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Superellipse — Definition, Formula & Examples

A superellipse is a curve defined like an ellipse but with the exponent 2 replaced by a variable exponent nn. By changing nn, the shape smoothly transitions between a diamond (n<2n < 2), an ellipse (n=2n = 2), and a rounded rectangle (n>2n > 2).

A superellipse centered at the origin with semi-axes aa and bb is the set of points (x,y)(x, y) satisfying ∣xa∣n+∣yb∣n=1\left|\frac{x}{a}\right|^n + \left|\frac{y}{b}\right|^n = 1, where n>0n > 0. When n=2n = 2, this reduces to the standard ellipse equation. Values n>2n > 2 produce curves with flattened sides, while 0<n<20 < n < 2 produce curves that pinch inward toward a diamond or astroid shape.

Key Formula

∣xa∣n+∣yb∣n=1\left|\frac{x}{a}\right|^n + \left|\frac{y}{b}\right|^n = 1
Where:
  • aa = Semi-axis length along the x-direction
  • bb = Semi-axis length along the y-direction
  • nn = Exponent controlling the shape (n > 0)

How It Works

Start with the standard ellipse equation x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 and generalize it by raising the absolute-value terms to the power nn instead of 2. When n=1n = 1, you get a diamond (rotated square if a=ba = b). When n=2n = 2, you recover the ordinary ellipse. As nn increases beyond 2, the curve bulges outward, approaching a rectangle with rounded corners. The special case a=ba = b with n≈2.5n \approx 2.5 is called a "squircle" — a shape famously used by Danish designer Piet Hein for urban furniture and later by Apple for app icons.

Worked Example

Problem: Determine whether the point (1,1)(1, 1) lies inside, on, or outside the superellipse with a=2a = 2, b=2b = 2, and n=4n = 4.
Step 1: Substitute into the superellipse equation.
∣12∣4+∣12∣4=(12)4+(12)4\left|\frac{1}{2}\right|^4 + \left|\frac{1}{2}\right|^4 = \left(\frac{1}{2}\right)^4 + \left(\frac{1}{2}\right)^4
Step 2: Evaluate each term.
116+116=216=18\frac{1}{16} + \frac{1}{16} = \frac{2}{16} = \frac{1}{8}
Step 3: Compare the result to 1. Since 18<1\frac{1}{8} < 1, the point lies inside the superellipse.
Answer: The point (1,1)(1,1) is inside the superellipse because the left side equals 18\frac{1}{8}, which is less than 1.

Why It Matters

Superellipses appear in industrial design, architecture, and computer graphics whenever designers need a shape between an ellipse and a rectangle. Understanding them connects high school algebra — absolute values, exponents, and implicit curves — to real-world design problems like creating smooth button shapes or optimizing table layouts.

Common Mistakes

Mistake: Forgetting the absolute value signs and getting incorrect results for points in quadrants where xx or yy is negative.
Correction: The absolute values ensure the curve is symmetric across both axes. Always apply ∣⋅∣|\cdot| before raising to the power nn, especially when nn is not an even integer.