Curve Shortening Flow

Watch curves shrink to circles and vanish — the Gage-Hamilton-Grayson theorem

Curve Shortening Flow

Let every point of a closed curve move inward at a speed set by how sharply it bends. Any simple closed curve, however wild, ends up a shrinking circle.

Curve Shortening Flow

Pick a starting curve and press play. Color shows curvature (blue low, orange/red high), so watch the red bends move fastest while the area, perimeter, and isoperimetric ratio update live.

Area
0.0
Perimeter
0.0
L²/(4πA)
1.0000
circle = 1.0000
Low κ
High κ
Go deeper:Curve Shortening Flow

Curve shortening flow (CSF) moves each point of a plane curve in the direction of its curvature normal at a speed proportional to the curvature. The evolution equation is∂γ/∂t = κN, where κ is the curvature and N is the inward unit normal.

Points of high curvature move faster, smoothing out sharp bends. The flow reduces the total length monotonically and drives any simple closed curve toward a perfectly round circle before it collapses to a single point.

Go deeper:The Gage-Hamilton Theorem (1986)

If γ is a convex simple closed curve in the plane, then under curve shortening flow:

1. The curve remains convex for all time.
2. The curve shrinks to a point in finite time T.
3. Rescaled to enclose constant area, the curve converges to a circle.

This was the first rigorous result showing that CSF drives convex curves to “round points.” The enclosed area satisfies A(t) = A(0) − 2πt, so extinction occurs at time T = A(0)/(2π).

Go deeper:Area Evolution
dA/dt = −∫ κ ds = −2π
  • Linear decrease: Area shrinks at a constant rate 2π
  • Extinction time: T = A(0)/(2π)
  • Universal: Rate is independent of shape

Grayson's Theorem

Start from a deeply non-convex shape and watch for the moment it becomes convex—after that it shrinks steadily to a round point.

Non-convex

Grayson's Theorem: every simple closed curve becomes convex under CSF, then shrinks to a round point.

Go deeper:Grayson's Theorem: The Full Picture

Grayson's theorem (1987) extends the Gage-Hamilton result to all simple closed curves—even highly non-convex ones. Any embedded closed curve first becomes convex, then shrinks to a round point.

The surprising part of Grayson's theorem is that non-convex curves don't develop singularities before becoming convex. No matter how wild the initial shape—deep indentations, narrow necks, complex spirals—the flow smooths them all out without self-intersection.

This is remarkable because in higher dimensions, mean curvature flow can develop singularities (neckpinches). The 2D case is special: the curve shortening flow is perfectly well-behaved for embedded curves.

Isoperimetric Ratio

Pick a shape and watch the roundness score L²/(4πA) on the graph. Once the curve turns convex, it falls steadily toward 1, the value for a circle.

Current Ratio
1.0000

Top: evolving curve under CSF. Bottom: isoperimetric ratio L²/(4πA) decreasing toward 1 over time.

Go deeper:Isoperimetric Ratio
L²/(4πA) ≥ 1, equality iff circle
  • Decreasing for convex curves: Once the curve is convex, the ratio monotonically approaches 1 under CSF (Gage's inequality)
  • Circle = 1: The minimum value, achieved only by circles
  • Measures roundness: Higher values mean less circular

Key Takeaways

  • Curve shortening flow moves each point inward at a speed equal to its curvature.
  • Convex curves stay convex and shrink to round points (Gage–Hamilton).
  • Every simple closed curve becomes convex first, then does the same (Grayson).
  • Area falls at the same rate 2π for every curve.

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