Transcript (M) A 1

Lecture 7: The Metric
problems
The metric problems
1- Introduction
2- The first problem
3- The second problem
4- The third problem
a- Rotation
b- Affinity
6- Examples
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1- Introduction
Metric problems deal with :
True lengths , true shapes , perpendicularity, the angles between two straight
lines or two planes Or a straight line and a plane and the rotation of Planes.
Theorem (1)
The right angle is projected into a right angle iff at least one of
its legs is parallel to the plane of projection. B
A
Bi
πi
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C
Ai
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Ci
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AB//1
AB//2
A2
B2
B2
A2
C2
C2
x12
A1
C1
C1
.
B
A1
1
The horizontal
projection of the angle
ABC is right angle
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B1
The vertical projection
of the angle ABC is
right angle
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x12
AB//π 3
A2
C2
B2
x12
C1
A1
B1
B3
T.L
A3
The side projection of
the angle ABC is right
angle
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C3
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Example (1) : Given the side AB of a square ABCD and the horizontal
Projection of a straight line m on which the side BC lies Represent this
square by its two projections
A2
B2
x12
A1
m1
B1
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D3
C2
K2
A2
Δz
B2
x12
D1
A1
C1
.
Δz
K1
B1
A3
.
[C]
B3
[K]
.
K3
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2- The first problem
The normal n through a given point M to a given plane.
To construct a straight line through a given point and perpendicular to a
given plane.
M
n
Theorem (2)
A straight line n is perpendicular to a
plane if it is perpendicular to two
intersecting straight lines h and f lying
in the plane.
f
h
h is taken a horizontal straight line and f is taken a frontal
straight line in the given plane. We use THEOREM (2) to
represent the normal n.
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i) the plane is given by its traces.
M2
n2
n2 passes thr, M2 and is
normal to vρ.
vρ
x12
hρ
n1 passes thr, M1 and is
normal to hρ.
n1
M1
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ii) the plane is given by two intersecting str. Lines a and b.
3\
M2
4\
f2
h2
2
1
n2
.
x12
f1
3
a1
4
h1
2\
1\
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.
n1
.
M1
We use a horizontal str. Line h and frontal
str. Line f in the plane.
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3- The second problem
To construct a plane ( normal plane) through a given point and
perpendicular to a given straight line.
i) The normal plane is determined by two straight lines h and f.
.
f2
m2
h2
M2
x12
f1
M1
m1
.
h1
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ii) The normal plane is determined by its traces.
.
vρ
m2
h2
M2
v = v2
v1
M1
m1
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.
h1
.
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hρ
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x12
4- The third problem ( The rotation)
i) The rotation of a plane about its horizontal trace till it coincides with
the Horizontal plane Π1.
vα
M2
zM
x12
M1
.
(M)
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hα
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[M]
4- The third problem ( Affinity)
Is one to one correspondence between points or straight lines. It is
defined by an axis o called the axis of affinity and a direction d called
the direction of affinity and two corresponding points M and M\ .
If a point A is given , to find A\.
Join AM
A
M
Q
Find Q on o
d
join QM \ and draw a
segment parallel to d
from A cutting QM\ in
the point A\.
o
M\
A\
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vα
M2
zM
A2
x12
A1
(A)
(M)
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M1
.
hα
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[M]
ii) The rotation of a plane about its vertical trace till it coincides with
the Vertical plane Π2.
.
[M]
(A)
A2
vα
M2
x12
A1
h
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α
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yM
M1
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v) The rotation of a plane about a horizontal straight line h till it
coincides with the horizontal plane Π passing through the horizontal
straight line h.
h2
{
M2
zM
x12
h1
. A1
M1
(A)
.
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[M]
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vi) The rotation of a plane about a frontal straight line f till it coincides
with the frontal plane passing through the frontal straight line h.
.
[M]
(A)
f2
.A
M2
2
x12
f1
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yM
M1
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Given two straight lines a and b intersecting in a point M.
Find the angle < (a, b) and represent its bisector.
a2
A2
{
M2
b2
B2
h2
R2
x12
M1
a1
α  (a, b)
(R)
A1
(R)  R1  h1
(a)
R1
b1
B1
(b)
α
(M)
R2  h2
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h1
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Represent a square ABCD lying in a given plane . If the vertical
projections of A and C are given. Hence find a point E such that:
AE = BE = CE= DE = 6 cms.
v
+ C2
ρ
A2 +
x12
hρ
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E
n2
The true length of
MK to get the
direction of true
length of n.
K2
E2
M
B2
[K]
[E]
*
.
//
A
vρ
*
+ C2
M2
A2 +
D2
x12
B1
hρ
M1
A1
C1
n1
(A)
D1
//
. KE
// (M)
//
(B)
1
(D)
1
(C)
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Given two planes by its traces find the dihedral angle between the
two planes  and ρ .
hρ
ρ
n
180  α

h2
n ρ
n  
n \2
n2
M
{
n\
M2
where
  (  , )
.
vρ
n1
(n) 180  α
(M) α
180  α  (n, n)
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x12
M1
h1
From M:
h
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n \1
(n\)
vσ