1. Exercise r2 q” Q2” Q1’ 1x2 r1 Q2’ Q1” q’
1. Determine the horizontal trace of the plane P which contains the straight line q.
1x2 q’ q”
Q2’
Q2” r2
Q1’ r1
Q1”

2. a) b) s2 a’’ a’’ s2 A2’’ A1’ A2’ A1’’ x x A1’’ A2’ A1’ s1 s1 a’ a’
2. Determine the vertical projection of the line a contained in the plane .
a’’ a) b)
a’’ s2 s2 s1 a’ x
A2’’
A1’
A2’
A1’’ x
A1’’
A2’
A1’ s1 a’
A2’’

3. c) d) s1 s2 x s1 s2 a’ x
= a’’ a’
Remark: if the plane  is a horizontal projection plane, then the vertical projection of the line a can not be determined.

4. r2 m’’ s2 a’’ A2” M1’ m’ A2’ x x M1” s1 r1 a’
3. Determine the vertical projection of the principal line.
b) Determine the vertical projection of the vertical principle line m of the plane P.
a) Determine the vertical projection of the horizontal principle line a of the plane . m’ r1 r2 x s2
m’’
a’’
A2”
M1’
M1”
A2’ x s1 a’

5. s2 . s2 p’’ p’ x a’’ P1’ P1” P2’ P2” A2’ A1” x s1 A1’ . s1 a’ A2”
4. Determine the vertical projection of the 1st steepest line a in the plane .
5. Detremine the traces of the plane  for which the line p is the 2nd steepest line of the plane. s2 . s2
p’’ p’ x
a’’
P1’
P1”
P2’
P2”
A2’
A1” x s1
A1’ . s1 a’
A2”

6. s2 T’’ b’’ s2 m’’ B1” x T’’ B2’ B1’ M1’ m’ . s1 T’ x T’ M1” b’ s1 B2”
6. Determine the projection of a point.
By using the 1st steepest line determine the vertical projection of the point T in the plane .
b) By using the vertical principle line determine the horizontal projection of the point T in the plane . s2
b’’
T’’ s1 s2
T’’ x
m’’
B1” x b’ .
B1’
B2’ T’
M1’ m’ s1 T’
M1”
B2”
Remark: a point in a plane is determined by any line lying in the plane that passes throught
the point

7. P1’ s2 A’ B” P2” s” A” p” 1x2 P1” P2’ s’ s1 B’ p’
7. Determine the horizontal projection of a line segment AB in the given plane .
P1’ p’ s2 A’ s” B”
P2” B’ p” A”
P2’
1x2
P1” s’ s1

8. Contruction of the traces of a plane determined by
a) two intersecting lines
b) two parallel lines
A2’’ x m’
m’’ n’
n’’ r1 r2
B2’’
a’’
b’’ S”
M1’
M2’’ r2 r1
N2’
A1’’
N1’’
B1’’ x
A2’
M1’’
A1’
B2’
M2’
B1’
N2’’
N1’ S’ a’ b’
A plane can determined also with a point and a line that are not incident, and with three non-colinear points. These cases are also solved as these two examples.

9. Intersection of two planesx b) q’ 
q’’ a) s2
Q2”
q’’
Q2’’ r2
Q2’
Q1’’
Q2’ q’ x
Q1’’ r1 s1
Q1’
Q1’
Q1  r1, Q1  s1  Q1 = r1  s1
Remark. The horizontal projection of the intersection line coincides with the 1st trace of the plane  (horizontal projection plane).
Q2  r2, Q2  s2  Q2 = r2  s2

10. Solved exercises m’’ r2 T’’ s2 M1’’ x s1 r1 m’ M1’ T’
1. Determine the traces of the plane  which is parallel with the given plane P and contains the point T.
m’’ m’ r2
T’’ s1 s2
M1’
M1’’ x r1 T’

11. r2 r1 x b’ b’’ a’’ a’ P’’ P’ P2” P2’ P1” P1’ p’ p’’ q’’ q’ Q1’ Q1”
2. Construct the traces of the plane which contains the point P and is parallel with lines a and b. r2 r1 x b’
b’’
a’’ a’
P’’ P’
P2”
P2’
P1”
P1’ p’
p’’
q’’ q’
Q1’
Q1”
Remark. A line is parallel with a plane if it is parallel to any line of the plane.
Instruction: Construct through the point P lines p and q so that p || b and q || a is valid.

12. 4. Construct the traces of the plane determined by the 3 non-colinear given points
3. Construct the traces of the plane determined by a given line and a point not lying on the line r1 r2 n” n’ m” m’
N2’
N2”
M2’
M2” x A’
A’’
C’’ C’ B’
B’’ m”
p’’
T’’ r2
M’’
M1’’
P2’ x s1
P1’’
N1’
N1”
P1’
M1’
P2’’
M1’
M1” m’ T’ M’ p’
Instruction. Place a line throught the point T that intersect (or is parallel with) the line p. Here the chosen line is the vertical principle line.

13. s2 . T2” p’’ p’ x P1’ P1” P2’ P2” s1 t’ T1’ T2’ 1 T20
5. Detremine the 1st angle of inclination of the plane  for which the line p is the 2nd steepest line of the plane. s2 .
T2”
p’’ p’ x
P1’
P1”
P2’
P2” s1 t’
T1’
T2’ 1
To determine the 1st angle of inclination we can use any 1st steepest line t of that plane.
T20

14. 6. Determine the intersection of planes P and .x z y r2 r1 s3 r3
t’’’ s1 s2
t’’ t’

15. 7. Construct the plane throught the point T parallel with the symmetry plane.
s1  s2  k1  k2 z y s3 T’ T” d3
T’’’
d1=d2