Defining Curve-Bounded Domains

This section discusses how curve-bounded domains, and curves, are defined in this package. A good demonstration of how these domains are worked with is in the curve-bounded tutorial.

Curves

To start, let us discuss curves. All functions that work with curves in this package treat them as being subtypes of the AbstractParametricCurve type. These curves must:

1. Be defined as parametric curves, parametrised over $0 \leq t \leq 1$.
2. Not be self-intersecting, with the exception of allowing for closed curves.
3. Implement differentiate, twice_differentiate, and thrice_differentiate.
4. Be defined as a callable struct.
5. Either implement point_position_relative_to_curve or get_closest_point. Alternatively, the struct should have a lookup_table field so that lookup_table[i] is the value of the curve at t = (i - 1) / (length(lookup_table) - 1).

With these specifications, a curve can be fully compatible with the functions in this package, as other functions such as arc_length, curvature, and total_variation are automatically defined for such a curve. The curves that are defined in this package can be found using subtypes:

using DelaunayTriangulation, InteractiveUtils
subtypes(DelaunayTriangulation.AbstractParametricCurve)

8-element Vector{Any}:
 BSpline
 BezierCurve
 CatmullRomSpline
 CircularArc
 DelaunayTriangulation.CatmullRomSplineSegment
 DelaunayTriangulation.PiecewiseLinear
 EllipticalArc
 LineSegment

Curve-Bounded Domains

For defining curve-bounded domains, defining them is similar to defining piecewise linear boundaries as described here. You still need to be careful about orientation, and you of course need to make sure that curves belonging to the same boundary connect appropriately. For defining curve-bounded domains that also have a piecewise linear section, the piecewise linear section should be defined as a vector of vertices just like here.