abcd.earth

Explore the rational trigonometry foundations used for ABCD and XYZ spatial coordinates, vector relationships and rotations with interactive demos below.

  1. Quadrance – rational coordinate distance, ‘how displaced am I?”
  2. Cross Product – spread & cross, the measure the separation of lines
  3. Weierstrass – circle parametrization
  4. Gravity Numberline – acceleration vs grid spacing
  5. Spread-Quadray Rotors – gimbal-lock-free rotation
  6. 4D± Gravity + Inversion – Quadray Janus drop
  7. Prime Projections – Gauss-Wantzel bypass presets

The links above will take you to an abcd.earth ‘lite’ version of the software that demonstrates some of the features, but is limited in xyz coordinate space due to web-rendering in THREE.js.

Pre-print whitepapers on the subjects featured above, can be found as PDFs on ResearchGate below:

  1. Extending the Janus Point from Temporal to Spatial Geometry via Tetrahedral (Quadray) Coordinates
  2. Spread-Quadray Rotors -v1.1 (Feb 2026) A Tetrahedral Alternative to Quaternions for Gimbal-Lock-Free Rotation Representation
  3. The 4D± Prime Projection Conjecture -v2.0 (Feb 2026) Rational-Spread Projections of Higher-Dimensional Polytopes as a Source of Non-Constructible Prime n-Gons
  4. The Quadray Bible (Pending)

Explore the rational trigonometry foundations used for ABCD and XYZ spatial coordinates, vector relationships and rotations with interactive demos below. The links above will take you to an abcd.earth ‘lite’ version of the software that demonstrates some of the features, but is limited in xyz coordinate space due to web-rendering…