Curve Shortening Flow

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

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.

Interactive: Curve Shortening Flow

Choose an initial curve and watch it evolve under CSF. Each vertex moves inward proportionally to its curvature κ. The curve is colored by local curvature—blue for low, orange/red for high. Track the area, perimeter, and isoperimetric ratio in real time.

Area
0.0
Perimeter
0.0
L²/(4πA)
1.0000
circle = 1.0000
Low κ
High κ

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).

Interactive: Grayson's Theorem

Grayson's theorem (1989) 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. Watch the moment of convexification.

Non-convex

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

Grayson's Theorem: The Full Picture

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.

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

Isoperimetric Ratio

L²/(4πA) ≥ 1, equality iff circle
  • Always decreasing: The ratio monotonically approaches 1 under CSF
  • Circle = 1: The minimum value, achieved only by circles
  • Measures roundness: Higher values mean less circular

Interactive: Isoperimetric Ratio

The isoperimetric ratio L²/(4πA) measures how far a curve deviates from a perfect circle. Under curve shortening flow, this ratio monotonically decreases toward 1. Watch the ratio evolve alongside the curve in real time.

Current Ratio
1.0000

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

Key Takeaways

  • Curve shortening flow evolves curves by ∂γ/∂t = κN, moving each point proportionally to curvature
  • The Gage-Hamilton theorem proves convex curves shrink to round points in finite time T = A(0)/(2π)
  • Grayson's theorem extends this to all simple closed curves: they first become convex, then shrink to round points
  • The isoperimetric ratio L²/(4πA) monotonically decreases toward 1 (the circle value) under CSF
  • Area decreases at the universal rate dA/dt = −2π, independent of shape

Next: Mean Curvature Flow — extending curve shortening to surfaces in 3D, where singularities can form.