Lecture 5: The Auxiliary projection Dr. Samah Mohamed Mabrouk By Dr. Samah Mohamed Mabrouk www.smmabrouk.faculty.zu.edu.eg

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.

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

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

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

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

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

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

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

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

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

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

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

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

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