Projection of Straight lines | projection of lines in engineering drawing | rules for projection of lines.

     PROJECTION OF STRAIGHT LINES

Line: A line or a straight line is defined as the the shortest distance between the two points. The projection of a straight line can be drawn by joining the projection of the end points of that line. In the projection of straight line always the front view (FV) of the straight line is shown above the XY line and the top view (TV) of the straight line i is always shown below the XY line.  

>The possible projection of straight line with respect to HP and VP in the first quandrant are:

1) line is perpendicular to one plane and parallel to other.

2)line is parallel to both the planes.

3)line is parallel to one plane and inclined to the other.

4) the line is inclined to both the planes.

Let us understand these concepts with the help of some problems

1- line is perpendicular to one plane and parallel to other:

In this category we have two possible position of the straight line i.e., 1-line is perpendicular to VP and parallel to HP and 2-line is parallel to VP and perpendicular to HP.

1) line perpendicular to VP and parallel to HP.

Problem:  A line AB 25mm long is perpendicular to VP and parallel to HP.Its end A is 20mm infront of VP and  the line is 40mm above HP. Draw it's projection.

Solution:

Steps:

1-draw XY line. Draw another line which is perpendicular to the XY line this perpendicular line will be the Projector.

2-According to the problem we can imagine that if we see the line from the top view we will get the true length of the line AB as the line is perpendicular to VP. The TV will be below XY line.

3- if we see the line from the front view we will only able to see a point where both the end points A and B of the line are coincide with each other. And the FV will be above the XY line.

4-take a point a'b'(front view of the line) on the projector line and 40mm above the XY (reference line) line as the line is 40mm above HP.

5- take a point a on the same projector line and 20mm below the XY line as the end A of line AB is 20mm infront of VP.

6- take another point b below the XY line and on the same projector line and 25mm away from the point a as the length of the line AB is 25mm.

7- mark the dimensions and the final projection is shown in the figure.

2) line perpendicular to HP and parallel to VP.

Problem: A line AB 25mm long is perpendicular to HP and parallel to VP.Its end A is 20mm above HP and the line is 40mm infront of VP. Draw it's projection.

Solution:

Steps:

1-draw XY line. Draw another line which is perpendicular to the XY line this perpendicular line will be the Projector.

2-According to the problem we can imagine that if we see the line from the front view (FV) we will get the true length of the line AB as the line is perpendicular to HP. The FV will be above the XY line.

3- if we see the line from the top view we will only able to see a point a'b' where both the end points A and B of the line are coincide with each other. And the top view will be below the XY line.

4-take a point a(b) (top view of the line) on the projector line and 40mm below the XY (reference line) line as the line is 40mm infront of VP.

5- take a point a'(front view) on the same projector line and 20mm above the XY line as the end A of the line AB is 20mm above HP.

6- take another point b' above the XY line and on the same projector line and 25mm away from the point a' as the length of the line AB is 25mm.

7- mark the dimensions and the final projection is shown in the figure.

2)line is parallel to  both the planes:

Problem: A line AB 25mm long is  parallel to both the planes HP and VP. The line  is 20mm above HP and 40mm infront of VP. Draw it's projection.

Solution:

Steps:

1-draw XY line. Draw two projectors on the XY line at a distance of 25mm from each other. On these two projectors the end points of the line AB is going to be lie.

2-According to the problem we can imagine that if we see the line from the top view (TV) we will get the true length of the line AB as the line is parallel to HP. The TV will be below XY line.

3- if we see the line from the front view (FV) then also we will get the true length of the line AB as the line is parallel to VP . And the FV will be above the XY line.

4- draw a line  a'b' (FV) of length 25mm  parallel to XY line by joining the two projectors and 20mm above the XY (reference line) line as the line AB is 20mm above HP.

5- draw another line ab (TV) of length 25mm and parallel to XY line by joining the two projectors and 40mm below the XY line as the line AB is 40mm infront of VP.

7- mark the dimensions and the final projection is shown in the figure.

3)line is parallel to one plane and inclined to the other:

In this category we have two possible position of the straight line i.e., 1-line is parallel to VP and inclined to HP and 2-line is parallel to HP and inclined to VP.

1)line is parallel to VP and inclined to HP.

Problem: A line AB 40mm long is parallel to VP and inclined at an angle of 30degree to HP. The end A is 20mm above HP and 25mm infront of VP. Draw it's projection.

Solution:

Steps:

1-draw XY line and mark HP and VP.

2-According to the problem we can imagine that if we see the line from the top view (TV) we will get the apparent length of the straight line AB as the line is inclined to HP. The TV will be below XY line and parallel to it.

3- if we see the line from the front view (FV) then  we will get the true length of the line AB as the line is parallel to VP . And the FV will be above the XY line and inclined to it at an angle of 30degree.

4- take a point a'  20mm above XY line as the end A of line AB is 20mm above HP.

5- through the point a' draw a locus parallel to XY line. From the point a' draw a line of length 40mm and inclined to the locus at an angle of 30 and name the other end of the line as b'.

6- draw two projectors through the point a' and b'.

7- take a point a 25mm below the XY line as the end A of line AB is 25mm infront of VP and on the projector of point a'. Similarly take another point b 25mm below the XY line and on the projector of point b'.

8- join the points a and b.This ab line is the top view of the line AB. Mark all the dimensions. 

2)line is parallel to HP and inclined to VP.

Problem: A line AB 40mm long is parallel to HP and inclined at an angle of 30degree to VP. The end A is 20mm above HP and 25mm infront of VP. Draw it's projection.

Solution:

Steps:

1-draw XY line and mark HP and VP.

2-According to the problem we can imagine that if we see the line from the top view (TV) we will get the true length of the straight line AB as the line is parallel to HP. The TV will be below XY line and inclined to it at an angle of 30degree.

3- if we see the line from the front view (FV) then we will get the apparent length of the line AB as the line is incline to VP . And the FV will be above the XY line and parallel to it.

4- take a point a 25mm below the XY line as the end A of line AB is 25mm infront of VP.

5- through the point a draw a locus parallel to XY line. From the point a draw a line of length 40mm and inclined to the locus at an angle of 30 and name the other end of the line as b.

6- draw two projectors through the point a and b.

7- take a point a' 20mm above the XY line and on the projector of point a. Similarly take another point b' 20mm above the XY line and on the projector of point b.

8- join the points a' and b'.This a'b' line is the front view of the line AB. Mark all the dimensions. 

4) the line is inclined to both the planes.

Problem: A line AB is 80mm long has it's end A 20mm above HP and 30mm infront of VP. It is inclined at 30degree to HP and 45degree to VP. Draw the projection of the line find apparent length and show the  apparent inclinations.

Solution:

Steps:

1- draw XY line and mention HP and VP. Draw a projector on the XY line perpendicular to it.

2- take a point a'(front view) 20mm above XY line as the end A of line AB is 20mm above HP.

3- take another point a (top view) 30mm below the XY line as the end A of line AB is 30mm infront of VP.

4- draw locus through the point a'(front view) and a(top view).

5- from the point a' draw a line of length 80mm(true length of the line) and inclined at an angle of 30degree to the locus a'. And mark the other end of the line as b1'.

6- from the end b1' draw a locus and name it as locus b1'.

NOTE - If the line is inclined to HP then the inclination will be shown in VP. And if the line is inclined to VP then the inclination will be shown in HP.

7- project the point b1' towards the locus a and the projector cuts the locus a at b1.

8- now from the point a draw a line of length 80mm(true length of the line) and inclined at an angle of 45degree to the locus a. And mark the other end of the line as b2.

9- from the end b2 draw a locus and name it as locus b2.

10- project the point b2 towards the locus a' and the projector cuts the locus a' at b2'.

11- now take a' as centre and a'b2' as radius draw an arc towards the locus b1' which cuts this locus at point b'. 

12- Join a' b' this line is the apparent length of the line AB from the front view. Darken this line as compared to the line a'b1'.

13- the inclination made by the line a'b' with respect to the XY line is called the apparent inclination of the line in front view and it is denoted by the Greek letter alpha.

14- similarly take a as centre and ab1 as radius draw an arc towards the locus b2 which cuts this locus at point b. 

15- Join ab this line is the apparent length of the line AB from the top  view. Darken this line as compared to the line ab2.

16- the inclination made by the line ab with respect to the XY line is called the apparent inclination of the line in top view and it is denoted by the Greek letter beta.

17- mention all the dimensions and measure the length ab and a'b' i.e, the lenth of ab is 70mm and the length of a'b' is 57mm.

SOME IMPORTANT RULES:

1- the inclination made by the apparent length of the line in front view is denoted by the Greek letter Alpha.

2- the inclination made by the apparent length of the line in top view is denoted by the Greek letter beta.

3- all the points above the XY line are denoted with a dashed line.

4- all the points below the XY line are denoted without any dashed line.

5- the points in the front view and  the points in the top view  are lies on the same and respective vertical lines or the projectors of that points.

6- by using i.e, by rotating the front view of the line vertically downward we will get the true length of the line in the top view OR by using i.e, by rotating the true length of the line in the top view in vertically upward direction we will get the front view of the line.

7-by using i.e, by rotating the top view of the line vertically upward we will get the true length of the line in the front view OR by using i.e, by rotating the true length of the line in front view vertically downward we will get the top view of the line.

8- the inclination of true length in front view is denoted by the letter theta. And the inclination of the true length in the top view is denoted by the letter phi.

9-If the line is inclined to HP then the inclination will be shown in VP. And if the line is inclined to VP then the inclination will be shown in HP. 

10- the true length of the line in front view and the true length of the line in top view are equal.

11- during solving problems of  straight line, in maximum number of problems we will get atleast 5 information about the line.

WORKING PROCEDURE:

1- First we have to write all the information about the line i.e, the TV length,FV length, position of end point,angle made by the TV line or front view line etc. 

2- then draw the XY line and mention HP and VP.

3- take the points as per the given position of the point i.e, the position of point in the FV or the position of point in the TV.

4- draw the line as per the given condition i.e, the length of the line and the inclination of the line with HP and VP.

5- To Draw The TV of The Line:

# project the true length of the line in  front view in downward direction which will give the TV of the line in HP.

6- To Draw The FV Of The Line:

# project the true length of the line in top view in upward direction which will give the FV of the line in VP.

7- To Draw The True Length Of The Line In FV:

# project the TV of the line in upward direction which will give the true length of the line in front view.

8- To Draw The True Length Of The Line In TV:

# project the FV of the line in downward direction which will give the true length of the line in top view.