HGeometry

HGeometry provides some basic geometry types, and geometric algorithms and data structures for them. The main two focusses are: (1) Strong type safety, and (2) implementations of geometric algorithms and data structures with 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.Seq2 which enforces that a polyline is a proper polyline, and thus has at least two vertices.

Please note that aspect (2), implementing good algorithms, is much work in progress. Only a few algorithms have been implemented, some of which could use some improvements. Currently, HGeometry provides the following algorithms:

• two (O(n \log n)) time algorithms for convex hull in \(\mathbb{R}^2\): the typical Graham scan, and a divide and conqueror 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 conqueror).
• 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.

It also has some geometric data structures. In particular, HGeometry contans an implementation of

• 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.

HGeometry also includes a datastructure/data type for planar graphs. In particular, it has a `EdgeOracle' data structure, that can be built in \(O(n)\) time that can test if the graph contains an edge in constant time.

Numeric Types

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.

A Note on the Ext (:+) data type

In many applications we do not just have geometric data, e.g. Point d rs or Polygon rs, 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 rs 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. 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.

Reading and Writing Ipe files

Apart from geometric types, HGeometry provides some interface for reading and writing Ipe (http://ipe.otfried.org). However, this is all very work in progress. Hence, the API is experimental and may change at any time! Here is an example showing reading a set of points from an Ipe file, computing the DelaunayTriangulation, and writing the result again to an output file

mainWith                          :: Options -> IO ()
mainWith (Options inFile outFile) = do
    ePage <- readSinglePageFile inFile
    case ePage of
      Left err                         -> print err
      Right (page :: IpePage Rational) -> case page^..content.traverse._IpeUse of
        []         -> putStrLn "No points found"
        syms@(_:_) -> do
           let pts  = syms&traverse.core %~ (^.symbolPoint)
               pts' = NonEmpty.fromList pts
               dt   = delaunayTriangulation $ pts'
               out  = [asIpe drawTriangulation dt]
           writeIpeFile outFile . singlePageFromContent $ out

See the examples directory for more examples.