1. Lecture 5: The Auxiliary projection Dr. Samah Mohamed MabroukBy
Dr. Samah Mohamed Mabrouk

2. The Auxiliary projection is defined by
The auxiliary projection planes are perpendicular planes on 1 or 2 and can be moved on parallel or perpendicular to a geometric objects in order to transform the positions of these geometric objects into more simple positions, from which the complex problems can be solved easily.

3. 1- Auxiliary projection in a plane 3  1.zA
x12 A1 zA
x13 A3
x35 A5

4. 2- Auxiliary projection in a plane 4  2.yA
x24 A2
x12 yA A1

5. Problem (1):The true length of a straight line in space .
x12 B1 A1
x13  A3 B3
T.L.

6. h // 1
f // 2 h2
f2 = T.L
x12  
h1 = T.L f1

7. Problem (1):The true length of a straight line in space . A4 B2
x24 A2
x12 B1 A1

8. Triangles of solution T.L. of m T.L. of m zAB yAB A1 B1 A2 B2
T.L. of m A1
zAB B2 
T.L. of m A2
yAB B3 
T.L. of m A3
xAB

9. Problem (2):Convert a straight line into a point .
x12 B1 A1
x13 A3 B3
T.L.
A5=B5
x35

10. The auxiliary projection of a plane.
Problem (3): Convert a plane into a line v A2
x12 A1 h 1 3 A3
x13

11. الفكرة الأساسية هى تحويل الخط المشترك (خط التقاطع) الى نقطة
Problem (4):The dihedral angle between two planes. C2 D2 B2 A2 C1 B1 A1 D1
الفكرة الأساسية هى تحويل الخط المشترك (خط التقاطع) الى نقطة

12. Problem (4):The dihedral angle between two planes.C2 D2 B2 A2
x12 C1
x35 B1
A5 =B5 A1 B3 D1
T.L.
x13 A3 C5 D5 D3 C3

13. Example (2):Given a point R and a line m{A,B}, find d(R, AB) .
T.L. A3 B3
x13
d(R, AB) R3 R5
x35

14. Example (4):Given the two projections of Parallelogram and the vertical projection of the point M. find the horizontal projection of point M if d(M,ABCD)=3cm also, find the true shape of the Parallelograms ABCD. M2 B2 K2 A2 zM C2
x12 D2 A1
x13 B1
L:M3 zM M2 K1
L:M3 M3
3 cm D1 C1 B3 A3 C3 D3

15. M2 B2 K2 A2 C2
x12 D2 A1
x13 B1 M1 K1 D1 C1 B3 A3
x35 C3 D3 C5 D5
T.S. B5 A5