Physics2D / transfer basis

One particle, nine grid nodes.

Drag the gold particle, or use the coordinate sliders. Colored node area follows its interpolation weight.

Coordinates are measured in grid-cell units. The local stencil origin follows base = floor(position − ½), so its shifted coordinate lies in [½, 1½]. This study holds that local origin fixed.
0.80
0.70
1.0
Σ weights
Grid mass / particle mass
Recovered position
Second moments xx, yy; xy
Recovered affine velocity
Inspect all nine weights
w₀(f) = ½(1½ − f)²
w₁(f) = ¾ − (f − 1)²
w₂(f) = ½(f − ½)²
wᵢⱼ = wᵢ(fₓ) wⱼ(fᵧ) x basis:
y basis:
The tensor-product weights sum to one, reproduce position and yield D = ¼I in grid-cell units. Restoring cell width Δx gives D = ¼Δx²I, the constant used in the MLS/APIC transfer.