Skip to main content

The shape invariant of triangles and trigonometry in two-point homogeneous spaces


We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces ℂP n and ℍP n (n≥2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces ℂH n and ℍH n (n≥2).

This is a preview of subscription content, access via your institution.


  1. Aslaksen, H., ‘Laws of trigonometry on SU(3)’, Preprint, Berkeley, 1988. (To appear in Transactions of the AMS.)

  2. Blaschke, W. and Terheggen, H., ‘Trigonometria Hermitiana’, Rend. Sem. Mat. Univ. Roma Ser. 4 (3) (1939), 153–161.

    Google Scholar 

  3. Hsiang, W.-Y., ‘On the trigonometry of two-point homogeneous spaces’, Preprint, Berkeley 1986. (To appear in Annals of Global Geometry and Analysis.)

  4. Rozenfeld, B. A., ‘On the theory of symmetric spaces of rank one’ (Russian), Mat. Sb. 41 (N.S. 83) (1957), 373–380; Math. Rev. 20 (1959), 2756a, b.

    Google Scholar 

  5. Sirokov, P. A., ‘On a certain type of symmetric spaces’ (Russian), Mat. Sb. 41 (N.S. 83) (1957), 361–372; Math. Rev. 20 (1959), 2755a, b.

    Google Scholar 

  6. Terheggen, H., ‘Zur analytischen Geometrie auf der Geraden von Hermite als Grenzfall der Geometrie der Hermiteschen Ebene und ihr Zusammenhang mit der gewöhnlichen sphärischen Trigonometrie’, Jahresber, deutsche Math. Verein. 50 (1940), 24–35.

    Google Scholar 

  7. Wang, H. C., ‘Two-point homogeneous spaces’, Ann. Math. 55 (1952), 172–191.

    Google Scholar 

  8. Wolf, J., ‘Elliptic spaces in Grassmann manifolds’, Illinois J. Math. 7 (1963), 447–462.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and Permissions

About this article

Cite this article

Brehm, U. The shape invariant of triangles and trigonometry in two-point homogeneous spaces. Geom Dedicata 33, 59–76 (1990).

Download citation

  • Received:

  • Issue Date:

  • DOI:


  • Projective Space
  • Homogeneous Space
  • Hyperbolic Space
  • Elementary Proof
  • Basic Invariant