Helmert Datum Transformation

Definition

The 7-parameter Helmert transformation converts geocentric Cartesian coordinates (X, Y, Z) from one datum to another. Also called similarity transformation or Bursa-Wolf model.

Parameters:

Symbol	Name	Unit
ΔX, ΔY, ΔZ	Translation (origin shift)	metres
Rₓ, Rᵧ, R_z	Rotation (small angles)	arcseconds (or radians)
ΔS	Scale factor change	ppm (parts per million)

Linearized Formula (Small Rotations)

When Rₓ, Rᵧ, R_z << 1 radian (typically < 10 arcsec for datum shifts):

⎡X'⎤   ⎡ΔX⎤          ⎡  1    R_z  −Rᵧ⎤ ⎡X⎤
⎢Y'⎥ = ⎢ΔY⎥ + (1+ΔS) ⎢−R_z   1    Rₓ ⎥ ⎢Y⎥
⎣Z'⎦   ⎣ΔZ⎦          ⎣  Rᵧ  −Rₓ   1  ⎦ ⎣Z⎦

ΔS is the fractional scale: ΔS = 0 means no scale change. Applied as (1 + ΔS × 10⁻⁶) when given in ppm.

Inverse Transformation

To reverse the transformation (source ← target), negate all 7 parameters:

(−ΔX, −ΔY, −ΔZ, −Rₓ, −Rᵧ, −R_z, −ΔS)

This is exact for the linearized form. For the full rotation matrix form, invert the rotation matrix.

WGS84 as Reference

WGS84 geocentric coordinates are the common hub. Typical workflow:

Source datum → WGS84 → Target datum

WGS84 ↔ WGS72 parameters are defined in the WGS84 standard. For national grids, parameters are published by the national geodetic authority (e.g., EPSG registry).

Sign Convention

Two conventions exist — confirm which is in use:

1. Coordinate Frame (used by EPSG): rotations rotate the coordinate frame
2. Position Vector (Bursa-Wolf): rotations rotate the position vector

The rotation matrix entries swap sign between conventions. The parameters are numerically identical but with opposite rotation signs.

Reference Files

File	Load When
references/formulas.md	Full rotation matrix form, ppm conversion, composition, inverse
references/numerical-stability.md	Angle unit conversion, scale near 1, ppm precision