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 union of 4 arcs of parabolas: parabola:   2/3 astroid: ditto  - 1 union 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 union of 8 arcs of parabolas: parabola:   2/3 and its symmetric image about y = x, of equation ditto; it is the union of two evolutes of hyperbola.  - 1 union 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. 