Semidiameter in the context of Chord (geometry)

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

The length of the semi-major axis a of an ellipse is related to the semi-minor axis's length b through the eccentricity e and the semi-latus rectum ${\displaystyle \ell }$, as follows:

A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but defined by an equation that allows for various shapes between a rectangle and an ellipse.

In two dimensional Cartesian coordinate system, a superellipse is defined as the set of all points ${\displaystyle (x,y)}$ on the curve that satisfy the equation$${\displaystyle \left|{\frac {x}{a}}\right|^{n}\!\!+\left|{\frac {y}{b}}\right|^{n}\!=1,}$$where ${\displaystyle a}$ and ${\displaystyle b}$ are positive numbers referred to as semi-diameters or semi-axes of the superellipse, and ${\displaystyle n}$ is a positive parameter that defines the shape. When ${\displaystyle n=2}$, the superellipse is an ordinary ellipse. For ${\displaystyle n>2}$, the shape is more rectangular with rounded corners, and for ${\displaystyle 0<n<2}$, it is more pointed.