### APPROXIMATION OF A CATENARY FORMED OUT OF A ROPE ACCORDING TO THE HEIGHTS OF ITS POINTS OF HOLD

Milivoj Vulic
unpublished
INTRODUCTION 1.An Elementary Definition of a Catenary When we talk about field observations obtained with the aid of a rope or measuring tape, held in two different points and at a particular height, it is necessary to acknowledge that the rope or measuring tape will sag at points where it is not firmly held, and therefore also assume a particular shape that will not be a straight line, which would invariably be the ideal in such a situation. This shape achieved through the sagging of a rope or
more » ... agging of a rope or measuring tape is referred to as a catenary in mathematics [1] [2], and is influenced primarily by the strain in the points that hold the rope or measuring tape in place, its weight, its weight distribution i.e. density of different segments of the rope, and the distance between the two points where the rope is held firmly i.e. the distance over which the sagging is permitted to happen. As any of these factors may influence field observations, we have put much effort into analysing them, as well as analysing the catenary formed by the rope in such a way as to minimise its effects on the measurements we obtain through the use of ropes and measuring tapes on the field. It is usually possible to completely discard at least the effect of a different weight distribution within the rope, as that is more of an anomaly than a regular and Abstract: When involved in field measurements, we frequently employ a kind of rope or measuring tape fastened to two different points and stretched out across a distance. In order for measurements using this tool to be more accurate, it is necessary to acknowledge and examine the sagging action of a rope or measuring tape when used in this way, as instead of acting as a straight line, it assumes the shape of a kind of curve, a catenary, the parameters of which are influenced by many factors, one of which is always the height difference or lack thereof between the points where the rope is held. In this paper, we assume that most ropes used for this purpose are of a uniform weight distribution, undamaged, and of no elasticity, as well as used in an ideal environment of uniform external impact upon measurements, and aim to examine the effect of different heights of points of hold of a rope upon the catenary its curve forms, as well as attempt to define and anylse this catenary by approximating it into a polynomial.