The Lamé Curve

A superellipse, or Lamé curve, generalizes the ellipse by raising coordinate ratios to an arbitrary exponent $n$. The standard form $\left|x\right|/<mi>a</mi>{|}^n$ plus the corresponding term for $y$ equals one. When $n=2$ we recover an ordinary ellipse; when $n=1$ the shape becomes a diamond. Large $n$ values yield shapes approaching rectangles with rounded corners, while fractional $n$ values produce star-like forms. Superellipses gained fame through the work of the Danish scientist Piet Hein, who popularized them in art and architecture, including the design of the Sergels Torg plaza in Stockholm.

The area enclosed by a superellipse is $4ab{\mathrm{\backslash Gamma}}^{1+\frac{1}{n}}{\mathrm{\backslash Gamma}}^{1+\frac{1}{n}}/{\mathrm{\backslash Gamma}}^{1+\frac{2}{n}}$, where $\mathrm{\backslash Gamma}$ is the Gamma function. This formula seamlessly connects geometry with special functions, reducing to $\mathrm{\backslash pi\; ab}$ when $n=2$. Because $\mathrm{\backslash Gamma}$ generalizes factorials, it smoothly handles fractional exponents beyond the scope of elementary calculus.

Derivation via Polar Coordinates

The area integral is easily expressed in polar form. Substitute $x=r\mathrm{\backslash cos\backslash theta}$ and $y=r\mathrm{\backslash sin\backslash theta}$, giving $r^n[{\mathrm{\backslash cos\backslash theta}}^n/a^n+{\mathrm{\backslash sin\backslash theta}}^n/b^n]=1$. Solving for $r$ and integrating $r\mathrm{dr}\mathrm{d\backslash theta}$ over $0$ to $\mathrm{2\backslash pi}$ yields the Gamma-function expression above. The derivation showcases how classical calculus techniques interact with special functions to reveal generalized geometric formulas.

Working with the Calculator

To use the tool, provide positive lengths $a$ and $b$ along the coordinate axes and choose an exponent $n$. The algorithm relies on math.js for the Gamma function. After computing the value $4ab{\mathrm{\backslash Gamma}}^{1+\frac{1}{n}}{\mathrm{\backslash Gamma}}^{1+\frac{1}{n}}/{\mathrm{\backslash Gamma}}^{1+\frac{2}{n}}$ the program outputs the result in square units. Experiment with different exponents to visualize how area changes from a circular ellipse ($n=2$) to a diamond ($n=1$) or beyond.

Context and History

Superellipses have found surprising applications in design because their intermediate shapes blend curvature and squareness elegantly. In architecture, they allow smooth corners without the abruptness of rectangular edges, while in data visualization they serve as visually pleasing boundaries for scatter plots. Mathematically, superellipses are related to p-norms and Minkowski distances. They generalize classical ellipses just as p-norms generalize Euclidean distance. The interplay between geometry and analysis continues to inspire new uses in computer graphics, optimization, and beyond.

By calculating area precisely with the Gamma function, this calculator bridges geometry and special function theory. It also illustrates how seemingly advanced mathematics—such as the analytic continuation of factorials—emerges naturally when exploring simple geometric variations. Use it to verify textbook examples or to explore your own shapes for art, architecture, or mathematical curiosity.