UNIT 4
SECTION OF SOLIDS
Position
of section plane
·
S.P
Perpendicular to HP and Parallel to VP
·
S.P
Perpendicular to VP and Parallel to HP
·
S.P
Perpendicular to VP and Inclined to HP
·
S.P
Perpendicular to HP and Inclined to VP
·
S.P
Perpendicular to both HP and VP
Position
of Solids
·
Axis
of Solid Perpendicular to HP
·
Axis
of Solid Perpendicular to VP
PROBLEMS
1) S.P
Perpendicular to HP and Parallel to VP
A hexagonal prism, side of
base 30 mm and axis 60 mm long, rests with its bas on HP such that one of its
rectangular faces is parallel to VP. A section plane perpendicular to HP and
parallel to VP cuts the prism at a distance of 10 mm from its axis. Draw its
top and sectional front views.
2) S.P
Perpendicular to VP and Parallel to HP
A
Cube
of 50 mm side rests with one of its edge on HP such that the square faces
containing that edge are making equal inclinations with HP. A horizontal
section plane cuts the cube at a distance of 18 mm below the horizontal edge
nearer to the observer. Obtain front and sectional top views.
3) S.P
Perpendicular to VP and Inclined to HP
A
Square prism, side of base 30 mm and axis 60 mm long, rests with its base on HP
and one of its rectangular faces is inclined at 30° to VP. A section plane
perpendicular to VP and inclined at 60° to HP cuts the axis of the prism at a
point 20 mm from its top end. Draw sectional top view and true shape of
section.
4) S.P Perpendicular to HP and Inclined to VP
A hollow hexagonal prism side
of base 30 mm and axis 60 mm long, has a square hole of side 20 mm through it
such that the axis of the hole coincides with that of the prism. The prism
rests with one of its rectangular face & rectangular face of square hole is
parallel to rectangle face of prism. It is cut by a plane perpendicular to HP.
Inclined at 45° to VP and passing through a point on the axis at a distance
of 20 mm from its top face. Draw the
sectional top view and the true shape of section.
HOME
WORK PROBLEMS:
1) A Square prism, side of base 40 mm and axis 60 mm long, rests with
its base on HP such that one of its rectangular faces is inclined at 30° to VP.
A section plane perpendicular to HP and inclined at 60° to VP passes through
the prism such that a rectangular face which is making 60° with VP is cut into
two halves. Draw the top view, sectional front view and true shape of section.
2) A Square prism of 32 mm side and 100
mm height is lying on its base on HP such that the edges of the base are
equally inclined to VP. The prism is cut by a section plane passing through the
mid-point of the axis such that the true shape of section is rhombus of
diagonals of 102 mm and 45 mm. Determine the inclination of the section plane
with HP.
3) A cube of 70 mm long edges
has its vertical forces equally inclined to the VP. It is cut by an auxiliary
inclined plane in such a way that the true shape of the cut part is a regular
hexagon. Determine the inclination of the cutting plane with HP. Draw front
view, sectional top view and true shape of the section.
SECTIONS
OF PYRAMIDS AND TETRAHEDRONS (section plane inclined to any one of the plane and
perpendicular to other plane)
1) A pentagonal pyramid, side of base 30
mm and axis 60 mm long, rests with its base on HP and an edge of its base is
parallel to VP. A section plane perpendicular to VP and inclined at 45° to HP
passes through the axis at point 35 mm above the base. Draw the sectional top
view.
2) A pentagonal pyramid, side
of base 30 mm and axis 60 mm long, rests with its base on HP such that one of
the edges of its base parallel to VP. It is cut by a section plane
perpendicular to HP, inclined to 45° to VP and passing through the pyramid at a
distance of 8 mm from its axis. Draw the sectional front view and the true
shape of section.
3) A hexagonal pyramid of base
side 25 mm and height 70 mm has the hexagonal end on VP, with two its edges
perpendicular to HP. A section plane perpendicular to VP and inclined to 30° to
HP, cuts the pyramid at a 8 mm from the axis and above it. Draw Sectional top
view and true shape of section.
SECTIONS
OF CYLINDERS (section
plane inclined to any one of the plane and perpendicular to other plane)
1) A cylinder of diameter 40 mm and
height 60 mm is having its axis vertical. It is cut by a plane perpendicular to
VP and inclined at 30° to HP. The plane bisects the axis of the cylinder. Draw
its front view, sectional top view and true shape of the section.
SECTIONS
OF CONES (section
plane inclined to any one of the plane and perpendicular to other plane)
1) A cone, 50 mm diameter and
axis 65 mm long, rests with its base on HP. It is cut by a section plane
perpendicular to VP, inclined 45° to HP and passing through a point on the axis
35 mm above the base. Draw the sectional top view, sectional side view and true
shape of the section.
2) A cone, base 40 mm diameter and axis
60 mm long, rests with its base on HP. It
is cut by a section plane perpendicular to VP, parallel to one of the
end generators and passing through a point on the axis 25 mm from the apex.
Draw the sectional top view and true shape of the section.
3) A cone, base 75 mm
diameter and axis 80 mm long is lying on its base on HP. It is cut by a plane
perpendicular to VP and parallel to and 12 mm away from one of its end
generators. Draw the sectional top view and true shape of the section.
DEVELOPMENT
OF SURFACES:
Development of surface of an object means
the unrolling (or) unfolding of all surfaces (or) sides of the object on an
imaginary plane.
Position
of solid in development of surfaces:
Simple position of solids
1.
Axis
of solid perpendicular to HP and parallel to VP
2.
Axis
of solid perpendicular to VP and parallel to HP
Methods of development of
lateral surfaces of a solid:
1.
Parallel
line method – prisms and cylinder
2.
Radial
line method – pyramids and cones.
Position of section plane:
1.
S.P
inclined to H.P and perpendicular to V.P
2.
S.P
inclined to V.P and perpendicular to H.P
PARALLEL
LINE DEVELOPMENT METHOD:
1.
Draw
the development of the lateral surfaces of a right square prism of edge of base
30 mm and axis 50 mm long.
2.
Draw
the development of the complete surface of a G.I cylindrical drum with
lid. Diameter is 30 cm and height is 1.6
times the diameter.
3.
A
hexagonal prism, edge of base 20 mm and axis 50 mm long, rests with its base on
HP such that one of its rectangular faces is parallel to VP. It is cut by a plane perpendicular to VP,
inclined at 45⁰ to HP and passing through the
right corner of the top face of the prism.
a.
Draw
the sectional top view.
b.
Develop
the lateral surfaces of the truncated prism
4.
A
pentagonal prism, side of base 25 mm and altitude 50 mm, rests on its base on
the HP such that an edge of the base is parallel to VP and nearer to the
observer. It is cut by a plane inclined
at 45⁰ to HP, perpendicular to VP and
passing through the centre of the axis.
a.
Draw
the true shape of section
b.
Draw
the development of the complete surfaces of the truncated prism
5.
A
cube of side 33 mm rests on its base on HP with a vertical face inclined at 30⁰
to VP. It is cut by a cutting plane
perpendicular to VP and inclined at 50⁰ to HP. Cutting plane bisects the axis of the
cube. Develop the complete surface of
the right portion of the cut cube.
6.
A
vertical chimney of 70 cm diameter joins a roof sloping at 35⁰
with horizontal. The shortest portion over the roof is 32 cm. Obtain the shape of the sheet metal from
which the chimney can be fabricated.
7.
Draw
the development of the lateral surface of the lower portion of a cylinder of
diameter 50 mm and axis 70 mm when sectioned by a plane inclined at 40⁰
to HP and perpendicular to VP and bisecting the axis.
RADIAL LINE DEVELOPMENT METHOD:
1. Draw the development of the lateral surfaces
of a square pyramid side of base 25 mm and height 50 mm, resting with its base
on HP and an edge of the base parallel to VP.
2.
Draw
the development of the lateral surface of a cone of base diameter 48 mm and
altitude 55 mm.
3.
A
regular hexagonal pyramid of side of base 30 mm and height 60 mm is resting
vertically on its base are perpendicular to VP.
It is cut by a plane inclined at 40⁰ to HP and
perpendicular to VP. The cutting plane
bisects the axis of the pyramid. Obtain
the development of the lateral surface of the truncated pyramid.
4.
A
pentagonal pyramid, side of base 30 mm and height 52 mm stands with its base on
HP and an edge of the base is parallel to VP and nearer to it. It is cut by a plane perpendicular to VP,
inclined at 40⁰ to HP and passing through a
point on the axis, 32 mm above the base.
Draw the sectional top view. Develop the lateral surface of the
truncated pyramid.
5.
A
vertical pentagonal pyramid of side of base 27 mm and altitude 50 mm rests with
a base edge parallel to VP and nearer to it.
It is cut by two planes perpendicular to VP. One is horizontal and cuts the portion of the
pyramid on the left of the axis at a height of 18 mm above the base of
pyramid. The other plane inclined at 45⁰
to HP cuts the portion of the pyramid to the right of the axis passing through
a point on it 18 mm above the base and leans upwards. Draw the development of the lateral surfaces
of the truncated pyramid.
6.
Determine
graphically the shortest length measured along the surface of a frustum of a
cone, between two points A and B. Point
A is on the base of the frustum which is 60 mm in diameter. Point on the top surface which is 30 mm in
diameter. Height of the frustum is 40
mm.
7.
A
cone of base diameter 50 mm and height 70 mm rests on its base on the
ground. A string is wound round the
curved surface of the cone starting from left extreme point and ending at the
same point. Find the shortest length of
the string required. Trace the path of
string in front and top views.
8.
A
lamp shade is formed by cutting a cone of base 144 mm diameter and 174 mm
height by a horizontal plane at a distance of 72 mm from the apex and another
plane inclined at 30⁰ to HP, passing through one
extremity of the base. Draw the
development.
DEVELOPMENT
OF SURFACES:
Development of surface of an object means
the unrolling (or) unfolding of all surfaces (or) sides of the object on an
imaginary plane.
Position
of solid in development of surfaces:
Simple position of solids
1.
Axis
of solid perpendicular to HP and parallel to VP
2.
Axis
of solid perpendicular to VP and parallel to HP
Methods of development of
lateral surfaces of a solid:
1.
Parallel
line method – prisms and cylinder
2.
Radial
line method – pyramids and cones.
Position of section plane:
1.
S.P
inclined to H.P and perpendicular to V.P
2.
S.P
inclined to V.P and perpendicular to H.P
PARALLEL
LINE DEVELOPMENT METHOD:
1.
Draw
the development of the lateral surfaces of a right square prism of edge of base
30 mm and axis 50 mm long.
2.
Draw
the development of the complete surface of a G.I cylindrical drum with
lid. Diameter is 30 cm and height is 1.6
times the diameter.
3.
A
hexagonal prism, edge of base 20 mm and axis 50 mm long, rests with its base on
HP such that one of its rectangular faces is parallel to VP. It is cut by a plane perpendicular to VP,
inclined at 45⁰ to HP and passing through the
right corner of the top face of the prism.
a.
Draw
the sectional top view.
b.
Develop
the lateral surfaces of the truncated prism
4.
A
pentagonal prism, side of base 25 mm and altitude 50 mm, rests on its base on
the HP such that an edge of the base is parallel to VP and nearer to the
observer. It is cut by a plane inclined
at 45⁰ to HP, perpendicular to VP and
passing through the centre of the axis.
a.
Draw
the true shape of section
b.
Draw
the development of the complete surfaces of the truncated prism
5.
A
cube of side 33 mm rests on its base on HP with a vertical face inclined at 30⁰
to VP. It is cut by a cutting plane
perpendicular to VP and inclined at 50⁰ to HP. Cutting plane bisects the axis of the
cube. Develop the complete surface of
the right portion of the cut cube.
6.
A
vertical chimney of 70 cm diameter joins a roof sloping at 35⁰
with horizontal. The shortest portion over the roof is 32 cm. Obtain the shape of the sheet metal from
which the chimney can be fabricated.
7.
Draw
the development of the lateral surface of the lower portion of a cylinder of
diameter 50 mm and axis 70 mm when sectioned by a plane inclined at 40⁰
to HP and perpendicular to VP and bisecting the axis.
RADIAL LINE DEVELOPMENT METHOD:
1. Draw the development of the lateral surfaces
of a square pyramid side of base 25 mm and height 50 mm, resting with its base
on HP and an edge of the base parallel to VP.
2.
Draw
the development of the lateral surface of a cone of base diameter 48 mm and
altitude 55 mm.
3.
A
regular hexagonal pyramid of side of base 30 mm and height 60 mm is resting
vertically on its base are perpendicular to VP.
It is cut by a plane inclined at 40⁰ to HP and
perpendicular to VP. The cutting plane
bisects the axis of the pyramid. Obtain
the development of the lateral surface of the truncated pyramid.
4.
A
pentagonal pyramid, side of base 30 mm and height 52 mm stands with its base on
HP and an edge of the base is parallel to VP and nearer to it. It is cut by a plane perpendicular to VP,
inclined at 40⁰ to HP and passing through a
point on the axis, 32 mm above the base.
Draw the sectional top view. Develop the lateral surface of the
truncated pyramid.
5.
A
vertical pentagonal pyramid of side of base 27 mm and altitude 50 mm rests with
a base edge parallel to VP and nearer to it.
It is cut by two planes perpendicular to VP. One is horizontal and cuts the portion of the
pyramid on the left of the axis at a height of 18 mm above the base of pyramid. The other plane inclined at 45⁰
to HP cuts the portion of the pyramid to the right of the axis passing through
a point on it 18 mm above the base and leans upwards. Draw the development of the lateral surfaces
of the truncated pyramid.
6.
Determine
graphically the shortest length measured along the surface of a frustum of a
cone, between two points A and B. Point
A is on the base of the frustum which is 60 mm in diameter. Point on the top surface which is 30 mm in
diameter. Height of the frustum is 40
mm.
7.
A
cone of base diameter 50 mm and height 70 mm rests on its base on the ground. A string is wound round the curved surface of
the cone starting from left extreme point and ending at the same point. Find the shortest length of the string
required. Trace the path of string in
front and top views.
8.
A
lamp shade is formed by cutting a cone of base 144 mm diameter and 174 mm
height by a horizontal plane at a distance of 72 mm from the apex and another
plane inclined at 30⁰ to HP, passing through one
extremity of the base. Draw the
development.
UNIT 5
ISOMETRIC
PROJECTION
ISOMETRIC PROJECTION OF PLANE
FIGURES
1) Draw the isometric projection
of a square lamina of side 35 mm when its surface
(a) vertical and
(b) horizontal.
2) Draw the isometric projection
of a regular hexagon of 25 mm side when its surface
(a) vertical and
(b) horizontal.
3) Draw isometric projection of a
circle with its surface parallel to HP.
ISOMETEIC PROJECTION OF SOLIDS
1) Draw the isometric projection
of a square prism of side of base 35 mm and height 65 mm when its
axis is (i) vertical and (ii) horizontal.
2) Draw the isometric projection
of a rectangular prism of base 50 mm × 40 mm and height 75 mm.
when it rests with its base on HP and one of its rectangular faces is parallel
to VP.
3) Draw the three possible ways
of representing the isometric projection of a hexagonal prism,
side of base 25 mm and height 60 mm.
4) A right regular
hexagonal prism of side of base 25 mm and altitude 56 mm has a square
hole of 20 mm side at the corner. The axes of the prism and the hole coincide.
One of the faces of the square hole is parallel to one of the
faces of the hexagon. Draw the isometric projection of the prism with the hole.
5) Draw the isometric projection
of a cylinder of base 50 mm diameter and 70 mm height it rests
with its base on HP.
PYRAMIDS AND CONES
1) Draw the isometric projection
of a hexagonal pyramid of side of base 30 mm and height 75 mm,
when it is resting on HP such that an edge of the base is parallel to VP.
2) Draw the isometric projection
of a cone of base 40 mm diameter and height 58 mm when it rests
with its base on HP.
SECTIONED SOLIDS – TRUNCATED AND
FRUSTUM
1) A pentagonal pyramid, base 30
mm and axis 65 mm long rests with its base on HP. An edge of the base is
parallel to VP and nearer to it. A horizontal section plane cuts the pyramid
and passes through a point on the axis at a distance of 25 mm from the apex.
Draw the isometric projection of the frustum of the pyramid.
2) A hexagonal prism, side of
base 25 mm and height 50 mm rests on HP and one of the edges of its base is
parallel to VP. A section plane perpendicular to VP and inclined at 50˚ to HP
bisects the axis of the prism. Draw the isometric projection of the truncated
prism, showing the cut surface.
ISOMETRIC PROJECTION OF SPHERES
1) Draw the isometric projection
of a sphere of diameter 50 mm resting centrally on the top of a cube
of side 60 mm.
2) A sphere of
radius 20 mm is kept on the top face of a square prism of side of
base 40 mm and height 20 mm. The latter is placed on the top face of a cylinder
of 65 mm diameter and 25 mm height. All the three solids have the common axis.
Draw the isometric projection of combination of solids.
PERSPECTIVE PROJECTION
PERSPECTIVE
PROJECTION OF SURFACE (visual ray method)
1)
A Square
lamina of 40 mm side lies on the ground plane. One of its corners is
touching the PP and an edge is inclined at 60⁰ to PP. The station
point is 40 mm in front of PP. 60 mm
above GP and lies in a central plane which is at a distance of 35 mm to the
right of the corner touching the PP. Draw the perspective projection of the
lamina.
PERSPECTIVE
PROJECTION OF SOLIDS (visual ray method)
1)
A square
prism side of base 40 mm and height 60 mm rests with its base on the
ground such that one of its rectangular faces is parallel to and 10 mm behind
the picture plane. The station point is 30 mm in front of PP, 80 mm above the
ground plane and lies in a central
plane 45 mm to the right of the centre of the prism. Draw the perspective
projection of the square prism.
2)
A square
pyramid of base edge 40 mm and altitude 50 mm rests with its base on
the ground plane such that all the edges of the base are equally inclined to
the PP. One of the corners of the base is touching the PP. The station point is
60 mm in front of the PP, 80 mm above the ground plane and lies in a central
plane which passes through the axis of the pyramid. Draw the perspective
projection.
3)
A frustum
of a hexagonal pyramid, base 40 mm side, top 20 mm side and height 50
mm rests with its base on the ground plane such that one of the edge of its
base is touching the PP. The station point is 40 mm in front of PP, 75 mm above GP and lies in a central
plane which passes through the centre of the frustum. Draw the perspective
projection.
4)
A cylinder
of 50 mm diameter and height 60 mm rests with its base on the ground plane such
that the axis is 30 mm behind the PP. A cone of base 50 mm diameter and axis 25
mm long is placed centrally on the top of the cylinder. The station point is 25
mm in front of the PP. 100 mm above the GP and lies in a central plane which is
65 mm to the right of the axes of the solids. Draw the perspective projection
of the arrangement.
5)
A model
of steps has three steps of 10 mm tread and 10 mm rise. The length of
the steps 60 mm. The model is placed with the vertical edge of the first step
touching the PP and its longer edge inclined at 30⁰ to PP.The
station point is 70 mm in front of PP, 55 mm above the ground plane and lies in
a central plane which is at 30 mm to the right of the vertical edge touching
the PP. Draw the perspective projection.
6)
A frustum
of a square pyramid, base 28 mm side, top 22 mm side and 36 mm height
is resting on its base on the GP such that the sides of base are equally
inclined to the picture plane. The axis of the frustum is 30 mm to the right of
the station point is 45 mm in front of the PP and 50 mm above the GP. The
nearest base corner is 10 mm behind the PP. Draw the perspective
projection.
