PLANS, MAPS AND SCALES
A plan is the graphical representation, to some scale, of the features on, near or below the surface of the earth as projected on a horizontal plane which is represented by plane of the paper on which the plan is drawn.
However, since the surface of the earth is curved and the paper of the plan or map is plane, no part of the surface can be represented on such maps without distortion.
In plane surveying, the areas involved are small, the earth’s surface may be regarded as plane and hence map is constructed by orthographic projection without measurable distortion.
The representation is called a map if the scale is small while it is called a plan if the scale is large.
On a plan, generally, only horizontal distances and directions are shown.
On a topographic map, however, the vertical distances are also represented by contour lines. hachures or other systems.
The area that is surveyed is vast and, therefore, plans are made to some scale.
Scale is the fixed ratio that every distance on the plan bears with corresponding distance on the ground.
Scale can be represented by the following methods :
1) One cm on the plan represents some whole number of metres on the ground, such as 1 cm = 10 m etc.
This type of scale is called engineer’s scale.
2) One unit of length on the plan represents some number of same units of length on the ground, such as 1/1000, etc.
This ratio of map distance to the corresponding ground distance is independent of units of measurement and is called representative fraction.
The representative fraction (abbreviated as R.F.) can be very easily found for a given engineer’s scale. For example, if the scale is 1 cm 50 m
R.F. = 1/(50 x 100) = 1/5000
The above two types of scales are also known as numerical scales.
3) An alternative way of representing the scale is to draw on the plan a graphical scale.
A graphical scale is a line sub-divided into plan distance corresponding to convenient units of length on the ground.
If the plan or map is to be used after a few years, the numerical scales may not give accurate results if the sheet or paper shrinks.
However, if a graphical scale is also drawn, it will shrink proportionately and the distances can be found accurately.
That is why scales are always drawn on all survey maps.
Types of Scales :
Scales may be classified as follows :-
1) Plain Scale :
A plain scale is one on which it is possible to measure two dimensions only, such as units and lengths, metres and decimetres, miles and furlongs, etc.
2) Diagonal Scale :
On a diagonal scale, it is possible to measure three dimensions such as metres, decime and centimetres; units, tenths and hundredths; yards, feet and inches etc.
A short length is divided into a number of parts by using the principle of similar triangles in which like sides are proportional.
3) Vernier Scale :
The vernier, invented in 1631 by Pierre Vernier, is a device for measuring the fractional part of one of the smallest divisions of a graduated scale.
It usually consists of a small auxiliary scale which slides along side the main scale.
The principle of vernier is based on the fact that the eye can perceive without strain and with considerable precision when two graduations coincide to form one continuous straight line.
The vernier carries an index mark which forms the zero of the vernier.
If the graduations of the main scale are numbered in one direction only, the vernier used is called a single vernier, extending in one direction.
If the graduations of the main scale are numbered in both the directions, the vernier used is called double vernier, extending in both the directions, having its index mark in the middle.
The divisions of the vernier are either just a little smaller or a little larger than the divisions of the main scale.
The fineness of reading or least count of the vernier is equal to the difference between the smallest division on the main scale and smallest division on the vernier.
Whether single or double, a vernier can primarily be divided into the following two classes :
(a) Direct Vernier
(b) Retrograde Vernier.
(a) Direct Vernier :-
A direct vernier is the one which extends or increases in the same direction as that of the main scale and in which the smallest division on the vernier is shorter than the smallest division on the main scale.
It is so constructed that (n – 1) divisions of the main scale are equal in length of n divisions of the vernier.
s = Value of one smallest division on main scale
v = Value of one smallest division on the vernier
n = Number of divisions on the vernier.
Since a length of (n-1) divisions of main scale is equal to n divisions of vernier, we have
nv = (n – 1) s
Least count =s – v = s/n
Thus, the least count (L.C.) can be found by dividing the value of one main scale division by the total number of divisions on the vernier.
(b) Retrograde Vernier :-
Retrograde vernier is the one which extends or increases in opposite direction as that of the main scale and in which the smallest division of the vernier is longer than the smallest division on the main scale.
It is so constructed that (n + 1) divisions of the main scale are equal in length of n divisions of the vernier.
nv = (n + 1)s
L.C = v – s = s/n
Special Forms of Verniers :
a) The Extended Vernier :-
It may happen that the divisions on the main scale are very close and it would then be difficult, if the vernier were of normal length. to judge the exact graduation where coincidence occurred.
In this case, an extended vernier may be used.
Here (2 n- 1) divisions on the main scale are equal to n divisions on the vernier so that
nv = (2n – 1)s
The difference between two main scale spaces and one vernier space
= 2s – v = s/n = L.C
b) The Double Folded Vernier :-
The double folded vernier is employed where the length of the corresponding double vernier would be so great as to make it impracticable.
This type of vernier is sometimes used in compasses having the zero in the middle of the length.
The full length of vernier is employed for reading angles in either direction.
The vernier is read from the index towards either of the extreme divisions and then from the other extreme division in the same direction to the centre.