# polyCub.midpoint: Two-Dimensional Midpoint Rule In polyCub: Cubature over Polygonal Domains

 polyCub.midpoint R Documentation

## Two-Dimensional Midpoint Rule

### Description

The surface is converted to a binary pixel image using the `as.im.function` method from package spatstat.geom (Baddeley et al., 2015). The integral under the surface is then approximated as the sum over (pixel area * f(pixel midpoint)).

### Usage

``````polyCub.midpoint(polyregion, f, ..., eps = NULL, dimyx = NULL,
plot = FALSE)
``````

### Arguments

 `polyregion` a polygonal integration domain. It can be any object coercible to the spatstat.geom class `"owin"` via a corresponding `as.owin`-method. Note that this includes polygons of the classes `"gpc.poly"` and `"SpatialPolygons"`, because polyCub defines methods `as.owin.gpc.poly` and `as.owin.SpatialPolygons`, respectively. sf also registers suitable `as.owin` methods for its `"(MULTI)POLYGON"` classes. `f` a two-dimensional real-valued function. As its first argument it must take a coordinate matrix, i.e., a numeric matrix with two columns, and it must return a numeric vector of length the number of coordinates. `...` further arguments for `f`. `eps` width and height of the pixels (squares), see `as.mask`. `dimyx` number of subdivisions in each dimension, see `as.mask`. `plot` logical indicating if an illustrative plot of the numerical integration should be produced.

### Value

The approximated value of the integral of `f` over `polyregion`.

### References

Baddeley A, Rubak E, Turner R (2015). Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press, London.

Other polyCub-methods: `polyCub.SV()`, `polyCub.exact.Gauss()`, `polyCub.iso()`, `polyCub()`

### Examples

``````## a function to integrate (here: isotropic zero-mean Gaussian density)
f <- function (s, sigma = 5)
exp(-rowSums(s^2)/2/sigma^2) / (2*pi*sigma^2)

## a simple polygon as integration domain
hexagon <- list(
list(x = c(7.33, 7.33, 3, -1.33, -1.33, 3),
y = c(-0.5, 4.5, 7, 4.5, -0.5, -3))
)

if (require("spatstat.geom")) {
hexagon.owin <- owin(poly = hexagon)

show_midpoint <- function (eps)
{
plotpolyf(hexagon.owin, f, xlim = c(-8,8), ylim = c(-8,8),
use.lattice = FALSE)
## add evaluation points to plot
points(expand.grid(xcol, yrow), col = t(m), pch = 20))
title(main = paste("2D midpoint rule with eps =", eps))
}

## show nodes (eps = 0.5)
show_midpoint(0.5)

## show pixel image (eps = 0.5)
polyCub.midpoint(hexagon.owin, f, eps = 0.5, plot = TRUE)

## use a decreasing pixel size (increasing number of nodes)
for (eps in c(5, 3, 1, 0.5, 0.3, 0.1))
cat(sprintf("eps = %.1f: %.7f\n", eps,
polyCub.midpoint(hexagon.owin, f, eps = eps)))
}
``````

polyCub documentation built on Oct. 25, 2023, 5:07 p.m.