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LAMÉ CURVE
Curves studied by Lamé
in 1818.
Gabriel Lamé (17951870): French mathematician and engineer. Other names for >2: superellipse, supercircle (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 halflines:  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.
See also the Lamé surfaces.
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© Robert FERRÉOL 2017