OBSERVATION CHAMBER / ACTIVE
Catenoid Minimal Surface Observatory
Separate two coaxial rings and observe two catenoid branches, their areas, curvatures, and the limit of a connected minimal surface.
LAB 88 / GEOMETRY / MINIMAL SURFACE · Geometry route
01 / OBSERVE THE SURFACE
Two rings, two possible necks
Drag to rotate. Choose “ring separation” to drag the rings apart vertically. With the canvas focused: arrows rotate, + / − change q, R resets.
Cyan: thick branch · violet: thin branch · amber: boundary rings / two disks. The view uses R as its length unit.
03 / READ THE GEOMETRY
Area and curvature are different tests
- Connected solutions
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- Full separation d
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- Neck radius a
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- Surface area A
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- A / (2πR²)
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- max |H_mean|
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- Gaussian K at neck
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- |R − a cosh(d/2a)| / R
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H_mean is recomputed from the profile derivatives at 33 locations. It vanishes because the two principal curvatures cancel. K is negative: zero mean curvature does not mean a flat surface.
MODEL / SCOPE
A membrane spanning fixed circles
Two parallel, coaxial rings have the same fixed radius R. Gravity and the pressure difference are zero. The connected surface is a surface of revolution, with no volume constraint. This is an equilibrium model, not a real-time fluid collapse simulation, a constant-volume liquid bridge, or a general Plateau solver.
Unlike the hanging chain in LAB 82, the observable here is a two-dimensional membrane: area, principal curvatures, and stability under fixed boundary perturbations.
FORMULA / TWO BRANCHES
One boundary, two stationary surfaces
R = a cosh(d/(2a)), u = d/(2a)
q = d/R = 2u / cosh(u)
A = 2πa²[u + sinh(u) cosh(u)]
H_mean = (1 + r′² − rr″) / [2r(1+r′²)^(3/2)]
K(z) = −1 / [a² cosh⁴(z/a)]
Bisection solves each monotone branch separately. At u tanh(u) = 1 the roots merge. Above q = 1.3254868386983631, no connected catenoid solves the boundary condition.
STABILITY / AREA
Two thresholds, not one
The thick branch first matches the area of two disconnected disks at q ≈ 1.055394794. It stays locally stable beyond this crossing until the fold at q ≈ 1.325486839. The thin branch is unstable; the merged solution is neutral at the stability limit. These labels use the known catenoid second-variation criterion, not area comparison alone.
The amber horizontal line is the disconnected disk competitor A = 2πR². The colored curves describe the family of equilibria, not a time trajectory.
HOW TO OBSERVE
Three controlled comparisons
- Switch between thick and thin at q = 0.8. Both satisfy the rings and have zero mean curvature, but their stability differs.
- Open area comparison. Visit the area crossing, then the solution limit. Move q above the limit and inspect the empty state.
- Double R from 1 to 2. Area grows fourfold; q and A/(2πR²) remain unchanged. Generate a report or PNG to record the observation.
04 / RECORD AND VERIFY
Record the current equilibrium
Download PNGGenerated report
Checks cover 2/1/0 branches, boundary residuals, curvature, independent area quadrature, scale, invalid input, and the double root. Reports and download links clear when the model or view changes.
Karcher & Polthier — Geometric Properties III
Jana & Kar — Perturbative stability of catenoidal soap films
OBSERVATION POLICY
This static mathematical model runs in your browser. Inputs, reports, and images stay local. It does not measure a real soap film, visitor, or material.
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