Title: Perimeter of the Elliptical Arc a Geometric Method
Abstract:There are well known formulas approximating the circumference of the Ellipse given in different periods in history.However there lacks a formula to calculate the Arc length of a given Arc segment of a...There are well known formulas approximating the circumference of the Ellipse given in different periods in history.However there lacks a formula to calculate the Arc length of a given Arc segment of an Ellipse.The Arc length of the Elliptical Arc is presently given by the Incomplete Elliptical Integral of the Second Kind, however a closed form solution of the Elliptical Integral is not known.The current solution methods are numerical approximation methods, based on series expansions of the Elliptical Integral.This paper introduces a Geometric Method (procedure) to approximate the Arc length of any given Arc segment of the ellipse.An analytical procedure of the defined geometric method is detailed.For a Circular Arc the Arc length of the Circular Segment is given through angular relations.However a similar formula for the Arc length of the Ellipse, using known trigonometric Identities is not known.The paper translates the defined geometric procedure to analytical, algebraic method using the known trigonometric identities.This method thus gives a method to estimate the Incomplete Elliptical Integral of the Second Kind using trigonometric identities.The formulae for Circular Arc can be considered to be based on the intuitive reasoning "Equal Arcs cuts Equal Angles at the center for a Circle".This paper presents a similar intuitive logic for the Elliptical Arc.The proposal is based on a geometric discovery based on intuitive reasoning and the algebraic theory explaining the intuitive logic is as such not yet developed.An analytical research into the intuitive logic behind the geometric approximation shall result in the development of the general theory and an analytical, algebraic proof of the proposed method.Read More