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geometry > Euclidean geometry > plane curve > Lamé curve

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Lamé curve

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 a different overall shape.
In the Cartesian coordinate system, the set of all points ${\\displaystyle (x,y)}$ on the curve satisfy the equation
${\\displaystyle \\left|{\ rac {x}{a}}\ ight|^{n}\\!\\!+\\left|{\ rac {y}{b}}\ ight|^{n}\\!=1,}$
where ${\\displaystyle n,a}$ and ${\\displaystyle b}$ are positive numbers, and the vertical bars around a number indicate the absolute value of the number. The 3-dimensional generalization is called superellipsoid (some literatures also name it superquadrics).
In the Polar coordinate system, the superellipse equation is (the set of all points ${\\displaystyle (r,\ heta )}$ on the curve satisfy the equation) :
${\\displaystyle r=\\left(\\left|{\ rac {\\cos(\ heta )}{a}}\ ight|^{n}\\!\\!+\\left|{\ rac {\\sin(\ heta )}{b}}\ ight|^{n}\\!\ ight)^{-1/n}\\!.}$
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Superellipse)

plane curve

squircle

superellipse

courbe de Lamé

francés
superellipse

URI

http://data.loterre.fr/ark:/67375/PSR-B827BGSZ-S

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