inertia_tensor — 3D `moment_invariant` op

• Data kinds: points → matrix

• Call: import fullseye as fs; fs.ledger.inertia_tensor(points) -> 'np.ndarray' (to call the implementation directly, import moments3d; moments3d.inertia_tensor(points) -> 'np.ndarray'; from the registry, ops3d.get("inertia_tensor"))

Usage

The inertia tensor (3,3) of a point set (from the central second moments; uniform mass, total mass 1).

> The detailed description below is the original text — the summary and the headings are translated.

I_xx = mean(y²+z²), I_yy = mean(x²+z²), I_zz = mean(x²+y²),

I_xy = -mean(xy), I_xz = -mean(xz), I_yz = -mean(yz)。

共分散 C を使うと I = tr(C)·E₃ − C(E₃ は単位行列)と等価。対称・半正定値。

重心中心化のため並進不変。

Returns

-------

np.ndarray, shape (3, 3)

対称な慣性テンソル。

補足:

• 単位は長さ²(質量 1 の等質量点とみなすので密度は入らない)。点群を回転で回すと `R I Rᵀ に写り、固有値(principal_moments)が回転不変量、固有ベクトルが主軸(moment_axes`)。

• 入力は (N,3)、N >= 1(1 点なら零行列)。形状不正・非有限は `ValueError`。

• 共分散 C とは `I = tr(C)·E₃ - C の関係で、C と I の固有ベクトルは同じ、固有値は tr(C) - c_i`。

• 実体(体積)のモーメントではなく サンプル点 のモーメントなので、同じ形でも点密度の偏りで値が変わる。密度を均すなら前段で `voxel_grid_downsample`。決定論的。

References (sample data, literature)

• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).

• Operator provenance and references — the sources of the research/methods this op family came from.

Runnable examples (verified samples that actually call this op)

• moment_invariants — py -3.11 examples_3d/moment_invariants.py

Ops the type connects to (they accept `matrix` as input)

fuse_to_voxel

Same category (`moment_invariant`)

moment_invariants · principal_moments · central_moments

*Provenance: moments3d.py — 3D operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*

inertia_tensor — 3D moment_invariant op