Regions

A region is the open set on one side of a closed curve or closed path.

Abstract interface

An AbstractRegion is expected to provide the following methods. Here R is an AbstractRegion and z is a number.

There are default implementations of the following methods.

There is also a parametric subtype AbstractConnectedRegion{N} used to represent a region of connectivity N. Any implementation is expected to provide the following methods.

Default implementations are provided for +, -, *, /, performing translation, rotation, and scaling.

AbstractSimplyConnectedRegion is an alias for AbstractConnectedRegion{1}.

Generic types

SimplyConnectedRegion

Type SimplyConnectedRegion is a union of parameterized types InteriorSimplyConnectedRegion{T} and ExteriorSimplyConnectedRegion{T}, where T is a subtype of a union of AbstractClosedCurve and AbstractClosedPath. The preferred method for constructing a value of one of these types is to use interior and exterior, respectively, with a closed curve or path as argument. If the given curve is bounded, these constructors ignore its orientation and select the bounded or unbounded region it defines, respectively. For an unbounded boundary curve, points "to the left" of it are considered the interior.

In addition to the methods of the Abstract interface, the SimplyConnectedRegion type provides the following methods.

The SimplyConnectedRegion type is parameterized by the type of curve bounding it in order to facilitate dispatch. Notably included are the definitions

AbstractDisk = SimplyConnectedRegion{T} where T<:Circle
AbstractHalfplane = SimplyConnectedRegion{T} where T<:Line
PolygonalRegion = SimplyConnectedRegion{Polygon}

There are also methods to facilitate construction of important common regions. For disks there are

• disk(C::Circle)
• disk(center, radius)
• unitdisk

For half-planes there are

• halfplane(L::Line)
• upperhalfplane
• lowerhalfplane
• lefthalfplane
• righthalfplane

ExteriorRegion

The parameterized type ExteriorRegion{N} represents a region lying exterior to N closed curves or paths. It is constructed by ExteriorRegion(inner), where inner is a vector of closed curves/paths. These are returned by innerboundary, and outerboundary returns nothing.

ConnectedRegion

The parameterized type ConnectedRegion{N} represents a region of connectivity N. You construct one by calling ConnectedRegion{N}(outer, inner), where outer (if given) is an outer boundary, possibly unbounded, and inner is a vector of disconnected inner boundary components. Some rudimentary checking is done that a valid region of connectivity N has been specified, but it should not be considered rigorous.

The particular case of a doubly connected region can be constructed by between(outer, inner), giving the two boundary components. The given orientation is ignored for any bounded component.

Specific subtypes

Annulus

An Annulus is the doubly connected region between two concentric circles. It is a subtype of AbstractConnectedRegion{2}. Construction is by Annulus(outer, inner), where Circle values are given explicitly, or by Annulus(outrad, inrad, center=0), giving the radii and optionally the center.

Examples

Here is a "dog bone" region.

a = Arc(-1,1,-1im);
right = Path([4+1im+a,4-1im-1im*a]);
s = Segment(-3+1im,3+1im);
p = ClosedPath([s,right...,-s,-right...]);
r = interior(exp(1im*pi/4)*p)

using ComplexPlots, Plots
default(linewidth=2,legend=:none)
plot(r)

Inspired by the Swiss flag?

box = [1-1im, 3-1im, 3+1im];
plus = Polygon([box; 1im*box; -box; -1im*box]);
r = between(rectangle(-6-4im, 6+4im), plus)
plot(r, color=:red, linewidth=0)

A multiply-connected exterior region:

c = Circle(0, 1)
t = n_gon(3)
s = n_gon(4)
plot( ExteriorRegion([c, 3+s, 6+t]) )