bibkey: ganeabecigneulhofmann2018cones authors: Octavian Ganea, Gary Becigneul, and Thomas Hofmann year: 2018 title: Hyperbolic Entailment Cones for Learning Hierarchical Embeddings doi: null url: https://proceedings.mlr.press/v80/ganea18a.html claim: The unit tangent hyperboloid geodesic has the classical cosh-sinh expression. strata_touched:

• D5/S3/Geometry/Hyperideal/ActualRaysJointAngle license: citation-only triage: anchor

Unit tangent rays on the hyperboloid

Source and scope

The paper appears in Proceedings of Machine Learning Research 80, 1646-1655. Appendix A of its supplementary material defines the future unit hyperboloid with time-first Lorentz signature and identifies its tangent space by Lorentz orthogonality. Theorem 6, equation (38), gives the unit-speed geodesic as x cosh(t) + v sinh(t), stated on [0,1]. It attributes the formula to Robbin and Salamon, equation (6.4.10). The original supplement is available at https://proceedings.mlr.press/v80/ganea18a/ganea18a-supp.pdf .

Exact application boundary

The repository uses time-last coordinates and the previously defined HyperbolicThreeSpace metric. The actual-ray construction verifies the future-sheet inequality for every real parameter and identifies its projection using the original lift, inverse-coordinate and cosh-distance contracts. It does not assume an identification of Riemannian and metric distances from the cited paper.

The joint comparison-angle limit is derived from these metric lines by a removable-slope extension of sinh at zero. The supplement’s Theorem 6 does not state that two-parameter limit or the repository’s precise projection contract. This attribution concerns the classical geodesic formula, with no novelty claim for the underlying hyperbolic geometry.

Verified locator

• Declared URL: https://proceedings.mlr.press/v80/ganea18a.html
• Primary supplementary PDF: https://proceedings.mlr.press/v80/ganea18a/ganea18a-supp.pdf
• Verified passage: Appendix A, Theorem 6, equation (38), PDF page 1. The unit-speed geodesic is stated on [0,1] and attributed there to Robbin and Salamon (2011), equation (6.4.10).
• Locator scope: the classical unit tangent geodesic formula. The global metric-ray construction and unrestricted joint comparison-angle limit are repository derivations within the exact application boundary above.