Mathematical Determination of Thread Length in the Knit Loop

Abstract

A method is proposed for determining the thread length in the knit loop using the law of normal probability distribution. An equation was obtained for determining the thread length in the loop for calculation with an integral method in the assumption that the shape of the loop is described by a Gaussian curve. The proposed method can be used to obtain sufficiently high results, since the deviation is no greater than 4%.It is first necessary to have a mathematical description of the structure of the designed knit which is distinguished by the variety and complexity of the weave, for implementing automated design of knits.Ensuring sufficient accuracy of the model means taking into consideration all important properties and links deriving from secondary unimportant properties and links in idealizing a real object. Solving this problem is not only a function of the character of the object but also of the problem posed. In representing a real object with sufficient accuracy, the mathematical model must at the same time be as simple as possible, since working with a complicated model is not only difficult but can also be almost impossible to do. The contradictory nature of these requirements frequently requires giving up accuracy in the interest of simplicity, but such a compromise is only acceptable if the model still reflects the important properties of the real object. The development of methods of simplifying real objects and systems to construct maximally simple mathematical models is a central problem in any sector. In modeling real objects, it is useful to be oriented toward mathematical models of standard form which are provided by the corresponding apparatus.We calculated the loop thread length of one of the basic elements in constructing knit fabrics. A relatively large number of equations has been proposed for solving this problem. Some of them are based on purely empirical dependences and other are based on approximating the results to the geometric shape of the loop.We propose describing the loop shape as a Gaussian curve. A comparison of the Gaussian curve with the shape of the loop photographed in the electron microscope ( Fig. 1) explains our hypothesis on the use of this curve to determine the knit loop length, i.e., ( )