Lecture - 11
Isometric Projection
The term ‘isometric’ comes from the Greek language which means ‘equal measure’, reflecting that the scale along each axis of the projection is the same. Isometric projection is used to create a pictorial drawing of an object.
·
Isometric projection
provides a three-dimensional view of an object on a two-dimensional
plane.
·
It’s a single-view drawing that
combines the length, breadth, and height into one illustration
·
Geometric principle
of isometric projection:
·
The X, Y, and Z axes (representing
width, depth, and height) are equally inclined (120° apart) from each
other.
·
·
Each axis is inclined at approximately
35.26° to the horizontal, but on paper we draw the horizontal axes at 30°
to the horizontal for simplicity.
·
·
View of a cube.
·
TERMINOLOGY
Referring to the figure
below, the important terms used in isometric projections are as follows:
1. Isometric
axes
The three lines CB, CD and CG, meeting at
point C and inclined at an angle of 120° with each other, are called isometric
axes.
2. Isometric
lines
The lines parallel to the isometric axes
are called isometric lines. Here lines AB, BF, FG, GH, DH and AD are isometric
lines.
3. Non-isometric
lines
The lines which are not parallel to
isometric axes are known as non-isometric lines. Here diagonals BD, AC, CF, BG,
etc., are non-isometric lines.
4. Isometric
plane
The plane representing any face of the
cube as well as other plane parallel to it is called an isometric plane. Here,
ABCD, BCGF, CGHD, etc., are isometric planes.
5. Non-isometric
plane
The plane which is not parallel to
isometric planes are known as non-isometric planes. Here, the plane ABGH, CDEF,
AFH, CFH, etc., are non-isometric planes.
6. Isometric
scale
It is the scale which is used to convert the true
length into isometric length. Mathematically, Isometric length = 0.816 x True
length
What Is Foreshortening?
Foreshortening
is a visual effect where an object appears shorter than its actual length
because it is angled relative to the viewer or projection plane.
In isometric projection,
an object is oriented such that all three axes (length, breadth, and height)
are inclined equally (typically 120° apart), and none of them is
parallel to the projection plane. As a result, the lengths along all three axes
appear shorter than their true lengths—this is foreshortening.
In isometric
projection, all three dimensions are equally foreshortened (about 82%
of the true size). So, to measure these correctly, we use a specially
constructed isometric scale.
Construction of an Isometric Scale
1)
Draw a horizontal line bo.
2)
Draw lines ba’ and ba inclined
at 45° and 30° with line bo, respectively.
3)
Mark off the true scale on the line ba’ as
0’, 10’, 20’, 30’, etc.
4)
Draw vertical lines from points 0’, 10’,
20’, 30’, etc., to meet line ba at points 0, 10, 20, 30, etc. The marked
divisions of ba represent the isometric lengths.
In triangle abo, ba/bo =
1/cos300 = 2/√3
In triangle a’bo, ba’/bo
= 1/cos450 =√2/1
Therefore, Isometric
length / True length
= ba/ba’
= 2/√3 x 1/√2
= √2/√3
= 9/11
= 0.816 (approx).
This reduction of the
true length can be obtained either by multiplying it by a factor 0.816 or by
taking the measurement with the help of an isometric scale.
Characteristics of Principal Lines in Isometric Projection:
1. Lines
that are parallel on the actual object remain parallel in the isometric
projection.
2. Any
vertical line on the object stays vertical in the isometric drawing.
3. Horizontal
lines on the object appear at a 30° angle to the horizontal axis in the
isometric view.
4. Lines
that are aligned with the principal axes, called isometric lines, appear
equally shortened.
5. Lines
that are not aligned with the principal axes, known as non-isometric lines, do
not have uniform foreshortening. For instance, in the front view, diagonals BD
and AC may be of the same length, but they appear differently in length in the
isometric projection. These non-isometric lines are represented by marking the
positions of their endpoints on the isometric planes.
Isometric Projection and Isometric View
In an isometric
projection, a scale factor of 0.816 is used to prepare the drawing whereas in
an isometric view the true length is used. Thus, the isometric view of an
object is larger than the isometric projection. Because of ease of construction
and advantage of measuring the dimensions directly from the drawing, it has
become a general practice to use the true lengths instead of isometric lengths.
Isometric Projection
· Isometric
projection is a method of visually representing three-dimensional objects
in two dimensions.
· In
this projection, the object is rotated in such a way that its three principal
axes (height, width, and depth) make equal angles of 120° with each
other.
· The
angles between the isometric axes on paper are 120°, and each axis is
inclined 30° to the horizontal.
· True
lengths are not used; instead, they are foreshortened
by approximately 82% of their actual size using an isometric scale.
· It
is considered a technical projection and more precise.
Isometric View
· An
isometric view is a visual drawing or sketch that represents an object
in three dimensions, just like isometric projection.
· However,
in an isometric view, the dimensions are drawn using their true lengths
without applying any foreshortening or isometric scale.
· This
is commonly used for presentation purposes as it gives a clearer and
more realistic view of the object.
· The
angles between axes remain the same as in isometric projection.
Fig.
(a) Orthographic projection (b) Isometric projection (c) Isometric view
Dimensioning in Isometric Projection:
1. Always
use the true lengths when specifying dimension values in isometric
drawings or views.
2. As
much as possible, extension lines and dimension lines should be drawn
along isometric directions, lying within the isometric planes.
3. Hidden
lines are generally avoided unless they are necessary
for clarity.
4. The
centerlines of circular features should be aligned parallel to the
isometric axes.
5. Dimensions
related to circular shapes should be placed in the plane where the feature
is most visible or prominent.
Problem
Draw the isometric view
of a square prism of base side 40 mm and axis 60 mm resting on the H.P. on the
(a) base with axis perpendicular to the H.P., (b) rectangular face with axis
perpendicular to the V.P., and (c) rectangular face with axis parallel to the
V.P.
Isometric View of Solid Containing Non-Isometric Lines
An isometric view of a
solid containing non-isometric lines is a type of 3D drawing where:
- The isometric view represents
the object using a 30° angle from the horizontal in both directions,
showing three sides of the solid in a single view.
- Isometric lines
are lines that run parallel to the isometric axes—typically aligned
at 30°, 150°, and vertical (90°).
- Non-isometric lines
are lines not parallel to any of the isometric axes. These lines do
not follow the standard isometric angle and thus are not drawn at
30°, 150°, or 90°.
Inclined or oblique edges
that do not align with the isometric axes are called non-isometric lines.
These lines are not drawn using the standard isometric angles (30°, 150°, or
90°) and cannot be measured directly with an isometric scale. Instead, they are
constructed by locating key points and connecting them. Two common methods are
used to accurately draw these lines:
1. Box Method
- In this method, the object is
imagined to be enclosed within a rectangular isometric box.
- Both isometric and non-isometric
lines are constructed by:
- Identifying where the object’s edges
intersect the faces, edges, or corners of the box.
- Plotting those points within the
isometric view of the box.
- Connecting the plotted points to
create non-isometric lines.
2. Offset Method
- The offset method involves projecting
coordinates (offsets) from a known point (usually a corner or
endpoint).
- Lines are drawn parallel to the
isometric axes to establish the x, y, and z offsets.
- The intersection of these offsets
determines the location of the next point.
- Finally, non-isometric lines are
drawn by joining these projected points.
Problem
Draw the isometric view
of a hexagonal prism of base side 30 and axis 70 mm. The prism is resting on
its base on the H.P. with an edge of the base parallel to the V.P
Construction Steps:
1. Start
by drawing a hexagon labeled abcdef to represent the
top view of the prism. Enclose this hexagon within a rectangle labeled pqrs,
as illustrated in Fig. (a).
2. Next,
construct the isometric view of the enclosing
rectangle, now labeled PQRS, ensuring that its sides are drawn at 30°
angles to the horizontal, consistent with standard isometric projection.
(a)
Box Method (Fig. (b))
Steps:
1. Draw
vertical lines PP₁, QQ₁, RR₁, and SS₁, each measuring 70 mm in height,
to represent the height of the prism. Then, connect the top ends to form the
rectangle P₁Q₁R₁S₁.
2. Locate
and mark the points A, B, C, D, E, and F on the lower face of the
isometric box, making sure their positions match those in the top view, such
that:
o PA
= pa,
PB = pb,
o QC
= qc,
o RD
= rd,
o SE
= se,
o PF
= pf.
3. Similarly,
locate and mark the points A₁, B₁, C₁, D₁, E₁, and F₁ on the top face of
the prism, maintaining the same offsets as the bottom:
o P₁A₁
= pa,
P₁B₁ = pb,
o Q₁C₁
= qc,
o R₁D₁
= rd,
o S₁E₁
= se,
o P₁F₁
= pf.
4. Finally,
connect all the corresponding points (A to A₁, B to B₁, etc.) and edges
to form the complete isometric view of the prism.
Offset Method (Fig. (c))
Steps:
1. In
the isometric projection, locate the base points A, B, C, D, E, and F
by measuring offsets from reference points, ensuring the following
relationships:
o PA
= pa,
PB = pb
o QC
= qc
o RD
= rd
o SE
= se
o PF
= pf
2. From
each of these base points, draw vertical lines—AA₁, BB₁, CC₁, DD₁,
EE₁, and FF₁—each 70 mm in height to represent the prism’s vertical
edges.
3. Connect
the top ends of these vertical lines—A₁ to B₁ to C₁
to D₁ to E₁ to F₁—in sequence to complete the top face and form the full
isometric view of the hexagonal prism.
Exercise
Draw
all the given isometric views on your paper.
