Transcript M 1

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.
A2
zA
x12
A1
A3
A5
2- Auxiliary projection in a plane 4  2.
A6
A4
A2
x12
yA
A1
Problem (1):The true length of a straight line in space .
B2
A2
B1
A1

B3
x12
h // 1
f // 2
h2
x12


f1
Problem (1):The true length of a straight line in space .
B4

A4
B2
A2
B1
A1
x12
Triangles of solution
zAB
yAB

A1
B1

A2
B2
xAB
A3

B3
Problem (2):Convert a straight line into a point .
B2
A2
B1
A1
B3
A5=B5
x12
The auxiliary projection of a plane.
Problem (3): Convert a plane into a line
v
A2
x12
A1
h
1
3
A3
‫‪Problem (4):The dihedral angle between two planes.‬‬
‫‪C2‬‬
‫‪D2‬‬
‫‪B2‬‬
‫‪A2‬‬
‫‪C1‬‬
‫‪B1‬‬
‫‪A1‬‬
‫‪D1‬‬
‫الفكرة األساسية هى تحويل الخط المشترك (خط التقاطع) الى نقطة‬
Problem (4):The dihedral angle between two planes.
C2
D2
B2
A2
C1
B1
A5 =B5
A1
B3
D1
A3
C5
D5
D3
C3
x12
Example (2):Given a point R and a line m{A,B}, find
d(R, AB) .
R2
A2
B2
x12
A1
R1
B1
A5 =B5
T.L.
B3
R3
x35
R5
A3
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.
B2
M2
K2
A2
zM
C2
x12
D2
A1
B1
M2
K1
D1
M3
3 cm
C1
B3
D3
C3
A3
C2
x12
D2
A1
B1
M1
K1
D1
C1
B3
D3
C3
C5
A3
D5
T.S.
B5
A5