Monge’s projection

Revolution

1. revolution

1.1. l

p

1

l

A'

A"

l'

l"

A

= 

p

2

' 

X

12

=

p

1

"

D: A, l

p

1

, 

j

1

, j

2

1. revolution

1.1. l

p

1

l

e

A

e

"

A

A'

A"

l'

l"

A

= 

p

2

' 

X

12

=

p

1

"

D: A, l

p

1

, 

j

1

, j

2

e

A



l

e

A



A

1. revolution

1.1. l

p

1

l

e

A

e

"

A

A'

A"

S"

A

l'

=S'

A

l"

S

A

A

= 

p

2

' 

X

12

=

p

1

"

D: A, l

p

1

, 

j

1

, j

2

e

A



l

e

A



A

S

A

=l

e

A

1. revolution

1.1. l

p

1

l

e

A

e

"

A

r

A

o'

A

r

A

A'

A"

S"

A

l'=S'

A

l"

S

A

A

o

A

= 

p

2

' 

X

12

=

p

1

"

D: A, l

p

1

, 

j

1

, j

2

e

A



l

e

A



A

S

A

=l

e

A



r

A



=



AS

A



1. revolution

1.1. l

p

1

l

e

A

e

"

A

j

1

r

A

o'

A

r

A

A'

1

A'

A"

1

A"

S"

A

l'=S'

A

j

1

l"

S

A

A

A

1

o

A

= 

p

2

' 

X

12

=

p

1

"

D: A, l

p

1

, 

j

1

, j

2

e

A



l

e

A



A

S

A

=l

e

A



r

A



=



AS

A



A

1

– revolved point A through   

j

1

1. revolution

1.1. l

p

1

l

e

A

e

"

A

j

1

j

2

r

A

o'

A

r

A

A'

1

A'

A'

2

A"

1

A"

A"

2

S"

A

l'=S'

A

j

1

j

2

l"

S

A

A

A

1

A

2

o

A

= 

p

2

' 

X

12

=

p

1

"

D: A, l

p

1

, 

j

1

, j

2

e

A



l

e

A



A

S

A

=l

e

A



r

A



=



AS

A



A

1

– revolved point A through   

j

1

A

2

– revolved point A through  

j

2

1. revolution

1.1. l

p

1

l

A

e

A

e

"

A

A

1

A

2

j

1

j

2

r

A

o

A

o'

A

r

A

A'

1

A'

A'

2

A"

1

A"

A"

2

S"

A

l'=S'

A

j

1

j

2

l"

S

A

= 

p

2

' 

X

12

=

p

1

"

D: A, l

p

1

, 

j

1

, j

2

e

A



l

e

A



A

S

A

=l

e

A



r

A



=



AS

A



A

1

– revolved point A through   

j

1

A

2

– revolved point A through  

j

2

Monge’s projection

True length by revolution

1.2. Obtain the true length of segment AB

D: A,B

A'

B"

A"

B'

= 

p

2

' 

X

12

=

p

1

"

D: A,B

l

p

1

B



l

A'

B"

A"

B'

=l'

l"

= 

p

2

' 

X

12

=

p

1

"

1.2. Obtain the true length of segment AB

e

"

A

D: A,B

l

p

1

B



l

o'

A

r

A

A'

1

A'

B"

A"

A"

1

B'=l'

l"

=B"

1

=B'

1

S"

A

=S'

A

B

1

=B

e

A



l

e

A



A



r

A



=



AS

A



A

1

– revolved point A

line A

1

B

1

is a frontal line, parallel to 

vertical projection plane   

~

~

= 

p

2

' 

X

12

=

p

1

"

1.2. Obtain the true length of segment AB

A"

1

e

"

A

D: A,B

l

p

1

B



l

B

1

=B

e

A



l

e

A



A



r

A



=



AS

A



o'

A

r

A

A'

1

A'

B"

A"

B'=l'

l"

=B"

1

=B'

1

S"

A

=S'

A

~

~

= 

p

2

' 

X

12

=

p

1

"

1.2. Obtain the true length of segment AB

A

1

– revolved point A

line A

1

B

1

is a frontal line, parallel to 

vertical projection plane   

1.3. Obtain the true size of quadrangle ABCD

D: 

a

=ABCD, 

ap

2

B'

A'

B"

A"

C"

D"

a

"

D'

C'

= 

p

2

' 

X

12

=

p

1

"

D: 

a

=ABCD, 

ap

2

l

p

2

A



l

B'

A'

B"

l'

A"

=l"

C"

D"

a

"

D'

C'

= 

p

2

' 

X

12

=

p

1

"

1.3. Obtain the true size of quadrangle ABCD

D: 

a

=ABCD, 

ap

2

l

p

2

A



l

A

1

=A

B'

A'

=A'

1

B"

l'

A"=l"

=A"

1

C"

D"

a

"

D'

C'

= 

p

2

' 

X

12

=

p

1

"

1.3. Obtain the true size of quadrangle ABCD

D: 

a

=ABCD, 

ap

2

l

p

2

A



l

A

1

=A

B'

1

B'

A'=A'

1

B"

l'

A"=l"=A"

1

C"

D"

a

"

D'

C'

B"

1

e

'

B

a

"

1

B

e

B

e

B



l

a

1

ll 

p

1

B

1

e

B

~

~

= 

p

2

' 

X

12

=

p

1

"

1.3. Obtain the true size of quadrangle ABCD

D: 

a

=ABCD, 

ap

2

l

p

2

A



l

A

1

=A

a

1

ll 

p

1

B

e

B

e

B



l

B

1

e

B

B'

1

B'

A'=A'

1

B"

l'

A"=l"=A"

1

C"

D"

a

"

C'

1

D'

C'

B"

1

C"

1

e

'

C

e

'

B

a

"

1

~

~

~

= 

p

2

' 

X

12

=

p

1

"

1.3. Obtain the true size of quadrangle ABCD

D: 

a

=ABCD, 

ap

2

l

p

2

A



l

A

1

=A

a

1

ll 

p

1

B

e

B

e

B



l

B

1

e

B

B'

1

B'

A'=A'

1

B"

l'

A"=l"=A"

1

C"

D"

a

"

C'

1

D'

1

D'

C'

B"

1

C"

1

D"

1

e

'

C

e

'

D

e

'

B

a

"

1

~

~

~

~

= 

p

2

' 

X

12

=

p

1

"

1.3. Obtain the true size of quadrangle ABCD

D: 

a

=ABCD, 

ap

2

l

p

2

A



l

A

1

=A

a

1

ll 

p

1

B

e

B

e

B



l

B

1

e

B

B'

1

B'

A'=A'

1

B"

l'

A"=l"=A"

1

C"

D"

a

"

C'

1

D'

1

D'

C'

B"

1

C"

1

D"

1

a

"

1

e

'

C

e

'

D

e

'

B

a

1

=A

1

B

1

C

1

D

1



ABCD

~

~

~

~

= 

p

2

' 

X

12

=

p

1

"

1.3. Obtain the true size of quadrangle ABCD

D: 

a

=ABCD, 

ap

2

l

p

2

A



l

A

1

=A

a

1

ll 

p

1

B

e

B

e

B



l

B

1

e

B

a

1

=A

1

B

1

C

1

D

1



ABCD

B'

1

B'

A'=A'

1

B"

l'

A"=l"=A"

1

C"

D"

a

"

C'

1

D'

1

D'

C'

B"

1

C"

1

D"

1

a

"

1

e

'

C

e

'

D

e

'

B

a

1

ll 

p

1

~

~

~

~

= 

p

2

' 

X

12

=

p

1

"

1.3. Obtain the true size of quadrangle ABCD

Monge’s projection

Polyhedrons 

– pyramid

Polyhedrons
Pyramid

C"

B"

A"

W"

C'

A'

W'

B'

= 

p

2

' 

X

12

=

p

1

"

C"

a

"

B"

A"

W"

C'

A'

W'

1'

2'

3'

1"

2"

3"

B'

= 

p

2

' 

X

12

=

p

1

"

Polyhedrons
Pyramid

C"

a

"

B"

A"

W"

C'

A'

W'

1'

2'

3'

1"

2"

3"

B'

= 

p

2

' 

X

12

=

p

1

"

Polyhedrons
Pyramid

C"

a

"

B"

A"

W"

C'

B'

A'

W'

1'

2'

3'

1"

2"

3"

l"

t

'

A'

C'

e

A

"

A"

C"

e

1

"

1"

3"

e

3

"

A

A

B

C

W

1

1

3

2

3

=l'

C

= 

p

2

' 

X

12

=

p

1

"

Polyhedrons
Pyramid

C"

a

"

B"

A"

W"

C'

B'

A'

W'

1'

2'

3'

1"

2"

3"

l"

t

'

A'

C'

e

A

"

A"

C"

e

1

"

1"

3"

e

3

"

A

A

B

C

W

1

1

3

2

3

=l'

C

= 

p

2

' 

X

12

=

p

1

"

Polyhedrons
Pyramid