Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-14T22:37:16.719Z Has data issue: false hasContentIssue false
  • English
  • Français

The volume of a lattice polyhedron

Published online by Cambridge University Press:  24 October 2008

I. G. Macdonald
I. G. Macdonald
Affiliation:
The University, Exeter
Get access Rights & Permissions [Opens in a new window]

Extract

Let L be the lattice of all points with integer coordinates in the real affine plane R2 (with respect to some fixed coordinate system). Let X be a finite rectilinear simplicial complex in R2 whose 0-simplexes are points of L. Suppose X is pure and the frontier Ẋ of X is a Jordan curve; then there is a well-known formula for the area of X in terms of the number of points of L which lie in X and Ẋ respectively, namely

where L(X) (resp. L(Ẋ)) is the number of points of L which lie in X (resp. Ẋ), and μ(X) is the area of X, normalized so that a fundamental parallelogram of L has unit area.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 4 , October 1963 , pp. 719 - 726
Copyright
Copyright © Cambridge Philosophical Society 1963

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Ehrhart, E.Sur les polyèdres rationnels homothétiques a n dimensions. C.R. Acad. Sci., Paris, 254 (1962), 616–618.Google Scholar
(2)Reeve, J. E.On the volume of lattice polyhedra. Proc. London. Math: Soc. (3), 7 (1957), 378–395.CrossRefGoogle Scholar
(3)Reeve, J. E.A further note on the volume of lattice polyhedra. J. London Math. Soc. 34 (1959), 57–62.CrossRefGoogle Scholar
(4)Sommerville, D. M. Y.The relations connecting the angle-sums and volume of a polytope in space of n dimensions. Proc. Roy. Soc. London, Ser. A, 115 (1927), 103–119.Google Scholar