Tool
Average over a sphere
Average a polynomial on the unit sphere with twelve icosahedral nodes of weight 1/12.
Normalized polynomial: 1 x^0 y^0 z^0.
| Quantity | Value | Arithmetic |
|---|---|---|
| Degree | 0 | exact |
| Continuous average | 1 | exact rational |
| Twelve-node average | 1 | binary64 sample |
| Sample minus exact | 0 | binary64 difference |
| Guarantee | exact through degree 5 for this rule | stated rule |
| # | x | y | z | weight |
|---|---|---|---|---|
| 1 | 0.000000 | -0.525731 | -0.850651 | 1/12 |
| 2 | -0.525731 | -0.850651 | 0.000000 | 1/12 |
| 3 | -0.850651 | 0.000000 | -0.525731 | 1/12 |
| 4 | 0.000000 | -0.525731 | 0.850651 | 1/12 |
| 5 | -0.525731 | 0.850651 | 0.000000 | 1/12 |
| 6 | 0.850651 | 0.000000 | -0.525731 | 1/12 |
| 7 | 0.000000 | 0.525731 | -0.850651 | 1/12 |
| 8 | 0.525731 | -0.850651 | 0.000000 | 1/12 |
| 9 | -0.850651 | 0.000000 | 0.525731 | 1/12 |
| 10 | 0.000000 | 0.525731 | 0.850651 | 1/12 |
| 11 | 0.525731 | 0.850651 | 0.000000 | 1/12 |
| 12 | 0.850651 | 0.000000 | 0.525731 | 1/12 |
Transfer field
Move samples between three exact rotations of the same twelve-node rule. Degree at most 2 is certified. A finer subdivision is not enabled.
The nodes are unnormalized, with squared radius φ+2. Unit-sphere coordinates are not in Q(φ). The basis is 1, x, y, z, xy, xz, yz, x²−y², 2z²−x²−y². The binary64 table above keeps its own arithmetic label. This panel uses exact elements of Q(φ).
| # | Coefficient | Target sample | Source residual |
|---|---|---|---|
| 1 | 1: 5/2 + (11/2) φ | 47/50 + (53/25) φ | -3/2 + (0/1) φ |
| 2 | x: 3/2 + (3/2) φ | 27/50 + (88/25) φ | -3/2 + (0/1) φ |
| 3 | y: 3/2 + (3/2) φ | -53/50 + (-12/25) φ | -3/2 + (0/1) φ |
| 4 | z: 3/2 + (3/2) φ | 197/50 + (203/25) φ | -3/2 + (-3/1) φ |
| 5 | xy: 0/1 + (0/1) φ | 9/10 + (26/5) φ | -3/2 + (-3/1) φ |
| 6 | xz: 0/1 + (0/1) φ | 157/50 + (198/25) φ | -3/2 + (-3/1) φ |
| 7 | yz: 0/1 + (0/1) φ | 17/50 + (38/25) φ | 3/2 + (3/1) φ |
| 8 | x^2-y^2: 1/2 + (1/4) φ | 57/10 + (53/5) φ | 3/2 + (3/1) φ |
| 9 | 2z^2-x^2-y^2: 0/1 + (-1/4) φ | 97/50 + (63/25) φ | 3/2 + (3/1) φ |
| 10 | 167/50 + (188/25) φ | 3/2 + (0/1) φ | |
| 11 | 207/50 + (163/25) φ | 3/2 + (0/1) φ | |
| 12 | 307/50 + (273/25) φ | 3/2 + (0/1) φ |
| Check | Result |
|---|---|
| Grids | 3 |
| Nodes | 12 |
| Basis recoveries | 9 |
| Gram entries | 81 |
| Round trip equals model projection | exact |
| Composition T(R²←R) T(R←I) = T(R²←I) | exact |
| Constants preserved | exact |
| Higher frequency | not enabled |
| Selected residual | nonzero |
| Projection idempotent on this sample | exact |