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helmert-datum-transformation

Abstract math for the 7-parameter (Helmert) similarity transformation used to convert coordinates between geodetic datums. USE FOR: implementing datum-to-datum 3D coordinate transformations; understanding the 7 Helmert parameters (3 translations, 3 rotations, 1 scale); applying the linearized vs. full rotation matrix form; composing transformations; transforming to/from WGS84. DO NOT USE FOR: geodetic coordinate validation (use geodetic-coordinate-bounds); reference ellipsoid parameters (use reference-ellipsoids); map projections.

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NetFabric/NetFabric.Numerics36 stars(GitHub 36 + cccskills 0)

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Files included (3)

SKILL.md2.7 KB
references/formulas.md2.3 KB
references/numerical-stability.md2.1 KB

SKILL.md (original)

Review the instructions the agent will receive before you install.

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:

Coordinate Frame (used by EPSG): rotations rotate the coordinate frame
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

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Install

Install with a command

npx skills add https://github.com/NetFabric/NetFabric.Numerics --skill helmert-datum-transformation

Paste it into your terminal and run it (requires Node.js). You can choose which agents to install it for, such as Claude Code or Codex.

Ask your agent

Please install the following Agent Skill.
https://github.com/NetFabric/NetFabric.Numerics/tree/main/.agents/skills/helmert-datum-transformation

First review SKILL.md and the bundled files and confirm there is nothing suspicious (such as sending data externally or reading credentials). Then place it in ~/.claude/skills/helmert-datum-transformation/ for Claude Code, or ~/.agents/skills/helmert-datum-transformation/ for Codex.

Send this text to Claude Code or Codex as is. It checks the contents before installing.

View source on GitHub

Repository: NetFabric/NetFabric.Numerics
Stars: 36
Forks: 1
Category: Other
License: MIT
Original language: English
Last updated: Aug 10, 2026

A skill is a set of instructions and scripts for an agent. cccskills does not guarantee that skills are safe. Review the contents before you install.