HGeometry
=========
[](https://hackage.haskell.org/package/hgeometry)
[](https://noinia.github.io/hgeometry/doc/)
HGeometry is a library for computing with geometric objects in
Haskell. It defines basic geometric types and primitives, and it
implements some geometric data structures and algorithms. The main two
focusses are:
- 1. Strong type safety, and
- 2. implementations of geometric algorithms and data structures that
have good asymptotic running time guarantees.
Design choices showing these aspects are for example:
- we provide a data type `Point d r` parameterized by a
type-level natural number `d`, representing d-dimensional points (in all cases
our type parameter `r` represents the (numeric) type for the (real)-numbers):
```haskell
newtype Point (d :: Nat) (r :: *) = Point { toVec :: Vector d r }
```
- the vertices of a `PolyLine d p r` are stored in a `Data.LSeq` which enforces
that a polyline is a proper polyline, and thus has at least two vertices.
Please note that aspect two, implementing good algorithms, is much work in
progress. Only a few algorithms have been implemented, some of which could use
some improvements.
HGeometry Packages
------------------
HGeometry is split into a few smaller packages. In particular:
- hgeometry : defines the actual geometric data types, data
structures, and algorithms,
- hgeometry-combinatorial : defines the non-geometric
(i.e. combinatorial) data types, data structures, and algorithms.
- hgeometry-ipe : defines functions for working with [ipe](http://ipe.otfried.org) files.
- hgeometry-svg : defines functions for working with svg files.
- hgeometry-web : defines functions for building an
interactive viewer using [miso](https://haskell-miso.org).
- hgeometry-interactive : defines functions for building an
interactive viewer using
[reflex-sdl2](https://hackage.haskell.org/package/reflex-sdl2).
In addition there are [hgeometry-examples](hgeometry-examples) and
[hgeometry-showcase](hgeometry-showcase) packages that define some
example applications, and a hgometry-test package that contains all
testcases. The latter is to work around a bug in cabal.
Available Geometric Algorithms
------------------------------
Apart from some basic geometric primitives such as intersecting
line segments, testing if a point lies in a polygon etc, HGeometry
implements some more advanced geometric algorithms. In particuar, the
following algorithms are currently available:
* two *O(n log n)* time algorithms for convex hull in
ℝ²: the typical Graham scan, and a divide and conquer algorithm,
* an *O(n)* expected time algorithm for smallest enclosing disk in ℝ²,
* the well-known Douglas Peucker polyline line simplification algorithm,
* an *O(n log n)* time algorithm for computing the Delaunay triangulation
(using divide and conquer),
* an *O(n log n)* time algorithm for computing the Euclidean Minimum Spanning
Tree (EMST), based on computing the Delaunay Triangulation,
* an *O(log n)* time algorithm to find extremal points and tangents on/to a
convex polygon,
* an optimal *O(n+m)* time algorithm to compute the Minkowski sum of two convex
polygons,
* an *O(1/εᵈn log n)* time algorithm for constructing a Well-Separated pair
decomposition,
* the classic (optimal) *O(n log n)* time divide and conquer algorithm to
compute the closest pair among a set of *n* points in ℝ²,
* an *O(nm)* time algorithm to compute the discrete Fréchet
distance of two sequences of points (curves) of length *n* and
*m*, respectively.
* an *O(n)* time single-source shortest path algorithm on triangulated polygons.
* an *O(n log n)* time algorithm for generating random convex polygons.
* an *O(n)* time algorithm for finding the convex hull of a simple polygon.
Available Geometric Data Structures
-----------------------------------
HGeometry also contains an implementation of some geometric data
structures. In particular,
* A one dimensional Segment Tree. The base tree is static.
* A one dimensional Interval Tree. The base tree is static.
* A KD-Tree. The base tree is static.
* An *O(n log n)* size planar point location data structure supporting
*O(log n)* queries.
There is also support for working with planar subdivisions. As a
result, [hgeometry-combinatorial] also includes a data structure for
working with planar graphs. In particular, it has an `EdgeOracle` data
structure, that can be built in *O(n)* time that can test if the
planar graph contains an edge in constant time.
Avoiding Floating-point issues
-------------------------------
All geometry types are parameterized by a numerical type `r`. It is well known
that Floating-point arithmetic and Geometric algorithms don't go well together;
i.e. because of floating point errors one may get completely wrong
results. Hence, I *strongly* advise against using `Double` or `Float` for these
types. In several algorithms it is sufficient if the type `r` is
`Fractional`. Hence, you can use an exact number type such as
`Data.RealNumber.Rational` or `Data.Ratio.Rational`.
Working with additional data
----------------------------
In many applications we do not just have geometric data, e.g. `Point d r`s or
`Polygon r`s, but instead, these types have some additional properties, like a
color, size, thickness, elevation, or whatever. Hence, we would like that our
library provides functions that also allow us to work with `ColoredPolygon r`s
etc. The typical Haskell approach would be to construct type-classes such as
`PolygonLike` and define functions that work with any type that is
`PolygonLike`. However, geometric algorithms are often hard enough by
themselves, and thus we would like all the help that the type-system/compiler
can give us. Hence, we choose to work with concrete types.
To still allow for some extensibility our types will use the Ext (:+)
type, as defined in the hgeometry-combinatorial package. For example,
our `LineSegment` data type, has an extra type parameter `p` that
allows the vertices of the line segment to carry some extra
information of type `p` (for example a color, a size, or
whatever). Polylines, Polylygons, Boxes, etc have similar such
parameters.
In all places this extra data is accessable by the (:+) type in Data.Ext, which
is essentially just a pair.