LAMÉ CURVE

 Curves studied by Lamé in 1818. Gabriel Lamé (1795-1870): French mathematician and engineer. Other names for >2: super-ellipse, super-circle (if a = b), squircle (contraction of the words square and circle).

 Cartesian equation of :  ;  :  with a, b > 0 , ;  Cartesian parametrization of  : . Area delimited by  :  where  , i.e.  if .

The Lamé curves and are defined by their Cartesian equation above.
For rational values of , the curve , the part of located in the quadrant, is a portion of an algebraic curve of degree pq ?, and equation ? (when p is even,  and  coincide); the same holds for the curves .

Examples of curves with a = b:

 Lamé curve associated algebraic curve figure: the Lamé curve in red, the associated algebraic curve in green. 1 square: line: 2 circle: ditto 3 Lamé cubic: 1/2 reunion of 4 arcs of parabolas: parabola: 2/3 astroid: ditto - 1 reunion of 4 branches of rectangular hyperbolas: rectangular hyperbola: -2 crosscurve: ditto

 Lamé curve associated algebraic curve figure: Lamé curve in red, the associated algebraic curve in green. 1 eight half-lines: line: 2 rectangular hyperbola: 1/2 reunion of 8 arcs of parabolas: parabola: 2/3 and its symmetric image about y = x, of equation ditto; it is the reunion of two evolutes of hyperbola. - 1 reunion of 8 branches of rectangular hyperbolas: rectangular hyperbola: -2 bullet nose curve:

When a = b =1 and a  = n is an integer,  is the Fermat curve.