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| Date post: | 16-Jul-2015 |
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Engineering |
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-The shape of the solid is described bydrawing its two orthographic views usuallyon the two principle planes i.e. H.P. & V.P.
PROJECTIONS OF SOLIDSDefinition of Solid:A solid is a three dimensional object havinglength, breadth and thickness. It iscompletely bounded by a surface or surfaceswhich may be curved or plane.
-For some complicated solids, in addition tothe above principle views, side view is alsorequired.
-A solid is an aggregate of points, lines andplanes and all problems on projections ofsolids would resolve themselves intoprojections of points, lines and planes.
Classification of Solids:
Solids may be divided into two maingroups;
(A) Polyhedra
(B) Solids of revolution
(A) Polyhedra :
A Polyhedra is defined as a solidbounded by planes called faces whichmeet in straight lines called edges.
There are seven regular Polyhedrawhich may be defined as stated below;
(3) Tetrahedron
(4) Cube or Hexahedron:
(5) Octahedron:
(6) Dodecahedron:
(7) Icosahedron:
(1) Prism
(2) Pyramid
(1) Prism:
It is a polyhedra having twoequal and similar facescalled its ends or bases,parallel to each other andjoined by other faces whichare rectangles.
-The imaginaryline joining theCentres of thebases or faces iscalled Axis ofPrism.
Axis
Faces
Edge
According to the shape of its base, prism can be sub classified intofollowing types:
(a) Triangular Prism:
(b) Square Prism:
(c) Pentagonal Prism:
(d) Hexagonal Prism:
(2) Pyramid:
This is a polyhedra having planesurface as a base and a numberof triangular faces meeting at apoint called the Vertex or Apex.
-The imaginaryline joining theApex with theCentre of thebase is calledAxis of pyramid.
Axis
Edge
Base
According to the shape of its base, pyramid can be sub classified intofollowing types:
(a) Triangular Pyramid:
(b) Square Pyramid:
(c) Pentagonal Pyramid:
(d) Hexagonal Pyramid:
(B) Solids of Revolutions:When a solid is generated by revolutionsof a plane figure about a fixed line (Axis)then such solids are named as solids ofrevolution.
Solids of revolutions may be of followingtypes;
(1) Cylinder
(2) Cone
(3) Sphere
(4) Ellipsoid
(5) Paraboloid
(6) Hyperboloid
(1) Cylinder:
A right regular cylinder is a solidgenerated by the revolution of arectangle about its vertical sidewhich remains fixed.
Rectangle
Axis
Base
(2) Cone:
A right circular cone is a solidgenerated by the revolution of a rightangle triangle about its vertical sidewhich remains fixed.
Right angle
triangle
Axis
Base
Generators
Important Terms Used in Projections of Solids:
(1) Edge or generator:
For Pyramids & Prisms, edges are thelines separating the triangular faces orrectangular faces from each other.
For Cylinder, generators are thestraight lines joining different pointson the circumference of the bases witheach other
Important Terms Used in Projections of Solids:
(2) Apex of solids:
For Cone andPyramids, Apexis the pointwhere all thegenerators orthe edges meet.
Apex
Apex
Edges
Generators
CONE
PYRAMID
Axis
Faces
Edge
PRISM
Rectangle
Axis
Base
Generators
CYLINDER
Important Terms Used in Projections of Solids:
(3) Axis of Solid:
For Cone and Pyramids, Axis is animaginary line joining centre ofthe base to the Apex.
For Cylinder and Prism, Axis is animaginary line joining centres ofends or bases.
Important Terms Used in Projections of Solids:
(4) Right Solid:
A solid is said tobe a Right Solidif its axis isperpendicular toits base.
Axis
Base
Important Terms Used in Projections of Solids:
(5) Oblique Solid:
A solid is saidto be a ObliqueSolid if its axisis inclined atan angle otherthan 90° to its
base.
Axis
Base
Important Terms Used in Projections
of Solids:
(6) Regular Solid:
A solid is said to be a Regular Solid if
all the edges of the base or the end
faces of a solid are equal in length and
form regular plane figures
Important Terms Used in Projections
of Solids:
(7) Frustum of Solid:
When a Pyramid or a
Cone is cut by a Plane
parallel to its base,
thus removing the top
portion, the remaining
lower portion is called
its frustum. FRUSTUM OF A
PYRAMID
CUTTING PLANE
PARALLEL TO
BASE
Important Terms Used in Projections
of Solids:
(8) Truncated Solid :
When a Pyramid or a
Cone is cut by a Plane
inclined to its base,
thus removing the top
portion, the remaining
lower portion is said to
be truncated.
Class A(1): Axis perpendicular to H. P. and hence
parallel to both V.P. & P.P.
X Y
a
b
d
c
c’,d’a’,b’
o’
o
Axis
c’,3’b’,2’
Class A(2): Axis perpendicular to V.P. and hence
parallel to both H.P. & P.P.
f’,6’
a
e’,5’
d’,4’a’,1’
b,f c,e d
43,52,61X Y
H
b”2”
1
a”1”1’2’
Class A(3): Axis perpendicular to P.P. and hence
parallel to both H.P. & V.P.
X Y
Lc”3”
a’,b’
c’
a
b
c 3
2
3’
Class B(1): Axis parallel to V.P. and inclined to
H.P. by θ & also inclined to P.P.
Exercise 1 :
A right regular pentagonal prism,
side of base 30 mm and height of
axis as 75mm rests on HP on one
of its base corners such that its
long edge containing the corner is
inclined to the HP at 60°. Draw its
projections.
a’
5
c1’
X Y31’21’
4
11
5141
21
31
a1
d1
c1
b1
e1
d1’ b1’e1’a1’
c’d’
b’e’
11’
51’41’
a
b
ed
32
1
60°
30
SCALE:-1:1
75
c
a’
5
c1’
X Y31’21’
4
11
5141
21
31
a1
d1
c1
b1
e1
d1’ b1’e1’a1’
c’d’
b’e’
11’
51’41’
a
b
ed
32
1
60°
30
SCALE:-1:1
75
c
Exercise 2 :
A tetrahedron of 40 mmlong edges, rests on HP onone of its edges such thatthe face containing thatedge is inclined to HP at30° and the same edge isinclined at 45° to VP. Draw
the projections of the solid.
30
a
b
c
d
a’
d’
b’c’
b1’c1’
a1’
d1’
b1
c1
a1 d1
a2’
d2’
b2’ c2’X Y
40
45 c2
b2
a2
d2
30
a
b
c
d
a’
d’
b’c’
b1’c1’
a1’
d1’
b1
c1
a1 d1
a2’
d2’
b2’ c2’X Y
40
45 c2
b2
a2
d2
A cone, diameter of base 60mm andheight 70mm, is resting on HP onthe point of periphery of the base.Axis of the cone makes 60 with HPand 30 with the VP. Draw theprojections of the cone, when theapex is nearer to the VP.
Exercise 3 :
O’
70
6
0
a’b’,h’ d’,f’
e’c’,g’ e1’
a1’
O1’
60O2
Locus of O2
30
a e
fg
h
b dc
O2’
a2’
e2’c2’
g2’
YX
O1a1
c1
e1
g1
O
g2
a2
c2
e2
O
Exercise 4 :A regular pentagonal prism of25mm long edges and axis70mm long rests on HP on oneof its corner of the base. Theslant edge passing throughcorner makes 45 with HP andthe side opposite to the samecorner makes 30 with VP.Draw its projections.
d1
e1
a1
b1
c1
b
a
c
d
e1
23
4
5
d’c’b’a’ e’
11
21
31
41
51
a1’b1’
c1’ d1’e1’
42’
12’22’
32’52’
d2’c2’
b2’ a2’
e2’
12
22
32 42
52
=45= 30
1’2’ 3’
5’4’
e2
d2c2
b2
a2
31’ 41’
51’11’
21’
X Y
d1
e1
a1
b1
c1
b
a
c
d
e1
23
4
5
d’c’b’a’ e’
11
21
31
41
51
a1’b1’
c1’ d1’e1’
42’
12’22’
32’52’
d2’c2’
b2’ a2’
e2’
12
22
32 42
52
=45= 30
1’2’ 3’
5’4’
e2
d2c2
b2
a2
31’ 41’
51’11’
21’
X Y
Exercise 5 :
A regular hexagonal prism of30mm sides and axis 80mmlong is resting on HP on oneof its corners of the base. Theaxis makes 30 with HP andplan of the axis makes 45with the VP. Draw itsprojections.
a2
d2b2
f2
c2
e2
12
22 42
62
32
52
d1’
b1’a1’
c1’e1’
f1’
21’
11’
31’41’
51’
61’
a1 d1
c1b1
e1f1
4’2’ 3’
1’5’6’
d’
f’ e’c’b’
a’
X Y
5
1 4
2 3
6e
a d
cb
f
11
21 31
41
5161
b2’
a2’
c2’ d2’
e2’
f2’
12’
32’
22’
42’
52’
62’
45
A square pyramid, side of base50mm and height 64mm, is freelysuspended from one of thecorners of the base. Draw itsprojections when vertical planecontaining axis makes an angle of45 with the VP.
Exercise 6 :
A cube of 40 mm edges, is restingon the H.P. on one of the edges ofthe base with face containing thatedge making 30 with the H.P.The edge on which the cube restson the H.P. is making 30 with theV.P. Draw the projections.
Exercise :
PROJECTIONS OF
SOLIDS
