Lecture 7: The Metric problems. The Main Menu اPrevious اPrevious Next The metric problems 1- Introduction 2- The first problem 3- The second problem

Lecture 7: The Metric problems

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The metric problems

1 -Introduction

2 -The first problem

3 -The second problem

4 -The third problem

a- Rotation

b- Affinity

6 -Examples

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.

The right angle is projected into a right angle iff at least one of its legs is parallel to the plane of projection.


A

B

C

Ai

Bi

Ciiπ

1 -Introduction

Theorem (1)


x12

C2

B1

C1

x12

A2 B2

A1

.

A2

B2

C2

A1 B1

C1

.

The horizontal projection of the angle ABC is right angle

The vertical projection of the angle ABC is right angle

T.L

T.L

AB//1 AB//2


3AB//π

A3

B3

C3

x12

A2

B2

C2

A1

B1

C1x13

.

The side projection of the angle ABC is right angle

T.L

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


x12

A2

B2

A1

B1

m1


x12

A2

x13

B2

A1

B1

A3 B3

T.L

.K1m 1

K3

.

K2

Δz

Δz

[K]/

/

T.L

//

.[C]

C1

C2

D1

D3

//

//

m3

The normal n through a given point M to a given plane.

h

f

nM

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.

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.


To construct a straight line through a given point and perpendicular to a given plane.

2 -The first problem

Theorem (2)

n1 passes thr, M1 and is normal to hρ.

i) the plane is given by its traces.


n2 passes thr, M2 and is normal to vρ.

x12

ρh

ρvM2

M1

n2

n1

ii) the plane is given by two intersecting str. Lines a and b.

We use a horizontal str. Line h and frontal str. Line f in the plane.


x12

a2

b 2

a1

b 1

M2

M1

h2

1\

1 2

2\

h1

f1

f2

3 4

3\

4\

n2

.

.n1

.

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.


h2

f1

x12

m2

m1

M2

M1

h1.

f2

.

3 -The second problem


ii) The normal plane is determined by its traces .

x12

m2

m1

M2

M1

h2

.h1

v1

v = v2

.

ρv

.ρh

i) The rotation of a plane about its horizontal trace till it coincides with the Horizontal plane Π1.


x12

αv

αh

M2

M1

.

// zM

// [M]*

*

)M(

4 -The third problem ( The rotation)

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\ .

M

d

A

If a point A is given , to find A\.

Q

join QM \ and draw a segment parallel to d from A cutting QM\ in the point A\.

Join AM

Find Q on o


oM\

A\

4 -The third problem ( Affinity)


x12

αv

αh

M2

M1

.

// zM

// [M]*

*)M(

A2

A1

)A(

ii) The rotation of a plane about its vertical trace till it coincides with the Vertical plane Π2.


x12

αh

αvM2

M1

.

//

yM

// [M]

*

*)M(

A2

A1

)A(

x12

2h

1h

M2

M1

.

// zM

//[M]*

*)M(

A1

)A(

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.


{

.


x12

1f

M2

M1

.

//

yM

//

[M]

*

*)M(

A2

)A(

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.

.2f

{

Given two straight lines a and b intersecting in a point M. Find the angle < (a, b) and represent its bisector.

)M(


x12

a2 b2

M2

M1 b1

a1

h2

h1

{

{

A2

A1

B2

B1

)b(

)a(

αb)(a,α

)R(

11 hR(R)

22 hR

R1

R2

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.


x12

ρv

ρh

C2

A2

+

+

x12

ρv

ρh

C2

A2

C1

A1

)A(

)C(

.)M(

M1

M2

D1

B1

B2

6 cm

M A

E

.


//

E1

K2

K1

The true length of MK to get the direction of true length of n.

[K]*

E2

[E] *

D2

)D(

)B(

n1

n2

//

//

//

//

+

+

Given two planes by its traces find the dihedral angle between the two planes and.

M

n

n\

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ρ

ρ

ρn n

From M:

)n(n,α180

where

),(

x12

ρhh

ρvσv

M2

M1

n2

n1

n\2

n\1

h2

h1

{{

.

)M( )n\()n( α180

α

α180

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Lecture 7: The Metric problems. The Main Menu اPrevious اPrevious Next The metric problems 1- Introduction 2- The first problem 3- The second problem