Metaball Isocontour Lab: Marching Squares Field Explorer

Presets

Live readouts

Graphs

Kernel profile: field of one blob vs distance (green = iso level T, vertical = outline radius)

Merge test: field at the midpoint of two equal blobs vs separation D

Separate bodies over time

Area where F >= T, as a function of the iso level

Numerical checks

running...

How to use

Drag a blob with the mouse or finger. In the pair scenario, dragging sets the separation directly.
Slide D / D_crit through 1.0 (or tick auto approach): the number of bodies flips from 2 to 1 at exactly that value.
Raise the iso level T and the blobs shrink; lower it and they bridge across gaps. The right-most graph shows the enclosed area as a function of T.
Make the grid coarse (cell size 0.3) and tick grid cells and sample points to see marching squares at work.
Tick gradient arrows to see grad F, and cursor is a blob to merge the blobs with your pointer.
Space pauses, R resets. This is the 2D companion of the 3D lava-lamp lab.

The science

A metaball is a level set. Each blob i has a centre c_i, an influence radius R_i and a strength s_i. The scalar field is the sum of compact kernels, and the visible outline is the contour where the field equals the threshold T:

F(x, y) = Σ_i s_i · f(q_i), q_i = |p - c_i|² / R_i² (f = 0 for q ≥ 1)
outline = { p : F(p) = T }

Kernels: Wyvill (1 - q)³, quadratic (1 - q)² and a clipped Gaussian (Blinn). The heat map shows F itself; the filled shape is the region F ≥ T.

Marching squares. The field is sampled on a grid. For every cell, the four corners are marked inside (F ≥ T) or outside, giving one of 16 cases. Each case has 0, 1 or 2 line segments whose endpoints lie on the cell edges where the sign changes. An endpoint is placed by linear interpolation, t = (T - F_a)/(F_b - F_a). The two ambiguous saddle cases (diagonal corners inside) are resolved with the average of the four corners. Neighbouring segments share edge points, so they link into closed loops. Use the Marching squares preset (coarse grid) to see the sample points, the 16 cases and the interpolated contour points.

When do two blobs fuse? For two equal blobs the field on the line between them has a saddle at the midpoint, F_mid = 2 s f((D/2)²/R²). The outlines merge exactly when F_mid = T; for Wyvill D_crit = 2R √(1 - (T/2s)^1/3). The graph shows F_mid against D, with D_crit and the live closest-pair distance marked. A neck therefore appears with zero width at D_crit and widens as the blobs approach, because the two fields keep adding up.

Gradient. ∇F = Σ s f'(q) · 2 (p - c)/R² points toward increasing field and is perpendicular to the outline. Cohesion in the drifting scenario is a = κ ∇F_others, a toy attraction toward the neighbours' field; a short-range core repulsion keeps blobs from collapsing onto one point. This is a model, not fluid dynamics: real droplets conserve area and obey surface tension, metaballs do not.

Numerical checks (run in this page). (1) the contour polygon area from the shoelace formula vs the closed-form disc area; (2) the contour length vs 2πr; (3) an independent estimate of the same area by counting grid points; (4) the connected-body count at 0.88 and 1.12 times the analytic D_crit must be 1 and 2; (5) the analytic gradient vs a central difference.