Transcript Slide 1

Represent the two projections of an equilateral triangle(‫)مثلث متساوى االضالع‬
ABC if its side AB is given and C(?, 1, ?).
B2
A2
Locus of c1 0
1
B1
A1
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x12
Δy
Locus of c2
Locus of c2
B2
A2
.
0
Vertical projection. of AC
Δy
Δy AC
c2
Locus of c1
c1
Vertical projection. of BC
Δy BC
B1
A1
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1
x12
B2
A Regular hexagon ABCDEF
A2
C2
s2
F2
D2
E2
D1
0
x12
C1
E1
Locus of s1
s1
B1
F1
A1
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A cube ABCDA\B\C\D\ is given by its vertex A\ , its face ABCD  π1
and its vertex C(?, 5, ?) . Represent this cube by its three projections.
A\2
D\
B\
C\
A\
D
B
C
A
x12
A\1

ABCD 
π1 
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zA = 0, zB = 0, zC = 0 zD = 0.
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_
AC
z
C‘
C
B‘
B
D‘
D
’
D‘ C
A’
A
O
D
C
A‘
B‘
A
B
x
A=A‘‘
D= D‘
5
B =B‘
y
C =C‘
A2A2\
C\2
B\2
A\2
T. L of edge
D\2
C2
D\
1
D2
D2
B2
A2A2\
A2
x12
= D1
A\1 = A1
5 cm
C\
.
1
Locus of C1 = C\1
= C1
B\1 = B1
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The plane is denoted by small Greek letter α,β, γ, δ, ζ, , σ,.....
The plane is determined in the space by:
1- Three non collinear points.
A2
B2
2- Two parallel straight lines.
a2
C2
b2
x12
C1
A1
b1
a1
B1
3- Two intersecting straight lines.
a2
x12
4- A straight line and a point outside it.
b2
m2
A2
x12
a1
b1
x12
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m1
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A1
h α  α π1
5- The traces of the plane.
s
α
hα
s α  α π 3
o
a
v
It is the side (profile) trace
of the plane
The plane is defined by

α

It is the horizontal trace of
the plane
x12
v α  α π 2
It is the vertical trace of
the plane
 (a,  ,   )
6- The intercepts on the principle axes by the plane.
The plane is defined by
sα
hα
c
b
o
b
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α(a,b, c)
a
x12
vα
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v
v
ss
c
a
s
x
x
b
o
h
y
Traces of a plane
A horizontal plane
z
z
v
s
o
x
s
v
s
x
o
y
h
A frontal plane
h
A vertical plane
x
o
y
h
h
A plane parallel to y-axis
1- The plane  // π1 :
2- The plane  // π 2 :
(  is a horizontal plane)
(  is a frontal plane)
s α B3
C3
C2
A3
B2
0
A2
A1
C1
v
sα
α
x12
A3
A2
B3
C3
T. S
B2
C2
0
hα
T. S
C1
B1
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A1
B1
x12
3- The plane  // π3 :
(  is a profile plane)
B3
A3
4- The plane  π1:
hα
B2
T. S
A2
C2
C3
π1
0
C1
s

A3
v
x12
0
B1
h
A1
vα
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A2
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A1
x12
5- The plane  π 2 :
s
6- The plane  π3 :
v
A3
A3
A2
v
s
A2
0
0
A1
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h

A1
x12
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x12
h
A2
m2
p2
q2
b2
a2
B2
C2
m2 p2
q2
x12
a1
B1
C1
m1
m1
q1
p1
p1
b2
A1
b1
a2
m2
q1
q2
p2
x12
m1
a1
b1
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p1
q1
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x12
m  α  Hm  h αandVm  vα
m2
V = V2
vα
m1
H2
V1
x12
H = H1
hα
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The horizontal straight line
h in the plane α
v
h2
The frontal straight line f in
the plane α
vα
f2
α
V = V2
H2
V1
x12
x12
f1
H = H1
h1
hα
hα
The profile straight line p in
the plane α
s
vα
α
V = V2
p2
p3
o
V1 H2
p1
x12
H = H1
hα
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A point that lies in a plane lies on a straight in that plane.
b2
a2
m2
p2
q2
a1
h2
M2
M2
x12
M1
m1
p1
h1
q1
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M1
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V = V2
V1
x12
b1
vα
hα
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Given the plane by its traces
It is required to:
1- Find a line m of steepest
slope w. r. t. π1
2- The angle of inclination
of m with π1
V = V2
vρ
m2
 is the angle of inclination of m
with π1
 is the angle of the plane with π1
H2
V1
x12
m1
[V]
[m]

H = H1
hρ
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v
v
V
s
s
s
s
s
s
H
h
Line of steepest slope w.r.t H.P.
h
Line of steepest slope w.r.t. V.P.
3- Find a line m of steepest slope
4- The angle of inclination
w. r. t. π 2
of m with π 2
β is the angle of inclination of m
with π 2
[m]
[H]
V2
β
vρ
β is the angle of the plane with π 2
m2
//
H2
V1
x12
m1
hρ
H1
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Given the plane by two straight line or three points.
1- Find a line m of steepest 1- Find a line m of steepest
It is required to:
slope w. r. t. π 2
A2 slope w. r. t. π1
A2
f2
m2
h2
.
B2
B2
C2
C2
m2
x12
x12
m1
m1
B1
C1
.
B1
f1
C1
h1
A1
A1
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Represent the planes α(3,30 ,60 )
β(4,45 ,90 )
vβ
vα
90
60
30
0
x12
0
45
hα
hβ
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x12
γ(5,4,3)
vγ
b = -4
δ(,2,3)
a=5
0
x12
vδ
hγ
C= -3
sδ c = 3
0
b=2
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x12
hδ
Given a plane ρ : {a : M/M  a}. Find the traces of the plane ρ .
a2
M2
x12
a1
M1
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V2 = Vb
V2 = Va
a2
vρ
M2
H2
V1
H2
V1
x12
M1
a1
H1 = Ha
h
ρ
H1 = Hb
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