Lecture 5 The Auxiliary projection By Dr Samah

Lecture 5: The Auxiliary projection 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. A 2 z. A x 12 A 1 z. A x 1 3 A 3 x 35 A 5

2 - Auxiliary projection in a plane 4 2. x 46 A 6 y. A A 2 A 4 x 2 4 x 12 y. A A 1

Problem (1): The true length of a straight line in space. B 2 A 2 x 12 B 1 A 1 B 3 T. L. A 3 x 13

h // 1 f // 2 f 2 h 2 . L T = x 12 h 1 f 1 =T . L

Problem (1): The true length of a straight line in space. B 4 T. L. A 4 B 2 x 2 4 A 2 B 1 A 1 x 12

Triangles of solution T. L z. AB . of T. L m y. AB A 1 T. L A 3 . of m A 2 B 1 x. AB . of B 2 m B 3

Problem (2): Convert a straight line into a point. B 2 A 2 x 12 B 1 A 1 x 13 B 3 A 5=B 5 T. L. A 3 x 35

The auxiliary projection of a plane. Problem (3): Convert a plane into a line v A 2 x 12 A 1 h 1 3 x 1 3 A 3

Problem (4): The dihedral angle between two planes. C 2 D 2 B 2 x 35 A 2 A 5 =B 5 C 1 B 3 x 12 D 1 T. L. A 1 A 3 D 5 C 3 D 3 x 13

Example (2): Given a point R and a line m{A, B}, find d(R, AB). R 2 A 2 B 2 A 1 R 1 B 1 A 3 T. L. B 3 d(R , A B) A 5 =B 5 R 3 x 35 R 5 x 12 x 13

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)=3 cm also, find the true shape of the Parallelograms ABCD. B 2 M 2 K 2 A 2 z. M C 2 x 12 D 2 L: M 3 3 B 1 z. M x 1 A 1 M 1 K 1 D 1 L: M 3 3 cm C 1 D 3 C 3 A 3 B 3

C 2 x 12 D 2 3 B 1 x 1 A 1 M 1 K 1 D 1 C 1 x 35 D 3 C 5 B 3 A 3 D 5 T. S. B 5 A 5