15mm above the HP and 15mm infront of VP. Draw it's projection and determine it's inclination with HP nd VP.

Solution:

Given:

 1- true length of the line--80mm(TL).  

 2- apparent length of the line in top view           --55mm

 3- apparent length of the line in front view       -- 70mm

 4- end A is 15mm above HP-front view(FV).

 5- and 15mm infront of VP (top view).     

Steps: 

1- from the given question we get five information means we can proceed to the further steps to draw the projection. 

2- draw XY line and mention HP and VP. Draw a projector on which the point A is going to be lie.

3- take a point a'(front view) on the projector and 15mm above the XY line as the end A is 15mm above HP. Through this point draw a locus and name the locus as locus a'.

4- take another point a(top view) on the same projector on which the point a' lies and 15mm below the XY line as the end A is 15mm infront of VP. Through this point draw a locus and name the locus as locus a.

5- we know that if we project the apparent length of the line of front view towards HP then we get the true length of the line in top view. So draw an arc from the point a' of arc radius 70mm as the apparent length of the line in front view is 70mm and project it downward towards HP.

6- now taking true length of the line i.e, 80mm from the point a cut the projected line and name the cutting point as b2 as shown in the figure. Join the point ab2 and this is the true length of the line in top view. And through the point b2 draw a locus and name the locus as locus b2.

7- now from the point a draw an arc of radius 55mm(Top view Length) which cuts the locus b2 at point b. Join ab this is the apparent length of the line in top view.

8- we know that the end points of the line in top and front view lies on the same projector. Therefore from the point b draw a perpendicular line which cuts the 70mm arc at point b'. Join the points a'b' and this is the apparent length of the line in front view. Through the point b' draw a locus and name the locus as locus b'.

9- again from the point a cut the locus b' by taking an arc of radius 80mm (TL) and name the cutting point as b1'. Join the point a'b1' and this is the true length of the line in front view.

10- mension all the dimensions and measure the inclination of the true and apparent length of the line in front and top view.

Problem:5- The end A of line AB is 10mm above HP and 15mm infront of VP while the other end B is 50mm above HP and 65mm infront of VP. The distance between the end projector of a line is 60mm. Draw the projection of line, determine it's true length and inclination of line with HP and VP.

Solution:

Given:

 1- end A is 10mm above HP (FV).

 2- end A is 15mm infront of VP (TV).

 3- end B is 50mm above HP (FV).

 4- end B is 65mm infront of VP (TV).

 5- distance between the end projectors-              -60mm.   

Steps: 

1- from the given question we get five information means we can proceed to the further steps to draw the projection. 

2- draw XY line and mention HP and VP. Draw  two projectors at a distance of 65mm from each other on which the end points of the line AB is going to be lie.

3- take a pont a' (FV) on the projector and 10mm above the XY line as the end A is 10mm above HP. Through the point a' draw a locus and name the locus as locus a'.

4- take another point a (TV) on the same projector on which the point a' lies and 15mm below the XY line as the end A is 15mm infront of VP. Through the point a draw a locus and name the locus as locus a.

5- on the other projector take a point b'(FV) 50mm above XY line as the end B is 50mm above HP. Through the point b' draw a locus and name the locus as locus b'.

6- take another point b (TV) on the same projector on which the point b' lies and 65mm below the XY line as the end B is 65mm infront of VP. Through the point b draw a locus and name the locus as locus b.

7- now join a'b'(front view) and ab (top view).

8- we know that if we project the front view of a line in vertically downward direction then we will get true length of the line in top view. So project the point b' (as shown in the figure) in downward direction which cuts the locus b at point b2.

9- join the point ab2 and this line will be the true length of the line in top view.

10- similarly if we project the top view of the line in upward direction then we will get the true length of the line in front view. So project the point b (as shown in the figure) in upward direction which cuts the locus b' at point b1' .

11- join the point a'b1' and this line will be the true length of the line in front view.

12- mension all the dimensions and measure the true length and inclination of the line with HP and VP.

Problem:6- The top view PQ of a straight line is 70mm and makes an angle of 60degree with XY line. The end Q is 10mm infront of VP and 30mm above the HP. The difference  between the distances P and Q above the HP is 45mm. Draw the projection and determine the true length and true inclination with HP and VP.

Solution:

Given:

 1- apparent length of the line in top view-         i.e, p'q'-70mm

 2- inclination made by apparent length of         the line in top view--60degree.

 3- end Q is 10mm infront of VP (TV).

 4- end Q is 30mm above HP (FV).

 5- difference between the distances P and         Q above the HP --45mm.   

Steps: 

1- from the given question we get five information means we can proceed to the further steps to draw the projection. 

2- draw XY line and mention HP and VP. Draw a projector on which the point Q is going to be lie. 

3- take a pont q' (FV) on the projector and 30mm above the XY line as the end Q is 30mm above HP. Through the point q' draw a locus and name the locus as locus q'.

4- take another point q (TV) on the same projector on which the point q' lies and 10 mm below the XY line as the end Q is 10mm infront of VP. Through the point q draw a locus and name the locus as locus q.

5- from the point q draw a line of length 70mm and it incline to the locus q at an angle of 60degree. And name the other end of the line as p and through the point p draw a locus and name the locus as locus p.

6- the difference between the distances P and Q is 45mm hence draw another locus above the locus q' such that the distance between the two locus is 45mm and name the locus as locus p'.

7- as we know the end point of the TV and FV of the line lies on the same projector therefore project the point p towards upward direction which cuts the locus p' at point p'. Join the points q'p', this is the front view of the line.

8- we know that if we project the front view of a line in vertically downward direction then we will get true length of the line in top view. So project the point p' (as shown in the figure) in downward direction which cuts the locus p at point p1. 

9- join the point qp1 and this line will be the true length of the line in top view.

10- similarly if we project the top view of the line in upward direction then we will get the true length of the line in front view. So project the point p (as shown in the figure) in upward direction which cuts the locus p' at point p2' .

11- join the point q'p2' and this line will be the true length of the line in front view.

12- mension all the dimensions and measure the true length and inclination of the line with HP and VP.