Superellipse (Lame Curve) Calculator

Semi-axis

a

Semi-axis

b

Exponent

n

Unit:

Decimal Places: 2 3 5 8

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Introduction

A superellipse, or Lame curve, is a closed planar figure defined by the semi-axes $a$ and $b$ together with the exponent $n$ , which collectively determine the curvature and overall morphology of the shape. Varying the exponent allows the curve to transition smoothly between near-rectangular outlines and forms approaching an ellipse, providing a flexible framework for analysing area, perimeter behaviour, and geometric structure through analytical and numerical methods.

What this calculator does

Based on three primary inputs - the semi-axis lengths and the shape-defining exponent - it performs a precise mathematical evaluation of the superellipse. It generates the total surface area using Gamma functions and approximates the perimeter through high-resolution numerical integration. Additionally, it outputs shape classification, the rectangularity fill factor, and the area ratio relative to a standard ellipse for comparative analysis.

Formula used

The area calculation utilises the properties of the beta function expressed through the Gamma function $Γ$ . The perimeter is determined by integrating the arc length of the parametric equations $x = a \cos^{2 / n} t$ and $y = b \sin^{2 / n} t$ over the interval $0$ to $2 π$ .

A = 4 a b \frac{{(Γ (1 + \frac{1}{n}))}^{2}}{Γ (1 + \frac{2}{n})}

P = 4 \int_{0}^{\frac{π}{2}} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t

How to use this calculator

1. Enter the numeric values for the semi-axes $a$ and $b$ .
2. Input the exponent $n$ to define the curvature of the shape.
3. Select the preferred measurement unit and decimal precision.
4. Execute the calculation to view the summary table, step-by-step logic, and visual plot.

Example calculation

Scenario: Analysing the geometric properties of a squircle-like structure in a mathematical modelling exercise to compare its area against a bounding rectangle.

Inputs: $a = 10$ , $b = 10$ , and $n = 4$ .

Working:

Step 1: $a \times b = 10 \times 10 = 100$

Step 2: $1 / n = 1 / 4 = 0.25$

Step 3: $Γ (1.25) \approx 0.9064$

Step 4: $A = 400 \times ({0.9064}^{2} / Γ (1.5))$

Result: Area ≈ 370.81, Perimeter ≈ 70.39.

Interpretation: The result confirms that an exponent of 4 creates a shape with an area significantly larger than a standard ellipse ( $n = 2$ ).

Summary: The squircle fills approximately 92.7% of its bounding box.

Understanding the result

The outputs reveal the efficiency of the shape in filling space. A fill factor approaching 100% indicates the curve is becoming more rectangular. The area ratio highlights how much the superellipse deviates from a standard ellipse, providing insight into the curvature's influence on the internal region.

Assumptions and limitations

The calculation assumes the semi-axes are positive non-zero values. The perimeter is limited by the precision of 2000-step numerical integration, and the exponent $n$ must remain within 0.01 and 10,000 to maintain numerical stability and avoid overflow during Gamma function evaluation.

Common mistakes to avoid

Common errors include entering negative values for semi-axes or using exponents close to zero, which can lead to extreme sensitivity and numerical errors. Additionally, users may confuse the semi-axes with the full widths of the shape, resulting in an area calculation that is four times larger than intended.

Sensitivity and robustness

The area calculation is stable for most typical exponents. However, the perimeter calculation and Gamma functions become highly sensitive to very small or very large values of $n$ . Small changes in the exponent significantly alter the shape classification and the resulting geometric properties, particularly when $n < 1$ .

Troubleshooting

If an error regarding numerical overflow occurs, ensure that the exponent and semi-axis inputs are within the specified stable ranges. If the results seem inconsistent, verify that the CSRF token has not expired by refreshing the page. Inaccurate results may also stem from inputs exceeding the $10^{12}$ limit.

Frequently asked questions

What is a squircle?

A squircle is a specific type of superellipse where the semi-axes are equal and the exponent $n$ is greater than 2, typically 4.

Why is the perimeter calculated in steps?

Unlike the area, the perimeter of a superellipse generally lacks a closed-form solution for most exponents, requiring numerical integration for an accurate result.

What happens if n equals 1?

When the exponent $n$ is exactly 1, the shape produced is a rhombus or a diamond.

Where this calculation is used

This mathematical model is frequently used in geometric design and mathematical modelling to study curves that lie between an ellipse and a rectangle. In environmental science, it may be used to model the cross-sections of certain biological structures or fluid conduits. In advanced geometry and calculus, it serves as a primary example of how the manipulation of exponents in algebraic equations can transform simple shapes into complex curves. It is also used in population studies to define boundary regions for non-standard distributions.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.