# Lamé Curve Calculator

Lamé curves are generalizations of ellipses. The equation of an ellipse in Cartesian coordinates is

(x/a)² + (y/b)² = 1,

where **a** and **b** are the half-width and half-length of the ellipse. The equation of a Lamé curve is

|x/a|^{n} + |y/b|^{n} = 1,

where **n** is greater than zero. As with the ellipse, **a** and **b** are the half-width and half-length, while the third parameter **n** governs the general shape of the curve. Ellipses and circles are special cases of Lamé curves where **n** = 2.

Lamé curves in which **n** is greater than 2 have the eye-pleasing shape of rectangles with rounded corners and are frequently used in design. Sometimes these curves are called superellipses. The city square *Sergels Torg* in Stockholm, Sweden was constructed in the shape of a superellipse with **a** = 5, **b** = 6, and **n** = 2.5.

When **n** is between 1 and 2, the curve resembles a diamond with rounded corners and sides that puff outward. In the image below are examples of Lamé curves with different values of **a**, **b**, and **n**. The larger the value of **n**, the more rectangular the figure becomes.

**n**= 4 and

**a**=

**b**, the figure is called a

*squircle*, a portmanteau of

*square*and

*circle*. The figure resembles are square with rounded corners. When

**n**= 2/3 and

**a**=

**b**, the figure is called an

*astroid*(not to be confused with asteroid).

### Area of a Lamé Curve

The area of a Lamé curve can be expressed in terms of the Gamma Function. The formula isArea = 4abΓ(1 +

^{1}/

_{n})

^{2}/Γ(1 +

^{2}/

_{n}).

There is no similarly neat formula for the perimeter of a Lamé curve, so numerical methods must be employed. You can also use the calculator above to compute the area and perimeter of a Lamé curve. The calculator is most accurate when

**n**is less than 10.

© *Had2Know 2010
*