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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

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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

//

//

**

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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 :

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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

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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

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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 . .

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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 . .

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xx

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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 . .

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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 // //

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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

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hh

kk

mm

mm

xx

kk

SECOND PROBLEM : Parallelism of two planesSECOND PROBLEM : Parallelism of two planes

THEOREMTHEOREM : :

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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

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MM

aa bb

aabb

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aabb

bbaa

cc

cc

iii) The plane is given by its tracesiii) The plane is given by its traces

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MM

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vv

hh

vv

hh

xx

xx

xx

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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

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H.PH.P..

VV . . PP..

THIRD PROBLEM: INTERSECTION OF TWOTHIRD PROBLEM: INTERSECTION OF TWO

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VV

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VV

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Some special casesSome special cases

One of the two planes is verticalOne of the two planes is vertical::

xx

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== 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

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rr

iii- One of the two planes is iii- One of the two planes is parallel to x- axisparallel to x- axis

xx

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vv

hhhh

rr

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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

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oo

Vi- One of the two planes is frontalVi- One of the two planes is frontal

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XX

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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 . .

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rr

vvvv

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vv

HH

vv

vv

hh

vv

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HH HH

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viii) Both vertical andviii) Both vertical andHorizontal tracesHorizontal traces

do not intersectdo not intersect . .

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hh

vv

hh

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SS

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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

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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

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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

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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

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vv

mm

mm

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xx