UTM縮尺係数の公式
k ≈ k₀(1 + q²/(2Rk₀)²)
where:
k₀ = 0.9996 (central meridian scale factor)
q = distance from central meridian (metres)
R = local radius of curvature (~6,371,000 m)
Simplified approximation:
k ≈ 0.9996 + 1.23 ±10⁻¹⁰ ±q²
参照テーブル
| Distance from CM (km) | Grid Scale Factor k | Distortion ppm | 1 km Ground Error |
|---|---|---|---|
| 0 km (Central Meridian) | 0.99960 | -400 ppm | -0.40 m |
| 90 km | 0.99976 | -240 ppm | -0.24 m |
| 180 km | 1.00000 | 0 ppm | 0.00 m (true scale) |
| 250 km | 1.00027 | +270 ppm | +0.27 m |
| 333 km (Zone Edge) | 1.00040 | +400 ppm | +0.40 m |
工学的影響
On a typical 5 km engineering traverse near the central meridian, applying grid distances directly to ground measurements introduces a systematic error of approx. 2 metres. This is far beyond the tolerance of:
- Legal boundary surveys (typically ±0.05 m)
- DOT highway staking (typically ±0.03 m)
- Construction layout (typically ±0.05 m)
The combined grid-to-ground correction factor C = (elevation factor) ±(grid scale factor k) must be computed and applied to all professional engineering surveys using UTM.
UTM vs State Plane Accuracy Comparisonよくあるご質問 (FAQ)
Why is UTM scale factor less than 1 at the central meridian?
UTM deliberately applies a 0.9996 'secant' scale factor at the CM. This makes the projection intersect the ellipsoid at two lines 180 km either side of the CM, rather than being tangent at only the CM. The result is that the maximum distortion across the entire 6-degree zone (400 ppm) is equal in magnitude to the minimum distortion (400 ppm), reducing worst-case errors compared to a tangent projection.
Should I use UTM or State Plane for construction?
State Plane always outperforms UTM for local precision construction. SPCS uses much narrower zones (distortion < 1:10,000 = 100 ppm), whereas UTM reaches 400 ppm at zone edges. For any project where ground-to-grid scale matters, which is all engineering work, State Plane is the professional standard.
誤った測地系（データム）の指定や、グリッド・地上補正（Grid-to-Ground）を適用しない座標変換は、1〜400mの位置ズレを引き起こします。これは測量過失請求や施工紛争の代表的な原因です。