Exercise 1 Determine the horizontal trace of the

Exercise 1. Determine the horizontal trace of the plane P which contains the straight line q. r 2 q” Q 2 ” Q 1 ’ 1 x 2 r 1 Q 2 ’ q’ Q 1 ”

2. Determine the vertical projection of the line a contained in the plane . a) a’’ b) a’’ s 2 A 2’’ A 2 ’ A 1’’ A 1 ’ s 1 x a’ x A 1 ’ A 2 ’ s 1 a’ A 2’’ s 2 A 1’’

c) s 2 = a’’ d) s 2 x x a’ s 1 Remark: if the plane is a horizontal projection plane, then the vertical projection of the line a can not be determined.

3. Determine the vertical projection of the principal line. a) Determine the vertical projection of the horizontal principle line a of the plane . a’’ x b) Determine the vertical projection of the vertical principle line m of the plane P. r 2 s 2 A 2 ” m’’ M 1 ’ m’ A 2 ’ x M 1 ” s 1 a’ r 1

4. Determine the vertical projection of the 1 st steepest line a in the plane . a’’ A 2 ’ s 1 A 1 ” A 1 ’. s 2 P 2 ”. x p’’ P 1 ” P 2 ’ s 1 a’ A 2 ” s 2 5. Detremine the traces of the plane for which the line p is the 2 nd steepest line of the plane. p’ P 1 ’ x

6. Determine the projection of a point. a) By using the 1 st steepest line determine the vertical projection of the point T in the plane . T’’ b’’ B 1 ” B 1 ’ B 2 ’ s 1 . b) By using the vertical principle line determine the horizontal projection of the point T in the plane . s 2 m’’ x T’’ m’ T’ b’ s 2 M 1 ’ T’ x M 1 ” B 2 ” Remark: a point in a plane is determined by any line lying in the plane that passes throught the point s 1

7. Determine the horizontal projection of a line segment AB in the given plane . P 1 ’ s 2 A’ p” A” B” P 2 ” s” 1 x 2 P 2 ’ P 1 ” s’ B’ p’ s 1

Contruction of the traces of a plane determined by b) two parallel lines a) two intersecting lines A 2’’ r 2 a’’ B 2’’ b’’ S” M 1 ’ A 1’’ B 1’’ A 2 ’ x N 2 ’ S’ r 1 a’ A 1 ’ M 2’’ m’’ r 2 N 1’’ x M 1’’ M 2’ B 2 ’ b’ B 1 ’ n’’ N 2’’ N 1 ’ r 1 n’ 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. m’

Intersection of two planes s 2 a) s 2 b) Q 2’’ r 2 q’’ Q 2 ’ Q 1’’ r 1 Q 2 ” r 2 Q 1’’ x q’ s 1 Q 1 r 1 , Q 1 s 1 Q 1 = r 1 s 1 Q 2 r 2 , Q 2 s 2 Q 2 = r 2 s 2 q’ r 1 Remark. The horizontal projection of the intersection line coincides with the 1 st trace of the plane (horizontal projection plane). x

Solved exercises 1. Determine the traces of the plane which is parallel with the given plane P and contains the point T. m’’ T’’ r 2 s 2 M 1’’ x s 1 r 1 m’ M 1 ’ T’

2. Construct the traces of the plane which contains the point P and is parallel with lines a and b. P 2 ” r 2 p’’ a’’ q’’ P 2 ’ b’’ P 1 ” Q 1 ” a’ q’ Q 1 ’ r 1 Remark. A line is parallel with a plane if it is parallel to any line of the plane. x P’ p’ P 1 ’ Instruction: Construct through the point P lines p and q so that p || b and q || a is valid. b’

3. Construct the traces of the plane determined by a given line and a point not lying on the line m” 4. Construct the traces of the plane determined by the 3 non-colinear given points N 2” M 2” r 2 C’’ p’’ n” A’’ T’’ M’’ r 2 P 1’’ m’ T’ P 1’ M’ M 1’’ P ’ 2 x P 2’’ M 1’ 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. s 1 B’’ m” N 2’ M 2’ x C’ M 1” A’ n’ m’ M 1’ N 1” r 1 B’ N 1’

5. Detremine the 1 st angle of inclination of the plane for which the line p is the 2 nd steepest line of the plane. T 2 ” s 2 P 2 ”. p’’ P 1 ” P 2 ’ s 1 To determine the 1 st angle of inclination we can use any 1 st steepest line t of that plane. p’ x T 2 ’ 1 t’ T 1’ P 1 ’ T 2 0

6. Determine the intersection of planes P and . z s 3 r 2 s 1 t’’’ r 3 t’’ s 2 x y t’ r 1 y

7. Construct the plane throught the point T parallel with the symmetry plane. z s 3 d 3 s 1 s 2 k 1 k 2 T” T’’’ T’ y d 1=d 2