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LAMÉ CURVE


cas alpha positif cas alpha négatif

 
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.

See also the Lamé surfaces.


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© Robert FERRÉOL 2017