PLANAR CURVE SHORTENING FLOW
A dense curve shrinking to the origin
Follow the same finite segment as X(s,t) = √(−t) X(s), for t < 0.
One full excursion Its successive continuations Inner and outer radii
Four times, one spatial scale
Select a frame to move the time slider. The orange segment keeps its identity throughout the flow.
Each excursion returns from the inner circle to the inner circle after touching the outer circle. The next is rotated by 2πα. An irrational α prevents closure; drawing more excursions reveals how the complete curve is dense in the annulus. Changing the excursion count changes the displayed parameter window, not time.
The dashed circles are the boundaries of the image closure, not additional flow components. The visible segment has artificial endpoints. As t → 0⁻, both radii tend to zero; t = 0 is not a regular time slice.
Source and numerical accuracy
Halldórsson, Self-similar solutions to the curve shortening flow, §2 and §5, Theorem 5.1 (pp. 11–13), Figures 4–7 (pp. 17–18). The source uses B = −1 in its proof; these profiles use B = −1/2.
The spatial profile solves u′ = uv, v′ = 1/2 − u², θ′ = u. We reconstruct X = −2i eⁱᶿ(u + iv) and shoot for the displayed irrational turning target. The ODE profile is numerical; the similarity time law is exact. Floating-point values and finite drawings cannot prove irrationality or show the full dense image.