lecture 6

Monge s projection
Rotation by right angle
1. Rotation of plane by right angle
1.1. a^餻1
a
A
A"
A'
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
a
A
A"
A'
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
a
A
A"
A'
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
a
A
A"
A'
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
a
A
A"
A'
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
a
A
A"
A'
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
A
A"
A'
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
a
A"
A'
A
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
A"
A'
a
A
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
A"
A'
ax
Ax
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
a
A
A"
A'
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
a
A
A"
A'
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
A'
a'=k'
1. Rotation of plane by right angle
1.1. a^餻1
a
A
A"
A'
Ax
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
金A'Ax金=金A"p"1金= 金Ap1金
A'
a'=k'
Ax
1. Rotation of plane by right angle
1.1. a^餻1
a
A
a
a"
A"
A'
Ax
p1
X12=p1"=p2 =k"
D: a, A勿a,
a^餻1
金A'Ax金=金A"p"1金= 金Ap1金
A'
a'=k'=a'
A勿a
Ax
1. Rotation of plane by right angle
1.1. a^餻1
a
A
a
a"
B=B'
A"
A
Ax
p1
B"
X12=p1"=p2 =k"
B'
D: a, A勿a,
a^餻1
金A'Ax金=金A"p"1金= 金Ap1金
A'
a'=k'=a'
A勿a
B勿a, B勿p1
Ax
1. Rotation of plane by right angle
1.1. a^餻1
a
A
a
a"
B=B'=
A"
=Bx
A'
Ax
p1
B"
X12=p1"=p2 =k"
B'=Bx
D: a, A勿a,
a^餻1
金A'Ax金=金A"p"1金= 金Ap1金
A'
a'=k'=a'
A勿a
B勿a, B勿p1
Ax
1. Rotation of plane by right angle
1.1. a^餻1
a
A
a
a"
B=B'=
A"
=Bx
A'
Ax
p1
ax
B"
X12=p1"=p2 =k"
B'=Bx
D: a, A勿a,
a^餻1
金A'Ax金=金A"p"1金= 金Ap1金
A'
a'=k'=a'
A勿a
B勿a, B勿p1
AxBx=ax
Ax
ax
1. Rotation of plane by right angle
1.1. a^餻1
a
A
a
a"
B=B'=
A"
=Bx
A'
Ax
p1
ax
B"
X12=p1"=p2 =k"
B'=Bx
D: a, A勿a,
a^餻1
金A'Ax金=金A"p"1金= 金Ap1金
A'
a'=k'=a'
A勿a
B勿a, B勿p1
AxBx=ax
Ax
ax
D: a=ABC,
1. Rotation of plane by right angle
1.1. a^餻1
a^餻1
Draw a centre of a circle inscribed into a triangle ABC
C"
B"
A"
X12=p1"=p2
A'
C'
B'
a'
D: a=ABC,
1. Rotation of plane by right angle
1.1. a^餻1
a^餻1
Draw a centre of a circle inscribed into a triangle ABC
金A'Ax金=金A"p"1金
C"
B"
A"
X12=p1"=p2
A'
C'
Ax
B'
a'
D: a=ABC,
1. Rotation of plane by right angle
1.1. a^餻1
a^餻1
Draw a centre of a circle inscribed into a triangle ABC
金A'Ax金=金A"p"1金
C"
金B'Bx金=金B"p"1金
B"
A"
X12=p1"=p2
A'
C'
Ax
B'
a'
Bx
D: a=ABC,
1. Rotation of plane by right angle
1.1. a^餻1
a^餻1
Draw a centre of a circle inscribed into a triangle ABC
金A'Ax金=金A"p"1金
C"
金B'Bx金=金B"p"1金
B"
金C'Cx金=金C"p"1金
A"
X12=p1"=p2
A'
C'
Ax
B'
a'
Cx
Bx
D: a=ABC,
1. Rotation of plane by right angle
1.1. a^餻1
a^餻1
Draw a centre of a circle inscribed into a triangle ABC
金A'Ax金=金A"p"1金
C"
金B'Bx金=金B"p"1金
B"
金C'Cx金=金C"p"1金
AxBxCx吼ABC
A"
X12=p1"=p2
A'
C'
Ax
B'
a'
Cx
Bx
D: a=ABC,
1. Rotation of plane by right angle
1.1. a^餻1
a^餻1
Draw a centre of a circle inscribed into a triangle ABC
金A'Ax金=金A"p"1金
C"
金B'Bx金=金B"p"1金
B"
金C'Cx金=金C"p"1金
AxBxCx吼ABC
S勿a
A"
Sx勿AxBxCx
X12=p1"=p2
A'
C'
Ax
B'
a'
Sx
Cx
Bx
D: a=ABC,
1. Rotation of plane by right angle
1.1. a^餻1
a^餻1
Draw a centre of a circle inscribed into a triangle ABC
金A'Ax金=金A"p"1金
C"
金B'Bx金=金B"p"1金
B"
金C'Cx金=金C"p"1金
AxBxCx吼ABC
S勿a
A"
Sx勿AxBxCx
X12=p1"=p2
S'勿a'
A'
C'
Ax
S'
B'
a'
Sx
Cx
Bx
D: a=ABC,
1. Rotation of plane by right angle
1.1. a^餻1
a^餻1
Draw a centre of a circle inscribed into a triangle ABC
金A'Ax金=金A"p"1金
C"
金B'Bx金=金B"p"1金
B"
金C'Cx金=金C"p"1金
S"
AxBxCx吼ABC
S勿a
A"
Sx勿AxBxCx
X12=p1"=p2
S'勿a'
A'
金S"p"1金=金S'Sx金
C'
Ax
S'
B'
a'
Sx
Cx
Bx
D: a=ABC,
1. Rotation of plane by right angle
1.1. a^餻1
a^餻1
Draw a centre of a circle inscribed into a triangle ABC
金A'Ax金=金A"p"1金
C"
金B'Bx金=金B"p"1金
B"
金C'Cx金=金C"p"1金
S"
AxBxCx吼ABC
S勿a
A"
Sx勿AxBxCx
X12=p1"=p2
S'勿a'
A'
金S"p"1金=金S'Sx金
C'
Ax
S'
B'
a'
Sx
Cx
Bx
D: a=aA,
1. Rotation of plane by right angle
Draw projections
1.3. a^餻2
a^餻2
of a square with a
vertex A and a
segment BC on
line a
a"=a"
A"
X12=p1"=p2
A'
a'
D: a=aA,
1. Rotation of plane by right angle
Draw projections
1.3. a^餻2
a^餻2
of a square with a
vertex A and a
segment BC on
line a
a"=a"
2"
A"
1"
X12=p1"=p2
A'
a'
2'
1'
D: a=aA,
1. Rotation of plane by right angle
Draw projections
1.3. a^餻2
a^餻2
of a square with a
vertex A and a
金A"Ax金=金A'p'2金
segment BC on
line a
a"=a"
2"
Ax
A"
1"
X12=p1"=p2
A'
a'
2'
1'
D: a=aA,
1. Rotation of plane by right angle
Draw projections
1.3. a^餻2
a^餻2
of a square with a
vertex A and a
金A"Ax金=金A'p'2金
segment BC on
金1"1x金=金1'p'2金
line a
a"=a"
1x
2"
Ax
A"
1"
X12=p1"=p2
A'
a'
2'
1'
D: a=aA,
1. Rotation of plane by right angle
Draw projections
1.3. a^餻2
a^餻2
of a square with a
2x
vertex A and a
金A"Ax金=金A'p'2金
segment BC on
金1"1x金=金1'p'2金
line a
a"=a"
1x
金2"2x金=金2'p'2金
2"
Ax
A"
1"
X12=p1"=p2
A'
a'
2'
1'
D: a=aA,
1. Rotation of plane by right angle
Draw projections
1.3. a^餻2
a^餻2
of a square with a
2x ax
vertex A and a
金A"Ax金=金A'p'2金
segment BC on
金1"1x金=金1'p'2金
line a
a"=a"
e"A
金2"2x金=金2'p'2金
1x
1x2x=ax
2"
Ax
A"
1"
X12=p1"=p2
A'
a'
2'
1'
D: a=aA,
1. Rotation of plane by right angle
Draw projections
1.3. a^餻2
a^餻2
of a square with a
2x ax
vertex A and a
金A"Ax金=金A'p'2金
Cx
segment BC on
Bx
金1"1x金=金1'p'2金
line a
a"=a"
金2"2x金=金2'p'2金
1x
Dx
1x2x=ax
2"
Ax
AxBxCxDx吼ABCD
A"
1"
X12=p1"=p2
A'
a'
2'
1'
D: a=aA,
1. Rotation of plane by right angle
Draw projections
1.3. a^餻2
a^餻2
of a square with a
2x ax
vertex A and a
金A"Ax金=金A'p'2金
Cx
segment BC on
Bx
金1"1x金=金1'p'2金
line a
a"=a"
e"A
金2"2x金=金2'p'2金
1x
Dx
C" 1x2x=ax
2"
Ax
D"
AxBxCxDx吼ABCD
B"
A"
1"
X12=p1"=p2
A'
a'
2'
1'
D: a=aA,
1. Rotation of plane by right angle
Draw projections
1.3. a^餻2
a^餻2
of a square with a
2x ax
vertex A and a
金A"Ax金=金A'p'2金
Cx
segment BC on
Bx
金1"1x金=金1'p'2金
line a
a"=a"
e"A
金2"2x金=金2'p'2金
1x
Dx
C" 1x2x=ax
2"
Ax D"
AxBxCxDx吼ABCD
B"
A"
1"
X12=p1"=p2
D'
A'
a'
2'
C'
B'
1'
D: a=aA,
1. Rotation of plane by right angle
Draw projections
1.3. a^餻2
a^餻2
of a square with a
2x ax
vertex A and a
金A"Ax金=金A'p'2金
Cx
segment BC on
Bx
金1"1x金=金1'p'2金
line a
a"=a"
e"A
金2"2x金=金2'p'2金
1x
Dx
C" 1x2x=ax
2"
Ax
D"
AxBxCxDx吼ABCD
B"
A"
ABCD勿a
1"
X12=p1"=p2
D'
A'
~
a'
~
2'
C'
B'
1'
D: a=aA,
1. Rotation of plane by right angle
Draw projections
1.3. a^餻2
a^餻2
of a square with a
2x ax
vertex A and a
金A"Ax金=金A'p'2金
Cx
segment BC on
Bx
金1"1x金=金1'p'2金
line a
a"=a"
e"A
金2"2x金=金2'p'2金
1x
Dx
C" 1x2x=ax
2"
Ax
D"
AxBxCxDx吼ABCD
B"
A"
ABCD勿a
1"
X12=p1"=p2
D'
A'
~
a'
~
2'
C'
B'
1'

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