4cf9development of Surfaces of Solids

8/3/2019 4cf9development of Surfaces of Solids

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1.1. SECTIONS OF SOLIDS.SECTIONS OF SOLIDS.

2.2. DEVELOPMENT.DEVELOPMENT.

3.3. INTERSECTIONS.INTERSECTIONS.

ENGINEERING APPLICATIONSENGINEERING APPLICATIONS

OFOF

THE PRINCIPLESTHE PRINCIPLESOFOF

PROJECTIONS OF SOLIDES.PROJECTIONS OF SOLIDES.

STUDY CAREFULLYSTUDY CAREFULLY

THE ILLUSTRATIONS GIVEN ONTHE ILLUSTRATIONS GIVEN ON

NEXTNEXT SIX SIX PAGES !PAGES !



SECTIONING A SOLID.SECTIONING A SOLID.

An object ( here a solid ) is cut byAn object ( here a solid ) is cut by

some imaginary cutting planesome imaginary cutting plane

to understand internal details of that object.to understand internal details of that object.

The action of cutting is calledThe action of cutting is called

SECTIONINGSECTIONING a solida solid

&&

The plane of cutting is calledThe plane of cutting is called

SECTION PLANE.SECTION PLANE.

Two cutting actions means section planes are recommendedTwo cutting actions means section planes are recommended..

A) Section Plane perpendicular to Vp and inclined to Hp.A) Section Plane perpendicular to Vp and inclined to Hp.

( This is a definition of an Aux. Inclined Plane i.e. A.I.P.)( This is a definition of an Aux. Inclined Plane i.e. A.I.P.)

NOTE:NOTE:-- This section plane appearsThis section plane appearsas a straight line in FV.as a straight line in FV.

B) Section Plane perpendicular to Hp and inclined to Vp.B) Section Plane perpendicular to Hp and inclined to Vp.

( This is a definition of an Aux. Vertical Plane i.e. A.V.P.)( This is a definition of an Aux. Vertical Plane i.e. A.V.P.)

NOTE:NOTE:-- This section plane appearsThis section plane appears

as a straight line in TV.as a straight line in TV.Remember:Remember:--1. After launching a section plane1. After launching a section plane

either in FV or TV, the part towards observereither in FV or TV, the part towards observeris assumed to be removed.is assumed to be removed.

2. As far as possible the smaller part is2. As far as possible the smaller part is

assumed to be removed.assumed to be removed.

OBSE

RVER OBSERVER

ASSUMEASSUME

UPPER PARTUPPER PART

REMOVEDREMOVED

OBSERVER OBSERVER

ASSUMEASSUME

LOWER PARTLOWER PART

REMOVEDREMOVED

(A)(A)

(B

)(B

)



ILLUSTRATION SHOWINGILLUSTRATION SHOWING

IMPORTANT TERMSIMPORTANT TERMS

IN SECTIONING.IN SECTIONING.

xx yy

TRUE SHAPETRUE SHAPE

Of SECTIONOf SECTION

SECTIONSECTION

PLANEPLANE

SECTION LINESSECTION LINES

(45(4500 to XY)to XY)

Apparent ShapeApparent Shape

of sectionof section

SECTIONAL T.V.SECTIONAL T.V.

For TVFor TV



Section PlaneSection Plane

Through ApexThrough Apex


Through GeneratorsThrough Generators

Section Plane ParallelSection Plane Parallel

to end generator.to end generator.


Parallel to Axis.Parallel to Axis.

TriangleTriangle EllipseEllipse

HyperbolaHyperbola

EllipseEllipse

Cylinder throughCylinder through

generators.generators.

Sq. Pyramid throughSq. Pyramid through

all slant edgesall slant edges

TrapeziumTrapezium

Typical Section PlanesTypical Section Planes

&&

Typical ShapesTypical Shapes

Of Of SectionsSections..



DEVELOPMENT OF SURFACES OF SOLIDS.DEVELOPMENT OF SURFACES OF SOLIDS.

MEANING: MEANING:--

ASSUME OBJECT HOLLOW AND MADEASSUME OBJECT HOLLOW AND MADE--UP OF THIN SHEET. CUT OPEN IT FROM ONE SIDE ANDUP OF THIN SHEET. CUT OPEN IT FROM ONE SIDE AND

UNFOLD THE SHEET COMPLETELY. THEN THEUNFOLD THE SHEET COMPLETELY. THEN THE SHAPE OF THAT UNFOLDED SHEET IS CALLEDSHAPE OF THAT UNFOLDED SHEET IS CALLED

DEVELOPMENT OF LATERLAL SUEFACESDEVELOPMENT OF LATERLAL SUEFACES OF THAT OBJECT OR SOLID.OF THAT OBJECT OR SOLID.

LATERLAL SURFACELATERLAL SURFACE IS THE SURFACE EXCLUDING SOLID¶S TOP & BASE.IS THE SURFACE EXCLUDING SOLID¶S TOP & BASE.

ENGINEERING APLICATION ENGINEERING APLICATION ::

THERE ARE SO MANY PRODUCTS OR OBJECTS WHICH ARE DIFFICULT TO MANUFACTURE BYTHERE ARE SO MANY PRODUCTS OR OBJECTS WHICH ARE DIFFICULT TO MANUFACTURE BY

CONVENTIONAL MANUFACTURING PROCESSES, BECAUSE OF THEIR SHAPES AND SIZES.CONVENTIONAL MANUFACTURING PROCESSES, BECAUSE OF THEIR SHAPES AND SIZES.

THOSE ARE FAB

RICATED IN SHEET METAL INDUSTRYB

Y USINGTHOSE ARE FAB

RICATED IN SHEET METAL INDUSTRYB

Y USINGDEVELOPMENT TECHNIQUE. THERE IS A VAST RANGE OF SUCH OBJECTS.DEVELOPMENT TECHNIQUE. THERE IS A VAST RANGE OF SUCH OBJECTS.

EXAMPLES: EXAMPLES:--

Boiler Shells & chimneys, Pressure Vessels, Shovels, Trays, Boxes & Cartons, Feeding Hoppers,Boiler Shells & chimneys, Pressure Vessels, Shovels, Trays, Boxes & Cartons, Feeding Hoppers,

Large Pipe sections, Body & Parts of automotives, Ships, Aeroplanes and many more.Large Pipe sections, Body & Parts of automotives, Ships, Aeroplanes and many more.

WHAT ISWHAT IS

OUR OBJECTIVEOUR OBJECTIVEIN THIS TOPIC ?IN THIS TOPIC ?

To learn methods of development of surfaces of To learn methods of development of surfaces of

different solids, their sections and frustumsdifferent solids, their sections and frustums..

1. Development is different drawing than PROJECTIONS.1. Development is different drawing than PROJECTIONS.

2. It is a shape showing AREA, means it¶s a 22. It is a shape showing AREA, means it¶s a 2--D plain drawing.D plain drawing.

3. Hence all dimensions of it must be TRUE dimensions.3. Hence all dimensions of it must be TRUE dimensions.

4. As it is representing shape of an un4. As it is representing shape of an un--folded sheet, no edges can remain hiddenfolded sheet, no edges can remain hidden

And hence DOTTED LINES are never shown on development.And hence DOTTED LINES are never shown on development.

But before going ahead, But before going ahead,

note following note following

Important Important points points..

Study illustrations given on next page carefully.Study illustrations given on next page carefully.



TD

H

D

SS

H

U

U = R L 3600

R=Base circle radius.L=Slant height.

L= Slant edge.

S = Edge of base

H= Height S = Edge of base

H= Height D= base diameter

Development of lateral surfaces of different solids.

(Lateral surface is the surface excluding top & base)

P risms: No.of Rectangles

Cylinder : A RectangleCone: (Sector of circle) P yramids: (No.of triangles)

Tetrahedron: Four Equilateral Triangles

All sides

equal in length

Cube: Six Squares.



U

U = R L

3600

R= Base circle radius of coneL= Slant height of cone

L1 = Slant height of cut part.

Base side

Top side

L= Slant edge of pyramid

L1 = Slant edge of cut part.

DEVELOPMENT OF

FRUSTUM OF CONE

DEVELOPMENT OF

FRUSTUM OF SQUARE PYRAMID

STUDY NEXTSTUDY NEXT N I NE N I NE PROBLEMS OFPROBLEMS OF

SECTIONS & DEVELOPMENTSECTIONS & DEVELOPMENT

FRUSTUMSFRUSTUMS



X Y

X1

Y1

A

B

C

E

D

a

e

d

b

c

A B C D E A

DEVELOPMENT

a´

b´

c´d´

e´

Problem 1:Problem 1: A pentagonal prism , 30 mm base side & 50 mm axisA pentagonal prism , 30 mm base side & 50 mm axis

is standing on Hp on it¶s base with one side of the base perpendicular to VP.is standing on Hp on it¶s base with one side of the base perpendicular to VP.

It is cut by a section plane inclined at 45It is cut by a section plane inclined at 45ºº to the HP, through mid point of axis.to the HP, through mid point of axis.

Draw Fv, sec.Tv & sec. Side view. Also draw true shape of section andDraw Fv, sec.Tv & sec. Side view. Also draw true shape of section and

Development of surface of remaining solid.Development of surface of remaining solid.

Solution Steps:Solution Steps: for sectional views: for sectional views:

Draw three views of standing prism.Draw three views of standing prism.

Locate sec.plane in Fv as described.Locate sec.plane in Fv as described.

Project points where edges are gettingProject points where edges are getting

Cut on Tv & Sv as shown in illustration.Cut on Tv & Sv as shown in illustration.

Join those points in sequence and showJoin those points in sequence and show

Section lines in it.Section lines in it.

Make remaining part of solid dark.Make remaining part of solid dark.

For True Shape:For True Shape:Draw xDraw x11yy11 // to sec. plane// to sec. plane

Draw projectors on it fromDraw projectors on it from

cut points.cut points.

Mark distances of pointsMark distances of points

of Sectioned part from Tv,of Sectioned part from Tv,

on above projectors fromon above projectors from

xx11yy11 and join in sequence.and join in sequence.

Draw section lines in it.Draw section lines in it.It is required true shape.It is required true shape.

For Development:For Development:Draw development of entire solid. Name fromDraw development of entire solid. Name from

cutcut--open edge I.e. A. in sequence as shown.open edge I.e. A. in sequence as shown.

Mark the cut points on respective edges.Mark the cut points on respective edges.

Join them in sequence in st. lines.Join them in sequence in st. lines.

Make existing parts dev.dark.Make existing parts dev.dark.



Q 14.11: A square pyramid, base 40 mm side and axis 65 mm long, has its base on the HP and all

the edges of the base equally inclined to the VP. It is cut by a section plane, perpendicular to the

VP, inclined at 45º to the HP and bisecting the axis. Draw its sectional top view, sectional side

view and true shape of the section. Also draw its development.

X45º

a

b

c

d

o

a¶b¶

c¶d¶

o¶

1

2

3

4

1¶

2¶

3¶

4¶11

41

21 31

Y

A

B

C

D

A

O

1

1

2

3

4



Q 15.17: A square pyramid, base 40 mm side and axis 65 mm long, has its base on the HP with

two edges of the base perpendicular to the VP. It is cut by a section plane, perpendicular to the

VP, inclined at 45º to the HP and bisecting the axis. Draw its sectional top view and true shape of

the section. Also draw its development.

X

o¶

Y

A

B

C

D

A

O

a b

cd

o

a¶ d¶ b¶ c¶

1

2

3

4

1¶ 4¶

2¶ 3¶

2

3

1

2True length

of slant

edge

1 4

1

1

4

2 3

2

3True length

of slant

edge



Q 14.14: A pentagonal pyramid , base 30mm side and axis 60 mm long is lying on one of its triangular faces

on the HP with the axis parallel to the VP. A vertical section plane, whose HT bisects the top view of the axis

and makes an angle of 30º with the reference line, cuts the pyramid removing its top part. Draw the top view,

sectional front view and true shape of the section and development of the surface of the remaining portion of

the pyramid.

YXa¶ b¶ e¶ c¶ d¶

a

b

c

d

e

o

o¶

6 0

c¶d¶ o¶

a¶

b¶e¶

3 0

a1

b1

c1

d1

e1

o1

1¶

2¶

3¶4¶

5¶

6¶

1

2

3

4

5

631¶

41¶

21¶

11¶

61¶

51¶

O

A

BC

D

E

A

12

3

4

5

6

1

5

6



X Y

1

2

34

5

6

7

8

910

11

12

Q 15.26: draw the projections of a cone resting on the ground on its base and show on them, the shortest path

by which a point P, starting from a point on the circumference of the base and moving around the cone will

return to the same point. Base of cone 65 mm diameter ; axis 75 mm long.

1¶ 2¶12¶

3¶11¶

4¶10¶

5¶9¶

6¶8¶ 7¶

2

3

4

5

6

7

8

9

10

11

12

1

=143º

O¶ =r/LX

360º

=32.5/81.74 X 360º

= 143º



Q 14.24: A right circular cone, base 25 mm radius and height 65 mm rests on its base on H.P. It is cut by a

section plane perpendicular to the V.P., inclined at 45º to the H.P. and bisecting the axis. Draw the projections

of the truncated cone and develop its lateral surface.

X Y

1

2

34

5

6

7

8

910

11

12

1 212

311

4 10

5 9

68 7

2

3

4

5

6

7

8

9

10

11

121

a

b

c

k

d

ef

g

h

il j

a

f

b

ck

d

e

g

hi

l

j

A

C

D

E

B

A

F

G

H

I

J

K

L

=103º



o¶

h

a

b

c

d

g

f

o e

a¶ b¶ c¶ g¶ d¶f¶ e¶h¶X Y

U = R L

3600


U

A

B

C

DE

F

G

H

AO

1

3

2

4

7

6

5

L

11

22

33

44

55

66

77

1¶1¶

2¶2¶

3¶3¶ 4¶4¶5¶5¶

6¶6¶

7¶7¶

Problem 6:Problem 6: Draw a semicircle 0f 100 mm diameter and inscribe in it a largestDraw a semicircle 0f 100 mm diameter and inscribe in it a largest

circle.If the semicircle is development of a cone and inscribed circle is somecircle.If the semicircle is development of a cone and inscribed circle is some

curve on it, then draw the projections of cone showing that curve.curve on it, then draw the projections of cone showing that curve.

Solution Steps:Solution Steps:

Draw semicircle of given diameter, divide it in 8 Parts and inscribe in it

Draw semicircle of given diameter, divide it in 8 Parts and inscribe in ita largest circle as shown.Name intersecting points 1, 2, 3 etc.a largest circle as shown.Name intersecting points 1, 2, 3 etc.

Semicircle being dev.of a cone it¶s radius is slant height of cone.( L )Semicircle being dev.of a cone it¶s radius is slant height of cone.( L )

Then using above formula find R of base of cone. Using this dataThen using above formula find R of base of cone. Using this data

draw Fv & Tv of cone and form 8 generators and name.draw Fv & Tv of cone and form 8 generators and name.

Take oTake o --1 distance from dev.,mark on TL i.e.o¶a¶ on Fv & bring on o¶b¶1 distance from dev.,mark on TL i.e.o¶a¶ on Fv & bring on o¶b¶

and name 1¶ Similarly locate all points on Fv. Then project all on Tvand name 1¶ Similarly locate all points on Fv. Then project all on Tv

on respective generators and join by smooth curve.on respective generators and join by smooth curve.

TO DRAW PRINCIPALTO DRAW PRINCIPAL

VIEWS FROM GIVENVIEWS FROM GIVEN

DEVELOPMENT.DEVELOPMENT.



Q.15.11: A right circular cylinder, base 50 mm diameter and axis 60 mm long, is standing on HP on its

base. It has a square hole of size 25 in it. The axis of the hole bisects the axis of the cylinder and is

perpendicular to the VP. The faces of the square hole are equally inclined with the HP.Draw its

projections and develop lateral surface of the cylinder.

Y

1

2

34

5

6

7

8

9

10

11

12

X

1¶

2¶

12¶

3¶

11¶

4¶

10¶5¶

9¶

6¶

8¶ 7¶

a¶

b¶

c¶

d¶

1 2 3 4 5 6 7 8 9 10 11 12 1

a

a

b d

b d

c

c

A

B

D

C C

B

D

A

a c



Q.15.21: A frustum of square pyramid has its base 50 mm side, top 25 mm side and axis 75 mm. Draw

the development of its lateral surface. Also draw the projections of the frustum (when its axis is vertical

and a side of its base is parallel to the VP), showing the line joining the mid point of a top edge of one

face with the mid point of the bottom edge of the opposite face, by the shortest distance.

YX

50 25

75

a b

cd

a1 b1

c1d1

a¶

d¶b¶

c¶

a1 ¶d1¶

b1 ¶c1¶

o

o¶

True

length of

slant

edge

A1

B1

C1

D1

A1

A

B

C

D

A

P

Q

R

S

p¶

p

q¶

q

r¶

r

s¶

s



Q: A square prism of 40 mm edge of the base and 65 mm height stands on its base on the HP with

vertical faces inclined at 45º with the VP. A horizontal hole of 40 mm diameter is drilled centrally

through the prism such that the hole passes through the opposite vertical edges of the prism, draw

the development of the surfaces of the prism.

YX

a

b

c

d

a¶ b¶d¶ c¶

a¶ b¶d¶ c¶

1¶

2¶

3¶4¶

5¶

6¶

7¶

8¶

9¶

10¶11¶

12¶

1

1

2

12

2

12

3

11

3

11

4 10

4 10

5

9

5

9

6

8

6

8

1 2

123

114

10AB

C

7

7

5

9

6

87 6

8

5

94

10

7 12

12

3

11 A

1

2

12

11

3

10

4

9

5

8

6

7 7

6

8

9 11

3

12

1

54

2

10

D



Y

h

a

b

c

d

e

g

f

X a¶ b¶ d¶ e¶c¶ g¶ f¶h¶

o¶

X1

Y1

g´ h´f´ a´e´ b´d´ c´

A

B

C

D

E

F

A

G

H

SECTIONAL T.V

SECTIONAL S.V

DEVELOPMENT

Problem 2:Problem 2: A cone, 50 mm base diameter and 70 mm axis is A cone, 50 mm base diameter and 70 mm axis is

standing on it¶s base on Hp. It cut by a section plane 45standing on it¶s base on Hp. It cut by a section plane 4500 inclinedinclined

to Hp through base end of end generator.Draw projections,to Hp through base end of end generator.Draw projections,

sectional views, true shape of section and development of surfacessectional views, true shape of section and development of surfaces

of remaining solid.of remaining solid.

Solution Steps:Solution Steps:for sectional views:for sectional views:

Draw three views of standing cone.Draw three views of standing cone.

Locate sec.plane in Fv as described.Locate sec.plane in Fv as described.

Project points where generators areProject points where generators are

getting Cut on Tv & Sv as shown ingetting Cut on Tv & Sv as shown in

illustration.Join those points inillustration.Join those points in

sequence and show Section lines in it.sequence and show Section lines in it.Make remaining part of solid dark.Make remaining part of solid dark.

For True Shape:For True Shape:Draw xDraw x11yy11 // to sec. plane// to sec. plane

Draw projectors on it fromDraw projectors on it from

cut points.cut points.

Mark distances of pointsMark distances of points

of Sectioned part from Tv,of Sectioned part from Tv,

on above projectors fromon above projectors from

xx11yy11 and join in sequence.and join in sequence.

Draw section lines in it.Draw section lines in it.It is required true shape.It is required true shape.

For Development:For Development:

Draw development of entire solid.Draw development of entire solid.

Name from cutName from cut--open edge i.e. A.open edge i.e. A.

in sequence as shown.Mark the cutin sequence as shown.Mark the cut

points on respective edges.points on respective edges.

Join them in sequence in curvature.Join them in sequence in curvature.

Make existing parts dev.dark.Make existing parts dev.dark.



X Ye¶a¶ b¶ d¶c¶ g¶ f¶h¶

o¶

h

a

b

c

d

e

g

f

O

DEVELOPMENT

A

B

C

D

E

F

A

G

H

OO

1122

33

44

66 5577

1¶1¶

2¶2¶

3¶3¶

4¶4¶

5¶5¶

6¶6¶

7¶7¶

11

22

33

44

556677

HELIX CURVEHELIX CURVE

Problem 9:Problem 9: A particle which is initially on base circle of a cone, standingA particle which is initially on base circle of a cone, standing

on Hp, moves upwards and reaches apex in one complete turn around the cone.on Hp, moves upwards and reaches apex in one complete turn around the cone.

Draw it¶s path on projections of cone as well as on it¶s development.Draw it¶s path on projections of cone as well as on it¶s development.

Take base circle diameter 50 mm and axis 70 mm long.Take base circle diameter 50 mm and axis 70 mm long.

It¶s a construction of curveIt¶s a construction of curveHelix of one turn on coneHelix of one turn on cone::Draw Fv & Tv & dev.as usualDraw Fv & Tv & dev.as usual

On all form generators & name.On all form generators & name.

Construction of curve H elix::Construction of curve H elix::

Show 8 generators on both viewsShow 8 generators on both views

Divide axis also in same parts.Divide axis also in same parts.

Draw horizontal lines from thoseDraw horizontal lines from those

points on both end generators.points on both end generators.1¶ is a point where first horizontal1¶ is a point where first horizontal

Line & gen. b¶o¶ intersect.Line & gen. b¶o¶ intersect.

2¶ is a point where second horiz.2¶ is a point where second horiz.

Line & gen. c¶o¶ intersect.Line & gen. c¶o¶ intersect.

In this way locate all points on Fv.In this way locate all points on Fv.

Project all on Tv.Join in curvature.Project all on Tv.Join in curvature.

For Development:For Development:

Then taking each points trueThen taking each points trueDistance From resp.generator Distance From resp.generator

from apex, Mark on developmentfrom apex, Mark on development

& join.& join.



o¶

h

a

b

c

d

g

f

o e

a¶ b¶ c¶ g¶ d¶f¶ e¶h¶X Y

U = R L

3600


U

A

B

C

DE

F

G

H

AO

1

3

2

4

7

6

5

L

11

22

33

44

55

66

77

1¶1¶

2¶2¶

3¶3¶ 4¶4¶5¶5¶

6¶6¶

7¶7¶

Problem 6:Problem 6: Draw a semicircle 0f 100 mm diameter and inscribe in it a largest Draw a semicircle 0f 100 mm diameter and inscribe in it a largest circle.If the semicircle is development of a cone and inscribed circle is somecircle.If the semicircle is development of a cone and inscribed circle is somecurve on it, then draw the projections of cone showing that curve.curve on it, then draw the projections of cone showing that curve.


Draw semicircle of given diameter, divide it in 8 Parts and inscribe in it

Draw semicircle of given diameter, divide it in 8 Parts and inscribe in ita largest circle as shown.Name intersecting points 1, 2, 3 etc.a largest circle as shown.Name intersecting points 1, 2, 3 etc.

Semicircle being dev.of a cone it¶s radius is slant height of cone.( L )Semicircle being dev.of a cone it¶s radius is slant height of cone.( L )

Then using above formula find R of base of cone. Using this dataThen using above formula find R of base of cone. Using this data

draw Fv & Tv of cone and form 8 generators and name.draw Fv & Tv of cone and form 8 generators and name.

Take oTake o --1 distance from dev.,mark on TL i.e.o¶a¶ on Fv & bring on o¶b¶1 distance from dev.,mark on TL i.e.o¶a¶ on Fv & bring on o¶b¶

and name 1¶ Similarly locate all points on Fv. Then project all on Tvand name 1¶ Similarly locate all points on Fv. Then project all on Tv

on respective generators and join by smooth curve.on respective generators and join by smooth curve.






h

a

b

c

d

g

f

e

o¶

a¶ b¶ d¶c¶ g¶ f¶h¶ e¶X Y

A

B

C

DE

F

G

H

AO L

U = R L

3600


U1¶1¶

2¶2¶ 3¶3¶

4¶4¶

5¶5¶6¶6¶

7¶7¶

1122

33

44

55

6677

Problem 7:Problem 7:Draw a semicircle 0f 100 mm diameter and inscribe in it a largestDraw a semicircle 0f 100 mm diameter and inscribe in it a largest

rhombusrhombus.If the semicircle is development of a cone and rhombus is some curve.If the semicircle is development of a cone and rhombus is some curve

on it, then draw the projections of cone showing that curve.on it, then draw the projections of cone showing that curve.





Similar to previousSimilar to previous

Problem:Problem:



A.V.P300 inclined to VpThrough mid-point of axis.

X Y1

2

3 4

5

6

78

b¶ f¶a¶ e¶c¶ d¶

a

b

c

d

e

f

a1

d1 b1

e1

c1

f 1

X1

Y1

AS SECTION PLANE IS IN T.V.,

CUT OPEN FROM BOUNDRY EDGE C 1 FOR DEVELOPMENT.

C D E F A B C

DEVELOPMENT

SECTIONAL F.V.

Problem 4:Problem 4: A hexagonal prism. 30 mm base side & A hexagonal prism. 30 mm base side &

55 mm axis is lying on Hp on it¶s rect.face with axis55 mm axis is lying on Hp on it¶s rect.face with axis

// to Vp. It is cut by a section plane normal to Hp and// to Vp. It is cut by a section plane normal to Hp and

303000 inclined to Vp bisecting axis.inclined to Vp bisecting axis.

Draw sec. Views, true shape & development.Draw sec. Views, true shape & development.

Use similar steps for sec.views & true shape.Use similar steps for sec.views & true shape.NOTE:NOTE: for development, always cut open object fromfor development, always cut open object from

From an edge in the boundary of the view in whichFrom an edge in the boundary of the view in which

sec.plane appears as a line.sec.plane appears as a line.

Here it is Tv and in boundary, there is c1 edge.HenceHere it is Tv and in boundary, there is c1 edge.Hence

it is opened from c and named C,D,E,F,A,B,C.it is opened from c and named C,D,E,F,A,B,C.

NoteNote the steps to locatethe steps to locate

Points 1, 2 , 5, 6 in sec.Fv:Points 1, 2 , 5, 6 in sec.Fv:

Those are transferred toThose are transferred to

11stst TV, then to 1TV, then to 1stst Fv andFv and

Then on 2Then on 2ndnd Fv.Fv.



1¶

2¶

3¶

4¶

5¶

6¶

7¶

7

1

5

4

3

2

6

7

16

5

4

32

a

b

c

d

e

f

g

4

4 5

3

6

2

7

1

A

B

C

D

E

A

F

G

O

O¶

d¶e¶ c¶f¶ g¶b¶ a¶X Y

X1

Y1

F.V.

SECTIONAL

TOP VIEW.

Problem 5:A solid composed of a half-cone and half- hexagonal pyramid is

shown in figure.It is cut by a section plane 450 inclined to Hp, passing through

mid-point of axis.Draw F.v., sectional T.v.,true shape of section and

development of remaining part of the solid.

( take radius of cone and each side of hexagon 30mm long and axis 70mm.)

Note:Note:Fv & TV 8f two solidsFv & TV 8f two solids

sandwichedsandwiched

Section lines style in both:Section lines style in both:

Development of Development of

half cone & half pyramid:half cone & half pyramid:



a¶a¶ b¶ b¶ c¶c¶ d¶d¶

o¶o¶

e¶e¶

aa

b b

cc

dd

oo ee

XX YY

AA

BB

CC

DD

EE

AA

OO

22

33

44

11

Problem 8:Problem 8: A half cone of 50 mm base diameter, 70 mm axis, is standing on it¶s half base on HP with it¶s flat faceA half cone of 50 mm base diameter, 70 mm axis, is standing on it¶s half base on HP with it¶s flat face

parallel and nearer to VP.An inextensible string is wound round it¶s surface from one point of base circle and parallel and nearer to VP.An inextensible string is wound round it¶s surface from one point of base circle and

brought back to the same point.If the string is of brought back to the same point.If the string is of shorte st lengt h shorte st lengt h, find it and show it on the projections of the cone., find it and show it on the projections of the cone.

11 22

33

44

1¶1¶

2¶2¶ 3¶3¶ 4¶4¶

TO DRAW A CURVE ONTO DRAW A CURVE ON

PRINCIPAL VIEWSPRINCIPAL VIEWSFROM DEVELOPMENT.FROM DEVELOPMENT. Concept:Concept: A string wound A string wound

from a point up to the samefrom a point up to the same

Point, of shortest lengthPoint, of shortest length

Must appear st. line on it¶sMust appear st. line on it¶s

Development.Development.

Solution steps:Solution steps:

Hence draw development,Hence draw development,

Name it as usual and joinName it as usual and join

A to A This is shortest A to A This is shortest

Length of that string.Length of that string.

Further steps are as usual.Further steps are as usual.

On dev. Name the points of On dev. Name the points of

Intersections of this line withIntersections of this line with

Different generators.BringDifferent generators.Bring

Those on Fv & Tv and joinThose on Fv & Tv and join

by smooth curves.by smooth curves.

Draw 4¶ a¶ part of string dottedDraw 4¶ a¶ part of string dotted

As it is on back side of cone. As it is on back side of cone.



X Ye¶a¶ b¶ d¶c¶ g¶ f¶h¶

o¶

o¶

Problem 3: A cone 40mm diameter and 50 mm axis is resting on one generator on Hp( lying on Hp)

which is // to Vp.. Draw it¶s projections.It is cut by a horizontal section plane through it¶s base

center. Draw sectional TV, development of the surface of the remaining part of cone.

A

B

C

D

E

F

A

G

H

O

a1

h1

g1

f 1

e1

d1

c1

b1

o1

SECTIONAL T.V

DEVELOPMENT

(SHOWING TRUE SHAPE OF SECTION)

HORIZONTAL

SECTION PLANE

h

a

b

c

d

e

g

f

O

F ollow similar solution steps for S ec.views F ollow similar solution steps for S ec.views - - T rue shape T rue shape ± ± Development as per previous problem! Development as per previous problem!

4cf9development of Surfaces of Solids

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