# Cubature over Polygonal Domains

### Description

The **R** package polyCub provides methods for **cubature**
(numerical integration) **over polygonal domains**.
The function `polyCub()`

is the main entry point of the package.
It is a wrapper around the specific cubature methods listed below.

### Details

`polyCub.midpoint`

:-
Two-dimensional midpoint rule. Polygons are converted to binary pixel images using the

`as.im.function`

method from package spatstat (Baddeley and Turner, 2005). The integral is then obtained as the sum over (pixel area * f(pixel midpoint)). `polyCub.SV`

:-
Product Gauss cubature as proposed by Sommariva and Vianello (2007).

`polyCub.iso`

:-
Efficient adaptive cubature for

*isotropic*functions via line`integrate()`

along the polygon boundary, see Meyer and Held (2014, Supplement B, Section 2.4). `polyCub.exact.Gauss`

:-
Quasi-exact method specific to the integration of the

*bivariate Gaussian density*over polygonal domains. It is based on formulae from Chapter 26 of the Abramowitz and Stegun (1972) handbook, i.e. triangulation of the polygonal domain (using`tristrip`

of package gpclib) and appropriate evaluations of`pmvnorm`

from package mvtnorm. Note that there is also a function`circleCub.Gauss`

to perform integration of the*isotropic*Gaussian density over*circular domains*.

See Section 3.2 of Meyer (2010) for a more detailed description and benchmark experiment of some of the above cubature methods (and others).

### References

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications.

Baddeley, A. and Turner, R. (2005).
spatstat: an **R** package for analyzing spatial point patterns.
*Journal of Statistical Software*, **12** (6), 1-42.

Meyer, S. (2010). Spatio-Temporal Infectious Disease Epidemiology based on Point Processes. Master's Thesis, LMU Munich. Available as http://epub.ub.uni-muenchen.de/11703/.

Meyer, S. and Held, L. (2014).
Power-law models for infectious disease spread.
*The Annals of Applied Statistics*, **8** (3), 1612-1639.

DOI-Link: http://dx.doi.org/10.1214/14-AOAS743,
arXiv:1308.5115

Sommariva, A. and Vianello, M. (2007).
Product Gauss cubature over polygons based on Green's integration formula.
*Bit Numerical Mathematics*, **47** (2), 441-453.

### See Also

The packages cubature and R2Cuba, which are more appropriate for cubature over simple hypercubes.