Measuring Curvature Method for the Exact Value of the Ellipse Perimeter

Abstract

This article introduces a novel geometric approach known as the Measuring Curvature Method MCM for precisely determining the perimeter of an ellipse. The MCM involves the construction of three centered circles: the Unit Circle, which has a radius equal to the major axis of the ellipse b; the Base Circle, with a radius of 2b/π; and the Measuring Circle. The radius of the Measuring Circle intersects with a newly defined curvature termed the Measuring Curvature. By utilizing this method, the circumference of the Measuring Circle provides the exact value of the ellipse perimeter. To establish a practical procedure for the MCM, data is generated, refined, and organized to create a chart and establish an empirical relationship between the radius of the Measuring Circle and the lengths of the ellipse's semi-axes. As a result, the perimeter can be readily determined by multiplying the radius obtained from the developed equation by the constant 2π. Additionally, a mathematical analysis is conducted to derive an equation that connects the radius of the Measuring Circle with four parameters: the major and minor axes of the ellipse, and the horizontal and vertical coordinates of the intersection point on the Measuring Curvature between the ellipse and its measuring circle. This equation enables the perimeter to be conventionally and simply calculated. To further validate the accuracy of the MCM theorem, numerical comparisons are performed using Ramanujan's method with the PRI test. The outcomes demonstrate that employing the MCM theorem yields highly reliable and precise calculations for determining the perimeter of an ellipse.