| Copyright | (C) Frank Staals |
|---|---|
| License | see the LICENSE file |
| Maintainer | Frank Staals |
| Safe Haskell | None |
| Language | Haskell2010 |
Data.Geometry
Description
Basic Geometry Types
Synopsis
- module Data.Geometry.Properties
- module Data.Geometry.Transformation
- module Data.Geometry.Point
- imap :: (Vector v a, Vector v b) => (Int -> a -> b) -> v a -> v b
- replicate :: Vector v a => a -> v a
- distanceA :: (Floating a, Foldable (Diff p), Affine p) => p a -> p a -> a
- qdA :: (Affine p, Foldable (Diff p), Num a) => p a -> p a -> a
- class Additive (Diff p) => Affine (p :: Type -> Type) where
- dot :: (Metric f, Num a) => f a -> f a -> a
- norm :: (Metric f, Floating a) => f a -> a
- signorm :: (Metric f, Floating a) => f a -> f a
- outer :: (Functor f, Functor g, Num a) => f a -> g a -> f (g a)
- unit :: (Additive t, Num a) => ASetter' (t a) a -> t a
- scaled :: (Traversable t, Num a) => t a -> t (t a)
- basisFor :: (Traversable t, Num a) => t b -> [t a]
- basis :: (Additive t, Traversable t, Num a) => [t a]
- (^/) :: (Functor f, Fractional a) => f a -> a -> f a
- (^*) :: (Functor f, Num a) => f a -> a -> f a
- (*^) :: (Functor f, Num a) => a -> f a -> f a
- sumV :: (Foldable f, Additive v, Num a) => f (v a) -> v a
- negated :: (Functor f, Num a) => f a -> f a
- newtype E (t :: Type -> Type) = E {}
- class Functor f => Additive (f :: Type -> Type) where
- data C (n :: Nat) = C
- type Arity d = (ImplicitArity (Peano d), KnownNat d)
- newtype Vector (d :: Nat) (r :: *) = MKVector {
- _unV :: VectorFamily (Peano d) r
- pattern Vector4 :: r -> r -> r -> r -> Vector 4 r
- pattern Vector3 :: r -> r -> r -> Vector 3 r
- pattern Vector2 :: r -> r -> Vector 2 r
- pattern Vector1 :: r -> Vector 1 r
- pattern Vector :: VectorFamilyF (Peano d) r -> Vector d r
- unV :: Lens (Vector d r) (Vector d s) (VectorFamily (Peano d) r) (VectorFamily (Peano d) s)
- readVec :: forall d r. (Arity d, Read r) => ReadP (Vector d r)
- vectorFromList :: Arity d => [r] -> Maybe (Vector d r)
- vectorFromListUnsafe :: Arity d => [r] -> Vector d r
- destruct :: (Arity d, Arity (d + 1)) => Vector (d + 1) r -> (r, Vector d r)
- element :: forall proxy i d r. (Arity d, KnownNat i, (i + 1) <= d) => proxy i -> Lens' (Vector d r) r
- element' :: forall d r. Arity d => Int -> Traversal' (Vector d r) r
- snoc :: (Arity (d + 1), Arity d) => Vector d r -> r -> Vector (d + 1) r
- init :: (Arity d, Arity (d + 1)) => Vector (d + 1) r -> Vector d r
- prefix :: forall i d r. (Arity d, Arity i, i <= d) => Vector d r -> Vector i r
- cross :: Num r => Vector 3 r -> Vector 3 r -> Vector 3 r
- isScalarMultipleOf :: (Eq r, Fractional r, Arity d) => Vector d r -> Vector d r -> Bool
- scalarMultiple :: (Eq r, Fractional r, Arity d) => Vector d r -> Vector d r -> Maybe r
- xComponent :: (1 <= d, Arity d) => Lens' (Vector d r) r
- yComponent :: (2 <= d, Arity d) => Lens' (Vector d r) r
- zComponent :: (3 <= d, Arity d) => Lens' (Vector d r) r
- module Data.Geometry.Line
- module Data.Geometry.LineSegment
- newtype PolyLine d p r = PolyLine {}
- points :: forall d p r d p r. Iso (PolyLine d p r) (PolyLine d p r) (LSeq 2 ((:+) (Point d r) p)) (LSeq 2 ((:+) (Point d r) p))
- fromPoints' :: Monoid p => [Point d r] -> PolyLine d p r
- fromLineSegment :: LineSegment d p r -> PolyLine d p r
- asLineSegment :: PolyLine d p r -> LineSegment d p r
- asLineSegment' :: PolyLine d p r -> Maybe (LineSegment d p r)
- type SomePolygon p r = Either (Polygon Simple p r) (Polygon Multi p r)
- type MultiPolygon = Polygon Multi
- type SimplePolygon = Polygon Simple
- data Polygon (t :: PolygonType) p r where
- data PolygonType
- bitraverseVertices :: (Applicative f, Traversable t) => (p -> f q) -> (r -> f s) -> t (Point 2 r :+ p) -> f (t (Point 2 s :+ q))
- outerBoundary :: forall t p r. Lens' (Polygon t p r) (CSeq (Point 2 r :+ p))
- polygonHoles :: forall p r. Lens' (Polygon Multi p r) [Polygon Simple p r]
- outerVertex :: Int -> Lens' (Polygon t p r) (Point 2 r :+ p)
- outerBoundaryEdge :: Int -> Polygon t p r -> LineSegment 2 p r
- holeList :: Polygon t p r -> [Polygon Simple p r]
- polygonVertices :: Polygon t p r -> NonEmpty (Point 2 r :+ p)
- outerBoundaryEdges :: Polygon t p r -> CSeq (LineSegment 2 p r)
- listEdges :: Polygon t p r -> [LineSegment 2 p r]
- withIncidentEdges :: Polygon t p r -> Polygon t (Two (LineSegment 2 p r)) r
- toEdges :: CSeq (Point 2 r :+ p) -> CSeq (LineSegment 2 p r)
- onBoundary :: (Fractional r, Ord r) => Point 2 r -> Polygon t p r -> Bool
- inPolygon :: forall t p r. (Fractional r, Ord r) => Point 2 r -> Polygon t p r -> PointLocationResult
- insidePolygon :: (Fractional r, Ord r) => Point 2 r -> Polygon t p r -> Bool
- area :: Fractional r => Polygon t p r -> r
- signedArea :: Fractional r => SimplePolygon p r -> r
- centroid :: Fractional r => SimplePolygon p r -> Point 2 r
- isCounterClockwise :: (Eq r, Fractional r) => Polygon t p r -> Bool
- toClockwiseOrder :: (Eq r, Fractional r) => Polygon t p r -> Polygon t p r
- toCounterClockWiseOrder :: (Eq r, Fractional r) => Polygon t p r -> Polygon t p r
- reverseOuterBoundary :: Polygon t p r -> Polygon t p r
- asSimplePolygon :: Polygon t p r -> SimplePolygon p r
- cmpExtreme :: (Num r, Ord r) => Vector 2 r -> (Point 2 r :+ p) -> (Point 2 r :+ q) -> Ordering
- extremesLinear :: (Ord r, Num r) => Vector 2 r -> Polygon t p r -> (Point 2 r :+ p, Point 2 r :+ p)
- numberVertices :: Polygon t p r -> Polygon t (SP Int p) r
Documentation
module Data.Geometry.Properties
module Data.Geometry.Transformation
module Data.Geometry.Point
imap :: (Vector v a, Vector v b) => (Int -> a -> b) -> v a -> v b #
Apply function to every element of the vector and its index.
replicate :: Vector v a => a -> v a #
Replicate value n times.
Examples:
>>>import Data.Vector.Fixed.Boxed (Vec2)>>>replicate 1 :: Vec2 IntfromList [1,1]
>>>replicate 2 :: (Double,Double,Double)(2.0,2.0,2.0)
>>>import Data.Vector.Fixed.Boxed (Vec4)>>>replicate "foo" :: Vec4 StringfromList ["foo","foo","foo","foo"]
distanceA :: (Floating a, Foldable (Diff p), Affine p) => p a -> p a -> a #
Distance between two points in an affine space
qdA :: (Affine p, Foldable (Diff p), Num a) => p a -> p a -> a #
Compute the quadrance of the difference (the square of the distance)
class Additive (Diff p) => Affine (p :: Type -> Type) where #
An affine space is roughly a vector space in which we have forgotten or at least pretend to have forgotten the origin.
a .+^ (b .-. a) = b@ (a .+^ u) .+^ v = a .+^ (u ^+^ v)@ (a .-. b) ^+^ v = (a .+^ v) .-. q@
Methods
(.-.) :: Num a => p a -> p a -> Diff p a infixl 6 #
Get the difference between two points as a vector offset.
(.+^) :: Num a => p a -> Diff p a -> p a infixl 6 #
Add a vector offset to a point.
(.-^) :: Num a => p a -> Diff p a -> p a infixl 6 #
Subtract a vector offset from a point.
Instances
dot :: (Metric f, Num a) => f a -> f a -> a #
Compute the inner product of two vectors or (equivalently)
convert a vector f a into a covector f a -> a.
>>>V2 1 2 `dot` V2 3 411
outer :: (Functor f, Functor g, Num a) => f a -> g a -> f (g a) #
Outer (tensor) product of two vectors
unit :: (Additive t, Num a) => ASetter' (t a) a -> t a #
Create a unit vector.
>>>unit _x :: V2 IntV2 1 0
scaled :: (Traversable t, Num a) => t a -> t (t a) #
Produce a diagonal (scale) matrix from a vector.
>>>scaled (V2 2 3)V2 (V2 2 0) (V2 0 3)
basisFor :: (Traversable t, Num a) => t b -> [t a] #
Produce a default basis for a vector space from which the argument is drawn.
basis :: (Additive t, Traversable t, Num a) => [t a] #
Produce a default basis for a vector space. If the dimensionality
of the vector space is not statically known, see basisFor.
(^/) :: (Functor f, Fractional a) => f a -> a -> f a infixl 7 #
Compute division by a scalar on the right.
(^*) :: (Functor f, Num a) => f a -> a -> f a infixl 7 #
Compute the right scalar product
>>>V2 3 4 ^* 2V2 6 8
(*^) :: (Functor f, Num a) => a -> f a -> f a infixl 7 #
Compute the left scalar product
>>>2 *^ V2 3 4V2 6 8
sumV :: (Foldable f, Additive v, Num a) => f (v a) -> v a #
Sum over multiple vectors
>>>sumV [V2 1 1, V2 3 4]V2 4 5
negated :: (Functor f, Num a) => f a -> f a #
Compute the negation of a vector
>>>negated (V2 2 4)V2 (-2) (-4)
newtype E (t :: Type -> Type) #
Basis element
Instances
class Functor f => Additive (f :: Type -> Type) where #
A vector is an additive group with additional structure.
Minimal complete definition
Nothing
Methods
The zero vector
(^+^) :: Num a => f a -> f a -> f a infixl 6 #
Compute the sum of two vectors
>>>V2 1 2 ^+^ V2 3 4V2 4 6
(^-^) :: Num a => f a -> f a -> f a infixl 6 #
Compute the difference between two vectors
>>>V2 4 5 ^-^ V2 3 1V2 1 4
lerp :: Num a => a -> f a -> f a -> f a #
Linearly interpolate between two vectors.
liftU2 :: (a -> a -> a) -> f a -> f a -> f a #
Apply a function to merge the 'non-zero' components of two vectors, unioning the rest of the values.
liftI2 :: (a -> b -> c) -> f a -> f b -> f c #
Apply a function to the components of two vectors.
- For a dense vector this is equivalent to
liftA2. - For a sparse vector this is equivalent to
intersectionWith.
Instances
A proxy which can be used for the coordinates.
Constructors
| C |
newtype Vector (d :: Nat) (r :: *) Source #
Datatype representing d dimensional vectors. The default implementation is based n VectorFixed. However, for small vectors we automatically select a more efficient representation.
Constructors
| MKVector | |
Fields
| |
Instances
unV :: Lens (Vector d r) (Vector d s) (VectorFamily (Peano d) r) (VectorFamily (Peano d) s) Source #
vectorFromListUnsafe :: Arity d => [r] -> Vector d r Source #
element :: forall proxy i d r. (Arity d, KnownNat i, (i + 1) <= d) => proxy i -> Lens' (Vector d r) r Source #
Lens into the i th element
element' :: forall d r. Arity d => Int -> Traversal' (Vector d r) r Source #
Similar to element above. Except that we don't have a static guarantee
that the index is in bounds. Hence, we can only return a Traversal
snoc :: (Arity (d + 1), Arity d) => Vector d r -> r -> Vector (d + 1) r Source #
Add an element at the back of the vector
init :: (Arity d, Arity (d + 1)) => Vector (d + 1) r -> Vector d r Source #
Get a vector of the first d - 1 elements.
prefix :: forall i d r. (Arity d, Arity i, i <= d) => Vector d r -> Vector i r Source #
Get a prefix of i elements of a vector
cross :: Num r => Vector 3 r -> Vector 3 r -> Vector 3 r Source #
Cross product of two three-dimensional vectors
isScalarMultipleOf :: (Eq r, Fractional r, Arity d) => Vector d r -> Vector d r -> Bool Source #
Test if v is a scalar multiple of u.
>>>Vector2 1 1 `isScalarMultipleOf` Vector2 10 10True>>>Vector2 1 1 `isScalarMultipleOf` Vector2 10 1False>>>Vector2 1 1 `isScalarMultipleOf` Vector2 11.1 11.1True>>>Vector2 1 1 `isScalarMultipleOf` Vector2 11.1 11.2False>>>Vector2 2 1 `isScalarMultipleOf` Vector2 11.1 11.2False>>>Vector2 2 1 `isScalarMultipleOf` Vector2 4 2True>>>Vector2 2 1 `isScalarMultipleOf` Vector2 4 0False
scalarMultiple :: (Eq r, Fractional r, Arity d) => Vector d r -> Vector d r -> Maybe r Source #
Get the scalar labmda s.t. v = lambda * u (if it exists)
module Data.Geometry.Line
module Data.Geometry.LineSegment
newtype PolyLine d p r Source #
A Poly line in R^d has at least 2 vertices
Instances
points :: forall d p r d p r. Iso (PolyLine d p r) (PolyLine d p r) (LSeq 2 ((:+) (Point d r) p)) (LSeq 2 ((:+) (Point d r) p)) Source #
fromPoints' :: Monoid p => [Point d r] -> PolyLine d p r Source #
pre: The input list contains at least two points. All extra vields are initialized with mempty.
fromLineSegment :: LineSegment d p r -> PolyLine d p r Source #
We consider the line-segment as closed.
asLineSegment :: PolyLine d p r -> LineSegment d p r Source #
Convert to a closed line segment by taking the first two points.
asLineSegment' :: PolyLine d p r -> Maybe (LineSegment d p r) Source #
Stricter version of asLineSegment that fails if the Polyline contains more than two points.
type SomePolygon p r = Either (Polygon Simple p r) (Polygon Multi p r) Source #
Either a simple or multipolygon
type MultiPolygon = Polygon Multi Source #
type SimplePolygon = Polygon Simple Source #
data Polygon (t :: PolygonType) p r where Source #
Constructors
| SimplePolygon :: CSeq (Point 2 r :+ p) -> Polygon Simple p r | |
| MultiPolygon :: CSeq (Point 2 r :+ p) -> [Polygon Simple p r] -> Polygon Multi p r |
Instances
data PolygonType Source #
We distinguish between simple polygons (without holes) and Polygons with holes.
bitraverseVertices :: (Applicative f, Traversable t) => (p -> f q) -> (r -> f s) -> t (Point 2 r :+ p) -> f (t (Point 2 s :+ q)) Source #
outerVertex :: Int -> Lens' (Polygon t p r) (Point 2 r :+ p) Source #
Access the i^th vertex on the outer boundary
outerBoundaryEdge :: Int -> Polygon t p r -> LineSegment 2 p r Source #
polygonVertices :: Polygon t p r -> NonEmpty (Point 2 r :+ p) Source #
The vertices in the polygon. No guarantees are given on the order in which they appear!
outerBoundaryEdges :: Polygon t p r -> CSeq (LineSegment 2 p r) Source #
The edges along the outer boundary of the polygon. The edges are half open.
running time: \(O(n)\)
listEdges :: Polygon t p r -> [LineSegment 2 p r] Source #
Lists all edges. The edges on the outer boundary are given before the ones on the holes. However, no other guarantees are given on the order.
running time: \(O(n)\)
withIncidentEdges :: Polygon t p r -> Polygon t (Two (LineSegment 2 p r)) r Source #
Pairs every vertex with its incident edges. The first one is its predecessor edge, the second one its successor edge.
>>>mapM_ print . polygonVertices $ withIncidentEdges simplePolyPoint2 [0 % 1,0 % 1] :+ SP LineSegment (Closed (Point2 [1 % 1,11 % 1] :+ ())) (Closed (Point2 [0 % 1,0 % 1] :+ ())) LineSegment (Closed (Point2 [0 % 1,0 % 1] :+ ())) (Closed (Point2 [10 % 1,0 % 1] :+ ())) Point2 [10 % 1,0 % 1] :+ SP LineSegment (Closed (Point2 [0 % 1,0 % 1] :+ ())) (Closed (Point2 [10 % 1,0 % 1] :+ ())) LineSegment (Closed (Point2 [10 % 1,0 % 1] :+ ())) (Closed (Point2 [10 % 1,10 % 1] :+ ())) Point2 [10 % 1,10 % 1] :+ SP LineSegment (Closed (Point2 [10 % 1,0 % 1] :+ ())) (Closed (Point2 [10 % 1,10 % 1] :+ ())) LineSegment (Closed (Point2 [10 % 1,10 % 1] :+ ())) (Closed (Point2 [5 % 1,15 % 1] :+ ())) Point2 [5 % 1,15 % 1] :+ SP LineSegment (Closed (Point2 [10 % 1,10 % 1] :+ ())) (Closed (Point2 [5 % 1,15 % 1] :+ ())) LineSegment (Closed (Point2 [5 % 1,15 % 1] :+ ())) (Closed (Point2 [1 % 1,11 % 1] :+ ())) Point2 [1 % 1,11 % 1] :+ SP LineSegment (Closed (Point2 [5 % 1,15 % 1] :+ ())) (Closed (Point2 [1 % 1,11 % 1] :+ ())) LineSegment (Closed (Point2 [1 % 1,11 % 1] :+ ())) (Closed (Point2 [0 % 1,0 % 1] :+ ()))
toEdges :: CSeq (Point 2 r :+ p) -> CSeq (LineSegment 2 p r) Source #
Given the vertices of the polygon. Produce a list of edges. The edges are half-open.
onBoundary :: (Fractional r, Ord r) => Point 2 r -> Polygon t p r -> Bool Source #
Test if q lies on the boundary of the polygon. Running time: O(n)
>>>point2 1 1 `onBoundary` simplePolyFalse>>>point2 0 0 `onBoundary` simplePolyTrue>>>point2 10 0 `onBoundary` simplePolyTrue>>>point2 5 13 `onBoundary` simplePolyFalse>>>point2 5 10 `onBoundary` simplePolyFalse>>>point2 10 5 `onBoundary` simplePolyTrue>>>point2 20 5 `onBoundary` simplePolyFalse
TODO: testcases multipolygon
inPolygon :: forall t p r. (Fractional r, Ord r) => Point 2 r -> Polygon t p r -> PointLocationResult Source #
Check if a point lies inside a polygon, on the boundary, or outside of the polygon. Running time: O(n).
>>>point2 1 1 `inPolygon` simplePolyInside>>>point2 0 0 `inPolygon` simplePolyOnBoundary>>>point2 10 0 `inPolygon` simplePolyOnBoundary>>>point2 5 13 `inPolygon` simplePolyInside>>>point2 5 10 `inPolygon` simplePolyInside>>>point2 10 5 `inPolygon` simplePolyOnBoundary>>>point2 20 5 `inPolygon` simplePolyOutside
TODO: Add some testcases with multiPolygons TODO: Add some more onBoundary testcases
insidePolygon :: (Fractional r, Ord r) => Point 2 r -> Polygon t p r -> Bool Source #
Test if a point lies strictly inside the polgyon.
area :: Fractional r => Polygon t p r -> r Source #
Compute the area of a polygon
signedArea :: Fractional r => SimplePolygon p r -> r Source #
Compute the signed area of a simple polygon. The the vertices are in clockwise order, the signed area will be negative, if the verices are given in counter clockwise order, the area will be positive.
centroid :: Fractional r => SimplePolygon p r -> Point 2 r Source #
Compute the centroid of a simple polygon.
isCounterClockwise :: (Eq r, Fractional r) => Polygon t p r -> Bool Source #
Test if the outer boundary of the polygon is in clockwise or counter clockwise order.
running time: \(O(n)\)
toClockwiseOrder :: (Eq r, Fractional r) => Polygon t p r -> Polygon t p r Source #
Orient the outer boundary to clockwise order
toCounterClockWiseOrder :: (Eq r, Fractional r) => Polygon t p r -> Polygon t p r Source #
Orient the outer boundary to counter clockwise order
reverseOuterBoundary :: Polygon t p r -> Polygon t p r Source #
asSimplePolygon :: Polygon t p r -> SimplePolygon p r Source #
Convert a Polygon to a simple polygon by forgetting about any holes.
cmpExtreme :: (Num r, Ord r) => Vector 2 r -> (Point 2 r :+ p) -> (Point 2 r :+ q) -> Ordering Source #
Comparison that compares which point is larger in the direction given by
the vector u.
extremesLinear :: (Ord r, Num r) => Vector 2 r -> Polygon t p r -> (Point 2 r :+ p, Point 2 r :+ p) Source #
Finds the extreme points, minimum and maximum, in a given direction
running time: \(O(n)\)
numberVertices :: Polygon t p r -> Polygon t (SP Int p) r Source #
assigns unique integer numbers to all vertices. Numbers start from 0, and are increasing along the outer boundary. The vertices of holes will be numbered last, in the same order.
>>>numberVertices simplePolySimplePolygon CSeq [Point2 [0 % 1,0 % 1] :+ SP 0 (),Point2 [10 % 1,0 % 1] :+ SP 1 (),Point2 [10 % 1,10 % 1] :+ SP 2 (),Point2 [5 % 1,15 % 1] :+ SP 3 (),Point2 [1 % 1,11 % 1] :+ SP 4 ()]