**Monthly Downloads**: 38

**Programming language**: Haskell

**License**: BSD 3-clause "New" or "Revised" License

**Latest version**: v0.11

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## README

## HGeometry

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):

```
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-combinatorial : defines some non-geometric (i.e. combinatorial) data types, data structures, and algorithms.
- hgeometry-ipe : defines functions for working with ipe files.
- hgeometry-svg : defines functions for working with svg files.
- hgeometry-interactive : defines functions for building an interactive viewer using miso.
- hgeometry : defines the actual geometric data types, data structures, and algorithms.

In addition there is a [hgeometry-examples](hgeometry-examples) package that defines 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
$\mathbb{R}
^{2$:}the typical Graham scan, and a divide and conquer algorithm, - an (O(n)) expected time algorithm for smallest enclosing disk in $\mathbb{R}
^{$2,} - 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
^{2}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/\varepsilon
^{dn\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 (\mathbb{R}
^{2).} - An (O(nm)) time algorithm to compute the discrete Fr\'echet distance of two sequences of points (curves) of length (n) and (m), respectively.

## 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.

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 `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 `Polygon`

data type, has an extra type parameter `p`

that allows
the vertices of the polygon to cary some extra information of type `p`

(for example a color, a size, or whatever).

```
data Polygon (t :: PolygonType) p r where
SimplePolygon :: C.CSeq (Point 2 r :+ p) -> Polygon Simple p r
MultiPolygon :: C.CSeq (Point 2 r :+ p) -> [Polygon Simple p r] -> Polygon Multi p r
```

In all places this extra data is accessable by the (:+) type in Data.Ext, which is essentially just a pair.