Geometric Properties of Euler’s Elastica

Abstract The aim of this article is to provide a concise and coherent summary of existing results on the geometric properties of the planar Euler elastica. No new governing equation or solution family is proposed; the emphasis is on presenting and synthesising established results. Taking the classical equilibrium equation for a planar, inextensible and unshearable elastic rod under terminal loads as a starting point, the article collects closed-form expressions in terms of Jacobi elliptic functions for the tangent angle, curvature, intrinsic coordinates, and internal force components. The elliptic modulus $k$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> is used as the organising parameter, enabling the inflectional and non-inflectional families, the homoclinic separatrix, and the straight and circular limiting cases to be described within a common notation. The roles of the solution parameters $k$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> , $\lambda $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> , $C$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> , $\alpha $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , and the coordinate constants are clarified in terms of shape, scale, phase, orientation, and position. Symmetries, periodicity, bounds, and characteristic points are summarised, while inflection, compression, and shear points are interpreted geometrically and mechanically. Conditions for self-intersections and self-touching points are presented, and threshold values of $k$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> are related to changes in topology. Formulae for characteristic dimensions are also collected.