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**COMPARATIVE STUDY OF STRENGTH OF FIGURE OF FOUR BRACED QUADRILATERAL USING A MINOR TRIANGULATION SCHEME**

**CHAPTER ONE**

**INTRODUCTION**

**1.1 Background of the study**

Triangulation is one of the methods of fixing accurate controls on the earth surface. It is based on the trigonometric proposition that if one side and two angles of a triangle are known, the remaining sides can be computed. A triangulation system consists of a series of overlapping triangles in which an occasional side called a baseline is measured and remaining sides are calculated from the angles measured at the vertices of the triangles, vertices being the control points are called triangulation stations (Chandra, 2012). When all sides of a triangulation are measured it is known as the triliteration system. However, the angular measurement define the shape of the triangulation system better than wholly linear measurements and so it is preferred that a number of angles are included in trilateration system. A combined triangulation and trilateration system in which all the angles and all the sides are measured, represents the strongest network for creating horizontal control and it is called triangulation. Because, at one time, it was easier to measure angles than it was for distance, triangulation was the preferred method of establishing the position of control points. Many countries used triangulation as the basis of their national mapping system. The procedure was generally to establish primary triangulation networks, with triangles having sides ranging from 30 to 50 km in length. The primary trigonometric points were fixed at the corners of these triangles and the sum of the measured angles was correct to ± 3″. These points were usually established on the tops of mountains to afford long, uninterrupted sight lines. The process of measuring the angles of a chain network of triangles of the earth is called triangulation. (Agor, 2010).

**1.1.1 Objective of triangulation surveys**

The main objective of any triangulation or trilateration surveys is to provide a number of stations whose relative and absolute positions, horizontal as well as vertical, are accurately established. More detailed location or engineering survey is then carried out from these stations. The triangulation surveys therefore are carried out for the following reasons:

(i) To create accurate control for plane and geodetic surveys of large areas, by terrestrial methods.

(ii) To establish precise control for photogrammetric surveys of large areas.

(iii) To support in the determination of the size and shape of the earth by making observations for latitude, longitude and gravity, and

(iv) To determine accurate locations of points in engineering works such as:

(a) Fitting centre line and abutments of long bridges over large rivers.

(b) Fitting centre line, terminal points, and shafts for extended tunnels.

(c) Transferring the control points across wide sea channels, large water bodies, etc.

(d) Detection of crustal activities, etc.

(e) Finding the direction of the movement of clouds.

**1.1.2 Strength of Figures **

The strength of figure is a thing to be considered in establishing a triangulation system to maintain the computations within a desired degree of precision. It plays also an important role in deciding the layout of triangulation system. The strength of figure is a function of: Angles change more rapidly than those of large angles. This suggests that smaller angles less than 30° should not be used in the computation of triangulation. If, due to unavoidable circumstances, angles less than 30° are used, then it must be ensured that this is not opposite the side whose length is required to be computed for carrying forward the triangulation series. The expression given by the U.S coast and Geodetic survey for evaluation of strength of figure is:

L2= d2R

Where,

L2= the square of the probable error that would occur in the sixth place of the logarithm of any side.

d= the probable error of an observed direction in seconds of arc

R= a term which represents the shape of a figure

R = )

D= the number of directions observed excluding the known side of the figure

= the difference in the sixth place of logarithm of the sine of the distance angles A, B, C, etc respectively

C = (nˈ-Sˈ+1) + (n-2S+3)

nˈ= the total number of sides including the known side of the figure,

n = the total number of sides observed in both directions including the known side

Sˈ= the number of stations occupied, and

S= the total number of stations.

**1.2 STATEMENT OF THE PROBLEM**

It has been established that the disregard for the consideration of strength of figure and also the absence of the use of four braced quadrilateral figure in triangulation work are the major factors resulting in in-accurate relative position fixing of control points which is the basis for all survey work. Since accuracy is a function of the strength of figure and four quadrilateral figures in a triangulation scheme, there is need to evaluate and consider a figure at the reconnaissance stage before triangulation. This is to ensure high accuracy of fixation of points.

**1.3 MOTIVATION OF RESEARCH**

This project is carried out due the degree of accuracy that will be required in minor triangulation scheme of four braced quadrilateral.

**1.4 AIM OF THE STUDY**

The plan of this project is to evaluate and examine the strength of figure of four braced quadrilateral using a minor braced triangulation scheme.

**1.5 OBJECTIVES OF THE STUDY**

This aim will be achieved through the following objectives:

i. To observe internal angles of the braced quadrilateral.

ii. To deduce the angles from the coordinates obtained.

iii. To determine the strength of figure of each braced quadrilateral that is to be considered.

iv. To compare results obtained from each figure of the four braced figure for minor triangulation scheme.

v. To determine the best angles in the triangle.

**1.6 STUDY AREA**

The area of study is located in part of federal university of technology, Minna; permanent campus Gidan-Kwano, it is located at new bus park. The major socio-economic activities of the study area is farming. It is located in Bosso local Government Area of Niger state. It lies on longitude 6o27’08.’’4E and Latitude 9o32’03’’46N and about 216.70m above Mean Sea Level.

JONNAH EHIS

Ajayi Crowther University, Oyo

MATTHEW NGBEDE

Ahmadu Bello University

Temitayo Ayodele

Obafemi Awolowo University

Stancy M

Abia State University, Uturu

Ibrahim Muhammad Muhammad

Usmanu danfodiyo university, sokoto

Dau Mohammed Kabiru

Kaduna State College of Education Gidan Waya

Joseph M. Yohanna

Merry From BSU

Samuel From Ajayi Crowther University

Musa From Ahmadu Bello University