Right Angles and Foot of Perpendicular on the Lines | Chain Surveying | Surveying

This article throws light upon the top five instruments that are used for setting out right angles and finding the foot of perpendicular from the object on the lines. The instruments are: 1. Cross Staff 2. Optical Square 3. Prism Square 4. Offset Rod 5. Measuring Tape.

Instrument # 1. Cross-Staff:

It is generally found in two patterns:

(i) Open cross- staff and

(ii) French cross-staff, the first one being in common use.

(i) Open Cross-Staff:

The simplest form of cross-staff is the open wooden cross-staff shown in fig. 3.7. It consists of a round or square piece of wood about 4 cm thick and varying form 15 cm to 30 cm in diameter or side mounted on an iron shod wooden staff about 2.5 cm diameter and 1.5 m long. The disc is provided with two saw cuts about 1 cm deep at right angles to each other, giving two lines of sight.

The modified form of the open cross-staff is the metal arm cross-staff (Fig. 3.8) in which the wooden head is replaced by four metal arms with vertical slits for sighting through at right angles to each other.

(ii) French Cross-Staff (Fig. 3.9.):

It consists of an octagonal brass tube with slits on all the eight sides. It has an alternate vertical sighting slit and an opposite vertical window with a vertical fine wire or horse hair on each of the four sides. These are used for setting out right angles. On the other sides are vertical slits, which are at 45° to those mentioned above for setting out angles of 45°. The sights are about 8 cm apart.

It carries a sockets at the base so that it may be mounted on the pointed staff when in use. The sights being too close, it is not as accurate as the open cross-staff.

Using the Cross-Staff:

For setting out a right angle at given point on a chain line, it is held vertically over the given point and turned until one of the saw cuts or (the pair of sights) is ranged along the chain line. Then the line of sights through the other saw cut or the pair of sights) is at right angles to the chain line and a ranging rod can be fixed in this direction at a convenient position.

To find the foot of the perpendicular from an object on a chain line, the cross staff is held vertically at an approximate position of the required object and turned until one of the saw cuts or (the pair of sights) is ranged along the chain line.

If on looking through the other saw cut or (the pair of sights) is seen the object from which the foot of the perpendicular is to be drawn, then the point where the cross-staff is held is the correct position of the foot of the perpendicular from the given object. If not, the cross- staff is moved forward or backward along the chain line until the line of sight through the saw cut or (the pair of sights) at right angles to the chain line bisects the given object.

Testing a Cross-Staff (Refer fig. 3.10):

Direct one of the saw cuts or (the pair of sights) say ab along the chain line XY and fix a ranging rod at R in the direction of the other saw cut or (the pair of sights) say ‘cd’. Now move the cross-staff horizontally through 90° so that the saw cut or (the pair of sights) ‘cd’ is along XY. If the instrument is correct, the other saw cut or (the pair of sights) ‘ba’ will point towards the ranging rod R, otherwise not, and then it should be rejected.

The cross-staff is a nonadjustable instrument and is not capable of high accuracy. However out of the above two types, the open one being light and strong is more commonly used.

Instrument # 2. Optical Square:

It is more accurate than the cross-staff and is used for setting out accurately the long offsets. It is a small compact hand instrument based upon the principle of reflection.

There are two forms of the optical square:

(i) Round or cylindrical and

(ii) Wedge shaped commonly known as Indian optical square.

(i) Round or Cylindrical Optical Square (fig 3.11):

It is a round brass box about 5 cm in diameter and 1.25 cm deep fig. 3.11 (a). It is protected by a metal cover fig. 3.11. (b) which can slide round the box as so as to cover the openings and thus prevents the interior from dust, moisture, etc. when the instrument is not in use. Fig. 3.12 (a) shows the sectional plan of the important parts of the instrument.

H and I are two mirrors placed at an angle of 45° to each other. The mirror H, known as the horizon mirror, is half silvered and half unsilvered and is rigidly attached to the base plate of the box. The mirror I, called the index mirror, is wholly silvered and is attached to the box in such a way that it can be adjusted to any position. In some forms of the instrument, the index mirror is also permanently fixed to the base plate.

Three openings are made in the rims of the box: a is a small hole for the eye ; b is a small rectangular slot for the horizon sight placed diametrically opposite to the eye-hole ; and c is a large rectangular slot for index sight placed at right angles to the line joining a and b.

Three similar openings are also made in the rims of the cover such that each of them is brought over the corresponding opening in the rims of the box when the instruments is in use. A small circular hole is also provided on the top of the cover for inserting the key by means of which the index mirror is adjusted.

The lines ab and cl are called the horizon and the index sights respectively both being at right angles to each other. The horizon mirror makes an angle of 120° to the horizon sight while the index mirror makes an angle of 105° to the index sight, the angle between both the mirror being 45°.

A ray of light from the object or ranging rod O strikes the index mirror (I) and is reflected along IH. Thus reflected ray again strikes the silvered portion of the horizon mirror (H) and is then reflected along HE. Thus the observer, looking through the eye-hole, can see the ranging rod at the far end B of the chain line directly through the unsilvered portion of the horizon mirror and at the same time he can see the image of the ranging rod at O in the silvered portion of the horizon mirror. 

If the ranging rod at O is at right angles to the horizon sight (chain line) exactly, then the ranging rod at B and the image of O are exactly coincident as in fig. 3.12 (b) otherwise they do not coincide as in fig. 3.12 (c).

Principle of Optical Square:

If there are two plane mirrors whose reflecting surfaces make a given angle with each other and if a ray of light is reflected successively from both of them, then the angle between the first incident ray and the last reflected ray is twice the angle between the two mirrors.

Since we want to set out right angles with the optical square, it follows that the two mirrors in it must make an angle with each other equal to half the right angle i.e 90°/2 = 45°

The principle underlying the construction of the optimal square can be explained as follows (fig 3.13):

EB is a chain line and O is an object on the line OI which is at right angles to EB.

A ray of light from O strikes the index mirror (I) and is reflected along (IH). The reflected ray again strikes the horizon mirror (H) and is then reflected towards Eye (E) of the observer.

i.e. the angle between the two mirrors (∠ILH)=45°=1/2 of 90°, which is the angle between the index sight and the horizon sight (chain line).

Use:

To find out the foot of the perpendicular from any object upon the chain line, place the eye opposite to the eye-hole so that the small slot (aperture) and the large slot (object-hole) are towards the far end of the chain line and the object respectively.

Then walking forward and backward along the chain line, the point where the reflection of the object as seer on the silvered portion of the horizon mirror appears to be coincident with the ranging rod at the farther extremity of the chain line as seen through the plane portion of the same mirror (fig. 3.12b) is the required point.

Note:

If the object lies on the left hand side of the chain line, the instrument is held in the right hand and vice-versa.

To set out a perpendicular from any point on the chain line, stand on the chain line holding the instruments exactly above the point. Now place the small slot (aperture) towards farther end of the chain line and the large slot (object-hole) towards the direction in which perpendicular is to be set out. Sight the end ranging rod through the aperture.

Then instruct the assistant to move to and fro until image of his ranging rod seen through the silvered portion of the horizon mirror appears to be coincident with the ranging rod at the farther end of the chain line as seen through the plane portion of the same mirror. The assistant is then directed to fix the ranging rod in that position. The line joining this position of the ranging rod and the point below the optical square is at right angles to the chain line.

(ii) Indian Optical Square (Fig 3.14):

It is a brass wedge shaped hollow box of about 5 cm sides and about 3 cm deep with a handle about 8 cm long fixed underneath, m₁ and m₂ are two mirrors fixed to the inclined sides of the box at an angle of 45°; ab and cd are two rectangular openings above these mirrors. PQRS is the open face which is to be turned towards the object to which the offset is to be taken.

Use (Fig. 3.15):

In taking an offset from an object say O, the observer holding the instrument in his hand stands on the chain line AB and turns the open face towards the object. He then sights the ranging rod B at the forward station by looking through the openings in the direction cb or ad, according as the object is to his left or right and moves along the chain line forward or backward until the image of the object appears exactly in line with the ranging rod B. The point vertically under the instrument is foot of the perpendicular from the object O and OC is perpendicular to the chain line AB.

In using the instrument, hold it quite erect and always apply the eye to the lower corner of the opening which is nearer the open face PQRS and look diagonally to the forward ranging rod. Thus if the object is to the left of the chain line, apply the eye at c and look in the direction of cb and if to the right, apply it at a and look in the direction of ad.

For setting out a right angle from any point ‘C’ on the chain line, stand on the chain line holding the instrument exactly above the point and turn the open face of the optical square towards the direction in which right angle is to be set out. Sight the end ranging rod ‘B’ through the openings.

Then instruct the assistant to move to and fro until image of his ranging rod ‘O’ seen through the mirror appears to be coincident with the ranging rod at the farther end of the chain line as seen through the openings as in fig. 3.16. (b). The line joining this position of the ranging rod and the point below the optical square is a right angles to the chain line.

Instrument # 3. Prism Square (Fig. 3.17):

It is based upon the same principle as that of the optical square. It is more reliable and bright than the optical square. It requires no adjustment since the angle of 45° between the reflecting surfaces of the prism is fixed.

It using the instrument hold it in your hand and see directly over the prism a ranging rod at B. Then walk along the chain line until the image of the object ‘O’ seen in the prism appears to coincide with the ranging rod at B.

Instrument # 4. Offset Rod:

When it is used for taking the offset, the perpendicular direction is first ascertained by judgement and then the offset rod is turned over end for end as many times consecutively as may be required to reach the object. This method of offsetting is not preferable as errors accumulate rapidly. This is used only for taking small offsets.

Instrument # 5. Measuring Tape:

When measuring tape is to be used for taking the offset hold its zero end at the point to which an offset is to be taken and swing the other end on the chain line and note the point of minimum reading on it, which denotes the length of the offset.