Descriptive Geometry

1. Descriptive Geometry
Dr. Tareq Al-
hababi
Sana’a University
Faculty of Engineering
Lecture 2
Projection of Points

2. Chapter 5
Projections of Points and Lines

3. Elements of descriptive geometry
Point
Plane Line
B
m = AB
A
point
line

4. Orthographic projection
VP
P’
P
Projector
An observer at
infinity in front of
a plane
FIGURE 5.1 A plane of projection

5. X
Y
Names and definitions
FIGURE 5.2’ The vertical and horizontal planes of projection
Vertical Plane (VP) = Y axis
Front view (FV) = Front elevation = elevation = 𝑝′
Horizontal Plane (HP) = X axis
Top view (TV) = Top plane = simply plane = 𝑝
The line of intersection of the HP and the VP is
known as hinge line or ground line or reference
line or (XY line).

6. X
Y
1
2
3
4
1
2
3 4
FIGURE 5.2 The vertical and horizontal planes of projection
The dihedral angles or quadrant numbers
Location
Dihedral Angle or
Quadrant Number
In front of the VP, above the HP First
Behind the VP, above the HP Second
Behind the VP, below the HP Third
In front of the VP, below the HP Fourth
TABLE 5.1 The dihedral angles or quadrant
numbers

7. X
Y
d
h
P
p
P’
h
d
X Y
FIGURE 5.3 (a) Pictorial view of the first-angle projections
of a point
(a) (b)
First-angle
projections
Clockwise (CW)
Counterclockwise
(CCW)
CW

8. q
q’
h
d
X Y
FIGURE 5.4 (a) The pictorial of the second-angle projections
of a point
X
Y
d
h
Q
(a) (b)
Second-angle
projections

9. X
Y
d
h
r
r’
h
d
X Y
FIGURE 5.5 (a) Pictorial of third-angle projections
of a point
(a) (b)
Third-angle
projections

10. X
Y
d
h
S
s
s’
s
s’
h
d
Y
X
(a) (b)
FIGURE 5.6 (a) Pictorial of fourth-angle projections of
a point
Fourth-angle
projections

11. TABLE 5.2 Conclusion about projections of points
1 The front view and the top view of a point are always on the same vertical line.
2
The distance of the front view of a point from the XY line is always equal to the
distance of the given point from the HP.
3
If a given point is above the HP, its front view is above the XY line. If the given
point is below the HP, its front view is below the XY line.
4
The distance of the top view of a point from the XY line is always equal to the
distance of the given point from the VP.
5
If a given point is in front of the VP, its top view is below the XY line. If the
given point is behind the VP, its top view is above the XY line.

12. Dihedral angle
or quadrant
Position of the
given point
Position
in
the front
view (a’)
Position
in the top
view (a)
First
Above the HP, in
front of the VP
Above XY Below XY
Second
Above the HP,
behind the VP
Above XY Above XY
Third
Below the HP,
behind the VP
Below XY Above XY
Fourth
Below the HP, in
front of the VP
Below XY Below XY
TABLE 5.3 Positions of a point and its projections
X
Y
1
2
3
4
1
2
3 4
Positions of a point and its
projections

13. f
d
c'
c
b
a'
a
Example 5.1 Draw the projections of the following points on the same ground line, keeping the
distance between projectors equal to 25 mm.
1
2
3 4
X Y
b'
d'
e'
f'
e
30
20
25
20
25
20
30
25
25
20
FIGURE 5.7 (a) Solution of
Example 5.1
(ii) Point B, 25 mm below the HP, 20 mm behind the VP.
(iv) Point D, 20 mm above the HP, 25 mm in front of the VP.
(iii) Point C, 20 mm below the HP, 30 mm in front of the VP.
(vi) Point F, on the VP, 30 mm above the HP.
(v) Point E, on the HP, 25 mm behind the VP.
(i) Point A, 20 mm above the HP, 25 mm behind the VP. A(25,20)
B(20,25)
C(30,20)
D(25,20)
E(25,0)
F(0,30)
X
Y
d
h
A

14. a’
a
b
b’
Example 2
A straight line AB, Point A is 10mm in front of
the VP and 20 mm above the HP; Point B is 30
mm in front of the VP and 50 mm above the HP;
the distance between aa’ and bb’ is 40 mm.
Draw the projections of the line AB.
Solution:
• A (10 , 20)
• B (30 , 50)
• aa’ bb’ = 40
40 mm
X Y
10
20
30
50
1
2
3 4
x
y
-x
-y

15. The End of Point
Projection
Thank You!