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The Main MenuThe Main Menu PreviousNextاPreviousا
Using the successive auxiliary projection, construct the development of the given regular oblique tetragonal prism ABCD A*B*C*D* whose base ABCD is a square 1π
B1
B*2 D*2A*2 C*2
x12B*1
A*1
D*1
C*1C1
D1
A1
C2B2D2A2
The Main MenuThe Main Menu PreviousNextاPreviousا
B1
C2A2 D2
B*2 D*2A*2 C*2
B*1
A*1
D*1
C*1C1
D1
A 1
B2 x12
x13C3A3 D3 B3
C*3A*3 B*3 D*3
x35
C5 = C*5
A5 = A*5
B5 = B*5
D5 = D*5
//
//
**
The Main MenuThe Main Menu PreviousاPreviousا
A*3
B*3D*3
//
C*3
x35
C5 = C*5
A5 = A*5
B5 = B*5
D5 = D*5
* *//
x13C3A3 D3 B3
B*
B
C*
C
D*
D
//
*
//
*
A*
A
A*
B*
..
A*
A
THE CIRCLE
•The orthogonal projection of a circle :
AA
BB
CCDD
CCDD
AA
BB
rrrr
rr
SS
ssSS
AAAABB
AA BB
CC
CC
DDDD
SS
SS
AB is aAB is a diameterdiameter ParallelParallel
tt oo thethe
Plane ofPlane of
ProjectionProjection. . . .
CD is a diamCD is a diam. . normalnormal
toto
ABAB . .
REMARKREMARK
bb
aa
MM
aa
ffii
nndd
To find the length of the To find the length of the semi minor axis if the semi minor axis if the
major axis and a point M major axis and a point M on the ellipse are givenon the ellipse are given
{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{{
xx
rr
rr
rr
SS==AA BB ==
Example 1Example 1 Represent a circle Represent a circle
lying in a plane lying in a plane perpendicular with perpendicular with
V.P. if its centre and V.P. if its centre and its radius are givenits radius are given . . SS
SS
EXA MPLE 2EXA MPLE 2
Represent a circle lying in a plane perpendicular Represent a circle lying in a plane perpendicular with S.P. ( i.e. parallel to the x-axis ) if its centre with S.P. ( i.e. parallel to the x-axis ) if its centre
and its radius are givenand its radius are given . .
SS SSrr
SS
xx
rr
SS
SS
SSSS
SS
rr
rr
rr
rr
xxOO
EXAMPLE 3EXAMPLE 3
Construct a circle lying in a plane (-7,8,6) ,its centre Construct a circle lying in a plane (-7,8,6) ,its centre s (1,4,?) and its rsdius is of length 3.5 cmss (1,4,?) and its rsdius is of length 3.5 cms . .
vv
hhss
ss
xx
SS
SS
POSITION PROBLEMS
11 . . INCIDENCEINCIDENCE
A point lying on a straight line. A point lying in a A point lying on a straight line. A point lying in a plane. A straight line lying ln a planeplane. A straight line lying ln a plane . .
MM
mm MM MM
mm
The position problems deal withThe position problems deal with : :
CHAPTER 8CHAPTER 8
22. . ParallelismParallelism : :
A straight liine is parallel to another straight lineA straight liine is parallel to another straight line , ,
a straight line is parallel to a planea straight line is parallel to a plane , ,
a plane is parallel to a given planea plane is parallel to a given plane . .
mm
33 . . IntersectionIntersection
The point of intersection of a straight line and a The point of intersection of a straight line and a plane . The straight line of intersection of two plane . The straight line of intersection of two
different planesdifferent planes
mm // //
mmmmrrMM
RR
R = mR = m rr = =
FIRST PROBLEM : Parallelism of a straight line and a FIRST PROBLEM : Parallelism of a straight line and a planeplane
THEOREMTHEOREM::A straight line m is parallel to a given plane iff m A straight line m is parallel to a given plane iff m
is parallel to a straight line lying in the given planeis parallel to a straight line lying in the given plane..
In figure the straight line k is lying in the planeIn figure the straight line k is lying in the plane The straight line m is parallel to the straight line kThe straight line m is parallel to the straight line k
vv
hh
kk
mm
mm
xx
kk
SECOND PROBLEM : Parallelism of two planesSECOND PROBLEM : Parallelism of two planes
THEOREMTHEOREM : :
aa
bb
bb
aa
A plane is said to be parallel to A plane is said to be parallel to another plane iff the plane another plane iff the plane
contains two intersecting contains two intersecting straight lines a and b, each of straight lines a and b, each of
them is parallel to the planethem is parallel to the plane. .
Given a plane and a point M out side it. It is Given a plane and a point M out side it. It is required to construct a plane passing through M required to construct a plane passing through M
and parallel to the given planeand parallel to the given plane
ii((
The plane is given by two intersecting str. LinesThe plane is given by two intersecting str. Lines
MM
MM
aa
aa
bb
bb
a and ba and b
ii) The plane is given by two parallel str. Lines a &bii) The plane is given by two parallel str. Lines a &b
MM
MM
aa bb
aabb
MM
MM
aabb
bbaa
cc
cc
iii) The plane is given by its tracesiii) The plane is given by its traces
vv
hh
MM
MM
MM
MM
vv
hh
vv
hh
xx
xx
xx
xx
The plane is perpendicular withThe plane is perpendicular with
The plane is perpendicular withThe plane is perpendicular with
The plane is parallel to x-axisThe plane is parallel to x-axis
vv
hh
MM
MM
vv
hh
xx
MM
MM
vv
hh
MM
MM
vv
hh
vvvv
vv
xx
hhhh
MM
MM
ss
vv
hh
MM
MM
xx
vv
hh
vv
hh
MM
MM
xx
MM
H.PH.P..
VV . . PP..
THIRD PROBLEM: INTERSECTION OF TWOTHIRD PROBLEM: INTERSECTION OF TWO
vvvv
hhhh
rr
vv
hh
vv
hh
VV
HH
rr
rr
VV
HH
rr
rr
Some special casesSome special cases
One of the two planes is verticalOne of the two planes is vertical::
xx
hh hh
vv vvrr
== rr
ii --
PLANESPLANES
Ii- One of the two planes is Ii- One of the two planes is perpendicular with V . Pperpendicular with V . P.. vv
vv
v=rv=r
hh
rr
iii- One of the two planes is iii- One of the two planes is parallel to x- axisparallel to x- axis
xx
vv
vv
hhhh
rr
rr
xx
Iv- The two planes are parallel to the x- axisIv- The two planes are parallel to the x- axis
vvvv
hh
hh
rr rr
rr
ss ssXX
xx
v- One of the two planes is horizontalv- One of the two planes is horizontal
vvvvvv == rr
rrrrhh
oo
Vi- One of the two planes is frontalVi- One of the two planes is frontal
hh
XX
rrrrrrrrvv
vv
hh
Vii- Two traces do not Vii- Two traces do not intersect intersect
We use an We use an auxiliary auxiliary
frontal or frontal or horizontal horizontal plane to plane to
find one find one point of point of
intersectionintersection . .
hh hh
vvvv vvvvvv
rr
vvvv
RR
RR
vv
HH
vv
vv
hh
vv
hh
rr
rr
HH HH
RR
RR
viii) Both vertical andviii) Both vertical andHorizontal tracesHorizontal traces
do not intersectdo not intersect . .
HHvv
hh
vv
hh
RR
RR
SS
SS
rr
rr
vv
hh
EXAMPLEEXAMPLE Construct the line of intersection of a plane given by Construct the line of intersection of a plane given by two intersecting str. Lines a&b with a plane given by two intersecting str. Lines a&b with a plane given by
two parallel str. Lines c& dtwo parallel str. Lines c& d . .
aa
bb
aa
bb
ccdd
dd cc
vv
vvrr
rr
SS
SS
RR
RR
Given a straight line mGiven a straight line m
mm
mm
xx
i. To pass a vertical plane i. To pass a vertical plane through the straight line mthrough the straight line m
vv
hh
mm
mm
ii. To pass a plane normal to ii. To pass a plane normal to V.P. through mV.P. through m
vv
hh
mm
mm
xx
xx
EXAMPLEEXAMPLE
FOURTH PROBLEM: POINT OF INTERSECTION OF A FOURTH PROBLEM: POINT OF INTERSECTION OF A STRAIGHT LINE m AND A PLANESTRAIGHT LINE m AND A PLANE
i) The plane is in a special positioni) The plane is in a special position: :
11 - -The plane isThe plane is horizontalhorizontal
22 . .The plane is frontalThe plane is frontal
mm
mm
mmhh
xx
mm
mm
vvRR
RR
RR
RR
33 . .The plane is verticalThe plane is vertical
44 . .The plane is normal to V.PThe plane is normal to V.P..
mm
mmxx
RR
RRhh
vv
mm
mm
vv
hh
xx