1. University of Warith AL-Anbiya’a
College of Engineering
Air-condition & Refrigeration Department
Desprictive geometry I
First stage
Asst.Lec. Aalaa Mohammed AL-Husseini

2. Straight line
In our study of the Straight line we use certain definitions and conventions as we need theories or results that have been studied within the subject of descriptive geometry
Line Definition: is a moving point path
Straight line: is a fixed-point path to a fixed direction
Straight line representation: We mean designation projection at the main projection levels
The slope of the straight line: is the angle between Straight line and its projection at that level

3. We will mark the angle that he made with the horizontal plane Z1 and the angle he made with the face plane Z2 and with the side (profile) plane of the projection Z3
The Straight line is in a general position: it is straight to any of the main projection levels
Horizontal Straight line : It is a straight parallel to the level of horizontal projection
Face Straight line : is a straight parallel to the level of face projection
Side Straight line : It is a straight parallel to the side projection level

4. Vertical straight: It is straight vertical on the level of horizontal projection
Theoretically, if the shape of a given plane is parallel with a given plane, the projection of that shape appears in real form
Result: If a straight line is parallel to a known level, its projection on that known level appears in real length (ie, the length of the projection equals the length of the straight line)

5. 1 - If the Straight line is horizontal, its horizontal projection appears in real length
 2 - If Straight line is face line , the appearance of the face shows real length
 3 - If the Straight line is side, its Straight line appearance appears in real length
The real shape denotes R_SH and the length of the real length is denoted by R -L

6. Represent of the straight line
If it is straight in the space, draw from a point a vertical beam at the main horizontal plane and cut it in any horizontal projection of point A and draw a beam from the point B vertically to the horizontal plane of the projection and cut it at the point B . Connect A to B so that we get the horizontal projection of the straight line. Also, we can Obtain straight line in a second way by moving the plane of the straight line so that it is vertically on the horizontal plane and cuts it in a straight line that is the horizontal projection of the straight line AB. It is clear that each point on AB is preceded by one point on AB because it is not possible to draw more than one column from a known point on a known level. If we draw the face projection of the two points of AB we get a AB and imagine that the plane has gone at a certain angle to get the three projection

7. Straight line positions:
General position as shown in figure below where it’s not parallel or vertical to any level of projection.
Example : Represent the straight line AB where A( 2,3,4) , B ( 1,4,2)

9. 2. Special cases:
a. Horizontal straight line parallel to the horizontal plane of projection:
Since the straight line parallel to the horizontal plane of projection, it’s horizontal projection will appear with R-L . The face projection of the horizontal straight line will be parallel to the earth line.
The angle made with the face plane of projection (Z2) is the same angle that made by the horizontal projection of the straight line with the earth line, and the angle made with the side plane of projection (Z3) is the same angle that made by the horizontal projection of the straight line with the vertical line between the horizontal and side planes of projection which is the same angle of the horizontal projection with the vertical folding after rotating the projection levels. so the summation of the slope angles will be
Z1=0
Z1+Z2+Z3=90 degree

10. Example : Represent the horizontal straight line AB where A( 3,4,2) , B (3,2,5) and find it’s angles with the projection planes
Sol:

11. b. Face straight line parallel to the face plane of projection:
Since the straight line parallel to the face plane of projection, it’s face projection will appear with R-L . The horizontal projection of the face straight line will be parallel to the earth line.
The angle made with the horizontal plane of projection (Z1) is the same angle that made by the face projection of the straight line with the earth line, and the angle made with the side plane of projection (Z3) is the same angle that made by the face projection of the straight line with the vertical folding line, so the summation of the slope angles will be
Z2=0
Z1+Z2+Z3=90 degree

12. Example : Represent the straight line AB where A( 2,3,5) , B (4,3,1) and find it’s angles with the projection planes
Sol:

13. c. Side straight line parallel to the side plane of projection:
Since the straight line parallel to the side plane of projection, it’s side projection will appear with R-L . The face projection of the side straight line will be parallel to the vertical folding line.
The angle made with the horizontal plane of projection (Z1) is the same angle that made by the side projection of the straight line with the earth line, and the angle made with the face plane of projection (Z2) is the same angle that made by the side projection of the straight line with the vertical folding line, so the summation of the slope angles will be
Z3=0
Z1+Z2+Z3=90 degree

14. Example : Represent the straight line AB where A( 2,6,6) , B (5,1,6) and find it’s angles with the projection planes
Sol:

15. d. Vertical straight line:
Vertical straight line is the line vertical to the horizontal plane of projection. This straight line will be parallel to each of the face and side plane of projection.
If we note the vertical straight line, we will conclude the following states:
The horizontal projection is point.
The face and side projection of the line will appear with the R-L .
Z1=90 and Z2=Z3=0.
Example : Represent the straight line AB where A( 1,3,4) , B (4,3,4) and find it’s angles with the projection planes
Sol:

17. e. Straight Line Vertical to the face plane of projection:
The projection of this straight line on the face plane of projection is point.
The horizontal and side projection of the line will appear with the R-L .
Z2=90 and Z1=Z3=0.
Example : Represent the straight line AB where A( 1,3,3) , B (1,0.5,3) and find it’s angles with the projection planes
Sol:

18. f. Straight Line Vertical to the side plane of projection:
The projection of this straight line on the side plane of projection is point.
This straight line parallel to the horizontal and face plane of projection, so the horizontal and face projection of the line will appear with the R-L .
Z3=90 and Z1=Z2=0.
Example : Represent the straight line AB where A( 2,4,3) , B (2,4,1) and find it’s angles with the projection planes
Sol:

20. g. Straight Line located within the horizontal plane of the projection:
When we study the representation of the point, we find that each point located at the horizontal plane of the projection have face projection applicable to the earth line. This fact can be derived from the theory that (if two levels are vertical to each other and drown one column from a point located on one of them vertical to the other, Cut it at a point on the intersection line).
In this study the two planes are the horizontal and side planes and the intersection line is the earth line so we concluded that:
If the Straight Line located within the horizontal plane of the projection, it’s face projection located on the earth line. By the same way we reached to the fact that the side projection located on the earth line
The horizontal projection will appear by R-L
Z1=0 , Z2+Z3=90

21. Example : Represent the straight line AB where A( 0,2, 6) , B (0,5,2) and find it’s angles with the projection planes
Sol:

22. h. Straight Line located within the face plane of the projection:
In this state the straight line coincide on it’s face projection, so the face projection will appear by R-L.
it’s horizontal projection located on the earth line and the side projection located on the vertical folding line .
Z2=0 , Z1+Z3=90
Example : Represent the straight line AB where A( 3,0, 2) , B (1,0,4) and find it’s angles with the projection planes
Sol:

23. I. Straight Line located within the side plane of the projection:
In this state the straight line coincide on it’s side projection, so the side projection will appear by R-L.
it’s face and horizontal projection located on the vertical folding line .
Z3=0 , Z1+Z2=90
Example : Represent the straight line AB where A( 2,4, 0) , B (5,1,0) and find it’s angles with the projection planes
J. Straight Line located on the intersection line of the horizontal and side plane:
In this state the horizontal and side projection of straight line will appear by R-L.
it’s face projection will appear as point in the origin point.
Z2=90 , Z1=Z3=0
Example : Represent the straight line AB where A( 0,4, 0) , B (0,1,0) and find it’s angles with the projection planes

24. K. Straight Line located on the earth line:
In this state the horizontal and face projection of straight line will located on the earth line and appear with R-L.
it’s side projection will appear as point in the origin point.
Z3=90 , Z1=Z2=0
Example : Represent the straight line AB where A( 0,0, 2) , B (0,0,6) and find it’s angles with the projection planes
L. Straight Line located on the vertical folding line:
In this state the face and side projection of straight line will located on the vertical folding line and appear with R-L.
it’s horizontal projection will appear as point in the origin point.
Z1=90 , Z2=Z3=0
Example : Represent the straight line AB where A( 2,0, 0) , B (5,0,0) and find it’s angles with the projection planes

25. Thank you For Your Attention