Copyright	(c) Justus Sagemüller 2015
License	GPL v3
Maintainer	(@) jsag $ hvl.no
Stability	experimental
Portability	portable
Safe Haskell	None
Language	Haskell2010

Data.Manifold.TreeCover

Contents

Shades
Shade trees
- View helpers
- Auxiliary types
Misc
- External

Description

Synopsis

data Shade x where
- Shade :: (Semimanifold x, SimpleSpace (Needle x)) => {..} -> Shade x
pattern (:±) :: () => (Semimanifold x, SimpleSpace (Needle x)) => Interior x -> [Needle x] -> Shade x
data Shade' x = Shade' {
- _shade'Ctr :: !(Interior x)
- _shade'Narrowness :: !(Metric x)
}
(|±|) :: forall x. WithField ℝ EuclidSpace x => x -> [Needle x] -> Shade' x
class IsShade shade
shadeCtr :: IsShade shade => Lens' (shade x) (Interior x)
shadeExpanse :: Lens' (Shade x) (Metric' x)
shadeNarrowness :: Lens' (Shade' x) (Metric x)
fullShade :: (Semimanifold x, SimpleSpace (Needle x)) => Interior x -> Metric' x -> Shade x
fullShade' :: WithField ℝ SimpleSpace x => Interior x -> Metric x -> Shade' x
pointsShades :: (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade x]
pointsShade's :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade' x]
pointsCovers :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade x]
pointsCover's :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade' x]
coverAllAround :: forall x s. (Fractional' s, WithField s PseudoAffine x, SimpleSpace (Needle x)) => Interior x -> [Needle x] -> Shade x
occlusion :: (IsShade shade, PseudoAffine x, SimpleSpace (Needle x), s ~ Scalar (Needle x), RealFloat' s) => shade x -> x -> s
prettyShowsPrecShade' :: LtdErrorShow m => Int -> Shade' m -> ShowS
prettyShowShade' :: LtdErrorShow x => Shade' x -> String
factoriseShade :: (IsShade shade, PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => shade (x, y) -> (shade x, shade y)
intersectShade's :: forall y. Refinable y => NonEmpty (Shade' y) -> Maybe (Shade' y)
linIsoTransformShade :: (IsShade shade, SimpleSpace x, SimpleSpace y, Scalar x ~ Scalar y, Num' (Scalar x)) => (x +> y) -> shade x -> shade y
embedShade :: (IsShade shade, Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SemiInner (Needle x), SimpleSpace (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> shade x -> shade y
projectShade :: (IsShade shade, Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SimpleSpace (Needle x), SemiInner (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> shade y -> shade x
class (WithField ℝ PseudoAffine y, SimpleSpace (Needle y)) => Refinable y
subShade' :: Refinable y => Shade' y -> Shade' y -> Bool
refineShade' :: Refinable y => Shade' y -> Shade' y -> Maybe (Shade' y)
convolveShade' :: Refinable y => Shade' y -> Shade' (Needle y) -> Shade' y
coerceShade :: (IsShade shade, Manifold x, Manifold y, LocallyCoercible x y, SimpleSpace (Needle y)) => shade x -> shade y
mixShade's :: forall y. (WithField ℝ Manifold y, SimpleSpace (Needle y)) => NonEmpty (Shade' y) -> Maybe (Shade' y)
type ShadeTree x = x `Shaded` ()
fromLeafPoints :: forall x. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => [x] -> ShadeTree x
fromLeafPoints_ :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => [(x, y)] -> x `Shaded` y
onlyLeaves :: WithField ℝ PseudoAffine x => (x `Shaded` y) -> [(x, y)]
onlyLeaves_ :: WithField ℝ PseudoAffine x => ShadeTree x -> [x]
indexShadeTree :: forall x y. (x `Shaded` y) -> Int -> Either Int ([x `Shaded` y], (x, y))
treeLeaf :: forall x y f. Functor f => Int -> (y -> f y) -> (x `Shaded` y) -> Either Int (f (x `Shaded` y))
positionIndex :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => Maybe (Metric x) -> (x `Shaded` y) -> x -> Maybe (Int, ([x `Shaded` y], (x, y)))
entireTree :: forall x y. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => (x `Shaded` y) -> LeafyTree x y
onlyNodes :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => ShadeTree x -> Trees x
trunkBranches :: (x `Shaded` y) -> NonEmpty (LeafIndex, x `Shaded` y)
nLeaves :: (x `Shaded` y) -> Int
treeDepth :: (x `Shaded` y) -> Int
type SimpleTree = GenericTree Maybe []
type Trees = GenericTree [] []
type NonEmptyTree = GenericTree NonEmpty []
newtype GenericTree c b x = GenericTree {
- treeBranches :: c (x, GenericTree b b x)
}
朳 :: c (x, GenericTree b b x) -> GenericTree c b x
class HasFlatView f where
- type FlatView f x
- flatView :: f x -> FlatView f x
- superFlatView :: f x -> [[x]]
shadesMerge :: forall x. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => ℝ -> [Shade x] -> [Shade x]
allTwigs :: forall x y. WithField ℝ PseudoAffine x => (x `Shaded` y) -> [Twig x y]
twigsWithEnvirons :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => (x `Shaded` y) -> [(Twig x y, TwigEnviron x y)]
type Twig x y = (Int, x `Shaded` y)
type TwigEnviron x y = [Twig x y]
seekPotentialNeighbours :: forall x y. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => (x `Shaded` y) -> x `Shaded` (y, [Int])
completeTopShading :: forall x y. (WithField ℝ PseudoAffine x, WithField ℝ PseudoAffine y, SimpleSpace (Needle x), SimpleSpace (Needle y)) => (x `Shaded` y) -> [Shade' (x, y)]
flexTwigsShading :: forall x y f. (WithField ℝ Manifold x, WithField ℝ Manifold y, SimpleSpace (Needle x), SimpleSpace (Needle y), Applicative f) => (Shade' (x, y) -> f (x, (Shade' y, LocalLinear x y))) -> (x `Shaded` y) -> f (x `Shaded` y)
traverseTrunkBranchChoices :: Applicative f => ((Int, x `Shaded` y) -> (x `Shaded` y) -> f (x `Shaded` z)) -> (x `Shaded` y) -> f (x `Shaded` z)
data Shaded x y
- = PlainLeaves [(x, y)]
- | DisjointBranches !LeafCount (NonEmpty (x `Shaded` y))
- | OverlappingBranches !LeafCount !(Shade x) (NonEmpty (DBranch x y))
fmapShaded :: (Semimanifold x, SimpleSpace (Needle x)) => (y -> υ) -> (x `Shaded` y) -> x `Shaded` υ
constShaded :: y -> (x `Shaded` y₀) -> x `Shaded` y
zipTreeWithList :: (x `Shaded` w) -> NonEmpty y -> x `Shaded` (w, y)
stiAsIntervalMapping :: (x ~ ℝ, y ~ ℝ) => (x `Shaded` y) -> [(x, ((y, Diff y), LinearMap ℝ x y))]
spanShading :: forall x y. (WithField ℝ Manifold x, WithField ℝ Manifold y, SimpleSpace (Needle x), SimpleSpace (Needle y)) => (Shade x -> Shade y) -> ShadeTree x -> x `Shaded` y
type DBranch x y = DBranch' x (x `Shaded` y)
data DBranch' x c = DBranch {
- boughDirection :: !(Needle' x)
- boughContents :: !(Hourglass c)
}
data Hourglass s = Hourglass {
- upperBulb, lowerBulb :: !s
}
unsafeFmapTree :: (NonEmpty (x, y) -> NonEmpty (ξ, υ)) -> (Needle' x -> Needle' ξ) -> (Shade x -> Shade ξ) -> (x `Shaded` y) -> ξ `Shaded` υ
type AffineManifold m = (Atlas m, Manifold m, AffineSpace m, Needle m ~ Diff m, HasTrie (ChartIndex m))
euclideanMetric :: EuclidSpace x => proxy x -> Metric x

Shades

data Shade x where Source #

A Shade is a very crude description of a region within a manifold. It can be interpreted as either an ellipsoid shape, or as the Gaussian peak of a normal distribution (use http://hackage.haskell.org/package/manifold-random for actually sampling from that distribution).

For a precise description of an arbitrarily-shaped connected subset of a manifold, there is Region, whose implementation is vastly more complex.

Constructors

Shade
Fields :: (Semimanifold x, SimpleSpace (Needle x)) => { _shadeCtr :: !(Interior x) , _shadeExpanse :: !(Metric' x) } -> Shade x

Instances

ImpliesMetric Shade Source #
Instance details Defined in Data.Manifold.Shade Associated Types type MetricRequirement Shade x :: Constraint Source # Methods inferMetric :: (MetricRequirement Shade x, LSpace (Needle x)) => Shade x -> Metric x Source # inferMetric' :: (MetricRequirement Shade x, LSpace (Needle x)) => Shade x -> Metric' x Source #
IsShade Shade Source #
Instance details Defined in Data.Manifold.Shade Methods shadeCtr :: Lens' (Shade x) (Interior x) Source # occlusion :: (PseudoAffine x, SimpleSpace (Needle x), s ~ Scalar (Needle x), RealFloat' s) => Shade x -> x -> s Source # factoriseShade :: (PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => Shade (x, y) -> (Shade x, Shade y) Source # coerceShade :: (Manifold x, Manifold y, LocallyCoercible x y, SimpleSpace (Needle y)) => Shade x -> Shade y Source # orthoShades :: (PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => Shade x -> Shade y -> Shade (x, y) Source # linIsoTransformShade :: (SimpleSpace x, SimpleSpace y, Scalar x ~ Scalar y, Num' (Scalar x)) => (x +> y) -> Shade x -> Shade y Source # projectShade :: (Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SimpleSpace (Needle x), SemiInner (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> Shade y -> Shade x Source # embedShade :: (Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SemiInner (Needle x), SimpleSpace (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> Shade x -> Shade y Source #
(Show (Interior x), Show (Metric' x), WithField ℝ PseudoAffine x) => Show (Shade x) Source #
Instance details Defined in Data.Manifold.Shade Methods showsPrec :: Int -> Shade x -> ShowS # show :: Shade x -> String # showList :: [Shade x] -> ShowS #
PseudoAffine x => Semimanifold (Shade x) Source #
Instance details Defined in Data.Manifold.Shade Associated Types type Needle (Shade x) :: Type # type Interior (Shade x) :: Type # Methods (.+~^) :: Interior (Shade x) -> Needle (Shade x) -> Shade x # fromInterior :: Interior (Shade x) -> Shade x # toInterior :: Shade x -> Maybe (Interior (Shade x)) # translateP :: Tagged (Shade x) (Interior (Shade x) -> Needle (Shade x) -> Interior (Shade x)) # (.-~^) :: Interior (Shade x) -> Needle (Shade x) -> Shade x # semimanifoldWitness :: SemimanifoldWitness (Shade x) #
LtdErrorShow x => Show (Shade x) Source #
Instance details Defined in Data.Manifold.Shade Methods showsPrec :: Int -> Shade x -> ShowS # show :: Shade x -> String # showList :: [Shade x] -> ShowS #
(WithField ℝ PseudoAffine x, Geodesic (Interior x), SimpleSpace (Needle x)) => Geodesic (Shade x) Source #
Instance details Defined in Data.Manifold.Shade Methods geodesicBetween :: Shade x -> Shade x -> Maybe (D¹ -> Shade x) Source # geodesicWitness :: GeodesicWitness (Shade x) Source # middleBetween :: Shade x -> Shade x -> Maybe (Shade x) Source #
type MetricRequirement Shade x Source #
Instance details Defined in Data.Manifold.Shade type MetricRequirement Shade x = (Manifold x, SimpleSpace (Needle x))
type Interior (Shade x) Source #
Instance details Defined in Data.Manifold.Shade type Interior (Shade x) = Shade x
type Needle (Shade x) Source #
Instance details Defined in Data.Manifold.Shade type Needle (Shade x) = Needle x

pattern (:±) :: () => (Semimanifold x, SimpleSpace (Needle x)) => Interior x -> [Needle x] -> Shade x infixl 6 Source #

Span a Shade from a center point and multiple deviation-vectors.

data Shade' x Source #

A “co-shade” can describe ellipsoid regions as well, but unlike Shade it can be unlimited / infinitely wide in some directions. It does OTOH need to have nonzero thickness, which Shade needs not.

Constructors

Shade'
Fields _shade'Ctr :: !(Interior x) _shade'Narrowness :: !(Metric x)

Instances

ImpliesMetric Shade' Source #
Instance details Defined in Data.Manifold.Shade Associated Types type MetricRequirement Shade' x :: Constraint Source # Methods inferMetric :: (MetricRequirement Shade' x, LSpace (Needle x)) => Shade' x -> Metric x Source # inferMetric' :: (MetricRequirement Shade' x, LSpace (Needle x)) => Shade' x -> Metric' x Source #
IsShade Shade' Source #
Instance details Defined in Data.Manifold.Shade Methods shadeCtr :: Lens' (Shade' x) (Interior x) Source # occlusion :: (PseudoAffine x, SimpleSpace (Needle x), s ~ Scalar (Needle x), RealFloat' s) => Shade' x -> x -> s Source # factoriseShade :: (PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => Shade' (x, y) -> (Shade' x, Shade' y) Source # coerceShade :: (Manifold x, Manifold y, LocallyCoercible x y, SimpleSpace (Needle y)) => Shade' x -> Shade' y Source # orthoShades :: (PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => Shade' x -> Shade' y -> Shade' (x, y) Source # linIsoTransformShade :: (SimpleSpace x, SimpleSpace y, Scalar x ~ Scalar y, Num' (Scalar x)) => (x +> y) -> Shade' x -> Shade' y Source # projectShade :: (Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SimpleSpace (Needle x), SemiInner (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> Shade' y -> Shade' x Source # embedShade :: (Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SemiInner (Needle x), SimpleSpace (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> Shade' x -> Shade' y Source #
LtdErrorShow x => Show (Shade' x) Source #
Instance details Defined in Data.Manifold.Shade Methods showsPrec :: Int -> Shade' x -> ShowS # show :: Shade' x -> String # showList :: [Shade' x] -> ShowS #
AffineManifold x => Semimanifold (Shade' x) Source #
Instance details Defined in Data.Manifold.Shade Associated Types type Needle (Shade' x) :: Type # type Interior (Shade' x) :: Type # Methods (.+~^) :: Interior (Shade' x) -> Needle (Shade' x) -> Shade' x # fromInterior :: Interior (Shade' x) -> Shade' x # toInterior :: Shade' x -> Maybe (Interior (Shade' x)) # translateP :: Tagged (Shade' x) (Interior (Shade' x) -> Needle (Shade' x) -> Interior (Shade' x)) # (.-~^) :: Interior (Shade' x) -> Needle (Shade' x) -> Shade' x # semimanifoldWitness :: SemimanifoldWitness (Shade' x) #
LtdErrorShow x => Show (Shade' x) Source #
Instance details Defined in Data.Manifold.Shade Methods showsPrec :: Int -> Shade' x -> ShowS # show :: Shade' x -> String # showList :: [Shade' x] -> ShowS #
(WithField ℝ AffineManifold x, Geodesic x, SimpleSpace (Needle x)) => Geodesic (Shade' x) Source #
Instance details Defined in Data.Manifold.Shade Methods geodesicBetween :: Shade' x -> Shade' x -> Maybe (D¹ -> Shade' x) Source # geodesicWitness :: GeodesicWitness (Shade' x) Source # middleBetween :: Shade' x -> Shade' x -> Maybe (Shade' x) Source #
type MetricRequirement Shade' x Source #
Instance details Defined in Data.Manifold.Shade type MetricRequirement Shade' x = (Manifold x, SimpleSpace (Needle x))
type Interior (Shade' x) Source #
Instance details Defined in Data.Manifold.Shade type Interior (Shade' x) = Shade' x
type Needle (Shade' x) Source #
Instance details Defined in Data.Manifold.Shade type Needle (Shade' x) = Needle x

(|±|) :: forall x. WithField ℝ EuclidSpace x => x -> [Needle x] -> Shade' x infixl 6 Source #

Similar to :±, but instead of expanding the shade, each vector restricts it. Iff these form a orthogonal basis (in whatever sense applicable), then both methods will be equivalent.

Note that |±| is only possible, as such, in an inner-product space; in general you need reciprocal vectors (Needle') to define a Shade'.

class IsShade shade Source #

Minimal complete definition

shadeCtr, occlusion, factoriseShade, coerceShade, orthoShades, linIsoTransformShade, projectShade, embedShade

Instances

IsShade Shade' Source #
Instance details Defined in Data.Manifold.Shade Methods shadeCtr :: Lens' (Shade' x) (Interior x) Source # occlusion :: (PseudoAffine x, SimpleSpace (Needle x), s ~ Scalar (Needle x), RealFloat' s) => Shade' x -> x -> s Source # factoriseShade :: (PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => Shade' (x, y) -> (Shade' x, Shade' y) Source # coerceShade :: (Manifold x, Manifold y, LocallyCoercible x y, SimpleSpace (Needle y)) => Shade' x -> Shade' y Source # orthoShades :: (PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => Shade' x -> Shade' y -> Shade' (x, y) Source # linIsoTransformShade :: (SimpleSpace x, SimpleSpace y, Scalar x ~ Scalar y, Num' (Scalar x)) => (x +> y) -> Shade' x -> Shade' y Source # projectShade :: (Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SimpleSpace (Needle x), SemiInner (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> Shade' y -> Shade' x Source # embedShade :: (Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SemiInner (Needle x), SimpleSpace (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> Shade' x -> Shade' y Source #
IsShade Shade Source #
Instance details Defined in Data.Manifold.Shade Methods shadeCtr :: Lens' (Shade x) (Interior x) Source # occlusion :: (PseudoAffine x, SimpleSpace (Needle x), s ~ Scalar (Needle x), RealFloat' s) => Shade x -> x -> s Source # factoriseShade :: (PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => Shade (x, y) -> (Shade x, Shade y) Source # coerceShade :: (Manifold x, Manifold y, LocallyCoercible x y, SimpleSpace (Needle y)) => Shade x -> Shade y Source # orthoShades :: (PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => Shade x -> Shade y -> Shade (x, y) Source # linIsoTransformShade :: (SimpleSpace x, SimpleSpace y, Scalar x ~ Scalar y, Num' (Scalar x)) => (x +> y) -> Shade x -> Shade y Source # projectShade :: (Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SimpleSpace (Needle x), SemiInner (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> Shade y -> Shade x Source # embedShade :: (Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SemiInner (Needle x), SimpleSpace (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> Shade x -> Shade y Source #

Lenses

shadeCtr :: IsShade shade => Lens' (shade x) (Interior x) Source #

Access the center of a Shade or a Shade'.

shadeExpanse :: Lens' (Shade x) (Metric' x) Source #

shadeNarrowness :: Lens' (Shade' x) (Metric x) Source #

Construction

fullShade :: (Semimanifold x, SimpleSpace (Needle x)) => Interior x -> Metric' x -> Shade x Source #

fullShade' :: WithField ℝ SimpleSpace x => Interior x -> Metric x -> Shade' x Source #

pointsShades :: (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade x] Source #

Attempt to find a Shade that describes the distribution of given points. At least in an affine space (and thus locally in any manifold), this can be used to estimate the parameters of a normal distribution from which some points were sampled. Note that some points will be “outside” of the shade, as happens for a normal distribution with some statistical likelyhood. (Use pointsCovers if you need to prevent that.)

For nonconnected manifolds it will be necessary to yield separate shades for each connected component. And for an empty input list, there is no shade! Hence the result type is a list.

pointsShade's :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade' x] Source #

pointsCovers :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade x] Source #

Like pointsShades, but ensure that all points are actually in the shade, i.e. if [Shade x₀ ex] is the result then metric (recipMetric ex) (p-x₀) ≤ 1 for all p in the list.

pointsCover's :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade' x] Source #

coverAllAround :: forall x s. (Fractional' s, WithField s PseudoAffine x, SimpleSpace (Needle x)) => Interior x -> [Needle x] -> Shade x Source #

Evaluation

occlusion :: (IsShade shade, PseudoAffine x, SimpleSpace (Needle x), s ~ Scalar (Needle x), RealFloat' s) => shade x -> x -> s Source #

Check the statistical likelihood-density of a point being within a shade. This is taken as a normal distribution.

prettyShowsPrecShade' :: LtdErrorShow m => Int -> Shade' m -> ShowS Source #

prettyShowShade' :: LtdErrorShow x => Shade' x -> String Source #

Misc

factoriseShade :: (IsShade shade, PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => shade (x, y) -> (shade x, shade y) Source #

intersectShade's :: forall y. Refinable y => NonEmpty (Shade' y) -> Maybe (Shade' y) Source #

linIsoTransformShade :: (IsShade shade, SimpleSpace x, SimpleSpace y, Scalar x ~ Scalar y, Num' (Scalar x)) => (x +> y) -> shade x -> shade y Source #

embedShade :: (IsShade shade, Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SemiInner (Needle x), SimpleSpace (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> shade x -> shade y Source #

Include a shade in a higher-dimensional space. Notice that this behaves fundamentally different for Shade and Shade'. For Shade, it gives a “flat image” of the region, whereas for Shade' it gives an “extrusion pillar” pointing in the projection's orthogonal complement.

projectShade :: (IsShade shade, Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SimpleSpace (Needle x), SemiInner (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> shade y -> shade x Source #

Squash a shade down into a lower dimensional space.

class (WithField ℝ PseudoAffine y, SimpleSpace (Needle y)) => Refinable y Source #

Class of manifolds which can use Shade' as a basic set type. This is easily possible for vector spaces with the default implementations.

Instances

Refinable ℝ Source #
Instance details Defined in Data.Manifold.Shade Methods debugView :: Maybe (DebugView ℝ) subShade' :: Shade' ℝ -> Shade' ℝ -> Bool Source # refineShade' :: Shade' ℝ -> Shade' ℝ -> Maybe (Shade' ℝ) Source # convolveMetric :: Functor p => p ℝ -> Metric ℝ -> Metric ℝ -> Metric ℝ Source # convolveShade' :: Shade' ℝ -> Shade' (Needle ℝ) -> Shade' ℝ Source #
Refinable ℝ⁰ Source #
Instance details Defined in Data.Manifold.Shade Methods debugView :: Maybe (DebugView ℝ⁰) subShade' :: Shade' ℝ⁰ -> Shade' ℝ⁰ -> Bool Source # refineShade' :: Shade' ℝ⁰ -> Shade' ℝ⁰ -> Maybe (Shade' ℝ⁰) Source # convolveMetric :: Functor p => p ℝ⁰ -> Metric ℝ⁰ -> Metric ℝ⁰ -> Metric ℝ⁰ Source # convolveShade' :: Shade' ℝ⁰ -> Shade' (Needle ℝ⁰) -> Shade' ℝ⁰ Source #
Refinable ℝ⁴ Source #
Instance details Defined in Data.Manifold.Shade Methods debugView :: Maybe (DebugView ℝ⁴) subShade' :: Shade' ℝ⁴ -> Shade' ℝ⁴ -> Bool Source # refineShade' :: Shade' ℝ⁴ -> Shade' ℝ⁴ -> Maybe (Shade' ℝ⁴) Source # convolveMetric :: Functor p => p ℝ⁴ -> Metric ℝ⁴ -> Metric ℝ⁴ -> Metric ℝ⁴ Source # convolveShade' :: Shade' ℝ⁴ -> Shade' (Needle ℝ⁴) -> Shade' ℝ⁴ Source #
Refinable ℝ³ Source #
Instance details Defined in Data.Manifold.Shade Methods debugView :: Maybe (DebugView ℝ³) subShade' :: Shade' ℝ³ -> Shade' ℝ³ -> Bool Source # refineShade' :: Shade' ℝ³ -> Shade' ℝ³ -> Maybe (Shade' ℝ³) Source # convolveMetric :: Functor p => p ℝ³ -> Metric ℝ³ -> Metric ℝ³ -> Metric ℝ³ Source # convolveShade' :: Shade' ℝ³ -> Shade' (Needle ℝ³) -> Shade' ℝ³ Source #
Refinable ℝ² Source #
Instance details Defined in Data.Manifold.Shade Methods debugView :: Maybe (DebugView ℝ²) subShade' :: Shade' ℝ² -> Shade' ℝ² -> Bool Source # refineShade' :: Shade' ℝ² -> Shade' ℝ² -> Maybe (Shade' ℝ²) Source # convolveMetric :: Functor p => p ℝ² -> Metric ℝ² -> Metric ℝ² -> Metric ℝ² Source # convolveShade' :: Shade' ℝ² -> Shade' (Needle ℝ²) -> Shade' ℝ² Source #
Refinable ℝ¹ Source #
Instance details Defined in Data.Manifold.Shade Methods debugView :: Maybe (DebugView ℝ¹) subShade' :: Shade' ℝ¹ -> Shade' ℝ¹ -> Bool Source # refineShade' :: Shade' ℝ¹ -> Shade' ℝ¹ -> Maybe (Shade' ℝ¹) Source # convolveMetric :: Functor p => p ℝ¹ -> Metric ℝ¹ -> Metric ℝ¹ -> Metric ℝ¹ Source # convolveShade' :: Shade' ℝ¹ -> Shade' (Needle ℝ¹) -> Shade' ℝ¹ Source #
(Refinable a, Refinable b, Scalar (DualVector (DualVector (Needle b))) ~ Scalar (DualVector (DualVector (Needle a)))) => Refinable (a, b) Source #
Instance details Defined in Data.Manifold.Shade Methods debugView :: Maybe (DebugView (a, b)) subShade' :: Shade' (a, b) -> Shade' (a, b) -> Bool Source # refineShade' :: Shade' (a, b) -> Shade' (a, b) -> Maybe (Shade' (a, b)) Source # convolveMetric :: Functor p => p (a, b) -> Metric (a, b) -> Metric (a, b) -> Metric (a, b) Source # convolveShade' :: Shade' (a, b) -> Shade' (Needle (a, b)) -> Shade' (a, b) Source #
(SimpleSpace a, SimpleSpace b, Refinable a, Refinable b, Scalar a ~ ℝ, Scalar b ~ ℝ, Scalar (DualVector a) ~ ℝ, Scalar (DualVector b) ~ ℝ, Scalar (DualVector (DualVector a)) ~ ℝ, Scalar (DualVector (DualVector b)) ~ ℝ) => Refinable (LinearMap ℝ a b) Source #
Instance details Defined in Data.Manifold.Shade Methods debugView :: Maybe (DebugView (LinearMap ℝ a b)) subShade' :: Shade' (LinearMap ℝ a b) -> Shade' (LinearMap ℝ a b) -> Bool Source # refineShade' :: Shade' (LinearMap ℝ a b) -> Shade' (LinearMap ℝ a b) -> Maybe (Shade' (LinearMap ℝ a b)) Source # convolveMetric :: Functor p => p (LinearMap ℝ a b) -> Metric (LinearMap ℝ a b) -> Metric (LinearMap ℝ a b) -> Metric (LinearMap ℝ a b) Source # convolveShade' :: Shade' (LinearMap ℝ a b) -> Shade' (Needle (LinearMap ℝ a b)) -> Shade' (LinearMap ℝ a b) Source #

subShade' :: Refinable y => Shade' y -> Shade' y -> Bool Source #

a subShade' b ≡ True means a is fully contained in b, i.e. from minusLogOcclusion' a p < 1 follows also minusLogOcclusion' b p < 1.

refineShade' :: Refinable y => Shade' y -> Shade' y -> Maybe (Shade' y) Source #

Intersection between two shades.

convolveShade' :: Refinable y => Shade' y -> Shade' (Needle y) -> Shade' y Source #

coerceShade :: (IsShade shade, Manifold x, Manifold y, LocallyCoercible x y, SimpleSpace (Needle y)) => shade x -> shade y Source #

mixShade's :: forall y. (WithField ℝ Manifold y, SimpleSpace (Needle y)) => NonEmpty (Shade' y) -> Maybe (Shade' y) Source #

Weakened version of intersectShade's. What this function calculates is rather the weighted mean of ellipsoid regions. If you interpret the shades as uncertain physical measurements with normal distribution, it gives the maximum-likelyhood result for multiple measurements of the same quantity.

Shade trees

type ShadeTree x = x `Shaded` () Source #

fromLeafPoints :: forall x. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => [x] -> ShadeTree x Source #

Build a quite nicely balanced tree from a cloud of points, on any real manifold.

Example: https://nbviewer.jupyter.org/github/leftaroundabout/manifolds/blob/master/test/Trees-and-Webs.ipynb#pseudorandomCloudTree

fromLeafPoints_ :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => [(x, y)] -> x `Shaded` y Source #

onlyLeaves :: WithField ℝ PseudoAffine x => (x `Shaded` y) -> [(x, y)] Source #

onlyLeaves_ :: WithField ℝ PseudoAffine x => ShadeTree x -> [x] Source #

Left (and, typically, also right) inverse of fromLeafNodes.

indexShadeTree :: forall x y. (x `Shaded` y) -> Int -> Either Int ([x `Shaded` y], (x, y)) Source #

The leaves of a shade tree are numbered. For a given index, this function attempts to find the leaf with that ID, within its immediate environment.

treeLeaf :: forall x y f. Functor f => Int -> (y -> f y) -> (x `Shaded` y) -> Either Int (f (x `Shaded` y)) Source #

positionIndex Source #

Arguments

:: (WithField ℝ Manifold x, SimpleSpace (Needle x))
=> Maybe (Metric x)	For deciding (at the lowest level) what “close” means; this is optional for any tree of depth >1.
-> (x `Shaded` y)	The tree to index into
-> x	Position to look up
-> Maybe (Int, ([x `Shaded` y], (x, y)))	Index of the leaf near to the query point, the “path” of environment trees leading down to its position (in decreasing order of size), and actual position+info of the found node.

“Inverse indexing” of a tree. This is roughly a nearest-neighbour search, but not guaranteed to give the correct result unless evaluated at the precise position of a tree leaf.

View helpers

entireTree :: forall x y. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => (x `Shaded` y) -> LeafyTree x y Source #

onlyNodes :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => ShadeTree x -> Trees x Source #

Imitate the specialised ShadeTree structure with a simpler, generic tree.

trunkBranches :: (x `Shaded` y) -> NonEmpty (LeafIndex, x `Shaded` y) Source #

nLeaves :: (x `Shaded` y) -> Int Source #

treeDepth :: (x `Shaded` y) -> Int Source #

Auxiliary types

type SimpleTree = GenericTree Maybe [] Source #

SimpleTree x ≅ Maybe (x, Trees x)

type Trees = GenericTree [] [] Source #

Trees x ≅ [(x, Trees x)]

type NonEmptyTree = GenericTree NonEmpty [] Source #

NonEmptyTree x ≅ (x, Trees x)

newtype GenericTree c b x Source #

Constructors

GenericTree
Fields treeBranches :: c (x, GenericTree b b x)

Instances

(Functor c, Functor b) => Functor (GenericTree c b) Source #
Instance details Defined in Data.Manifold.TreeCover Methods fmap :: (a -> b0) -> GenericTree c b a -> GenericTree c b b0 # (<$) :: a -> GenericTree c b b0 -> GenericTree c b a #
(Foldable c, Foldable b) => Foldable (GenericTree c b) Source #
Instance details Defined in Data.Manifold.TreeCover Methods fold :: Monoid m => GenericTree c b m -> m # foldMap :: Monoid m => (a -> m) -> GenericTree c b a -> m # foldr :: (a -> b0 -> b0) -> b0 -> GenericTree c b a -> b0 # foldr' :: (a -> b0 -> b0) -> b0 -> GenericTree c b a -> b0 # foldl :: (b0 -> a -> b0) -> b0 -> GenericTree c b a -> b0 # foldl' :: (b0 -> a -> b0) -> b0 -> GenericTree c b a -> b0 # foldr1 :: (a -> a -> a) -> GenericTree c b a -> a # foldl1 :: (a -> a -> a) -> GenericTree c b a -> a # toList :: GenericTree c b a -> [a] # null :: GenericTree c b a -> Bool # length :: GenericTree c b a -> Int # elem :: Eq a => a -> GenericTree c b a -> Bool # maximum :: Ord a => GenericTree c b a -> a # minimum :: Ord a => GenericTree c b a -> a # sum :: Num a => GenericTree c b a -> a # product :: Num a => GenericTree c b a -> a #
(Traversable c, Traversable b) => Traversable (GenericTree c b) Source #
Instance details Defined in Data.Manifold.TreeCover Methods traverse :: Applicative f => (a -> f b0) -> GenericTree c b a -> f (GenericTree c b b0) # sequenceA :: Applicative f => GenericTree c b (f a) -> f (GenericTree c b a) # mapM :: Monad m => (a -> m b0) -> GenericTree c b a -> m (GenericTree c b b0) # sequence :: Monad m => GenericTree c b (m a) -> m (GenericTree c b a) #
Eq (c (x, GenericTree b b x)) => Eq (GenericTree c b x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods (==) :: GenericTree c b x -> GenericTree c b x -> Bool # (/=) :: GenericTree c b x -> GenericTree c b x -> Bool #
Show (c (x, GenericTree b b x)) => Show (GenericTree c b x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods showsPrec :: Int -> GenericTree c b x -> ShowS # show :: GenericTree c b x -> String # showList :: [GenericTree c b x] -> ShowS #
Generic (GenericTree c b x) Source #
Instance details Defined in Data.Manifold.TreeCover Associated Types type Rep (GenericTree c b x) :: Type -> Type # Methods from :: GenericTree c b x -> Rep (GenericTree c b x) x0 # to :: Rep (GenericTree c b x) x0 -> GenericTree c b x #
MonadPlus c => Semigroup (GenericTree c b x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods (<>) :: GenericTree c b x -> GenericTree c b x -> GenericTree c b x # sconcat :: NonEmpty (GenericTree c b x) -> GenericTree c b x # stimes :: Integral b0 => b0 -> GenericTree c b x -> GenericTree c b x #
MonadPlus c => Monoid (GenericTree c b x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods mempty :: GenericTree c b x # mappend :: GenericTree c b x -> GenericTree c b x -> GenericTree c b x # mconcat :: [GenericTree c b x] -> GenericTree c b x #
(NFData x, Foldable c, Foldable b) => NFData (GenericTree c b x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods rnf :: GenericTree c b x -> () #
type Rep (GenericTree c b x) Source #
Instance details Defined in Data.Manifold.TreeCover type Rep (GenericTree c b x) = D1 (MetaData "GenericTree" "Data.Manifold.TreeCover" "manifolds-0.5.1.0-1Kt2fUEXtjvw812WS2MXA" True) (C1 (MetaCons "GenericTree" PrefixI True) (S1 (MetaSel (Just "treeBranches") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (c (x, GenericTree b b x)))))

朳 :: c (x, GenericTree b b x) -> GenericTree c b x Source #

U+6733 CJK UNIFIED IDEOGRAPH tree. The main purpose of this is to give GenericTree a more concise Show instance.

Misc

class HasFlatView f where Source #

Associated Types

type FlatView f x Source #

Methods

flatView :: f x -> FlatView f x Source #

superFlatView :: f x -> [[x]] Source #

shadesMerge Source #

Arguments

:: (WithField ℝ Manifold x, SimpleSpace (Needle x))
=> ℝ	How near (inverse normalised distance, relative to shade expanse) two shades must be to be merged. If this is zero, any shades in the same connected region of a manifold are merged.
-> [Shade x]	A list of n shades.
-> [Shade x]	m ≤ n shades which cover at least the same area.

Attempt to reduce the number of shades to fewer (ideally, a single one). In the simplest cases these should guaranteed cover the same area; for non-flat manifolds it only works in a heuristic sense.

allTwigs :: forall x y. WithField ℝ PseudoAffine x => (x `Shaded` y) -> [Twig x y] Source #

twigsWithEnvirons :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => (x `Shaded` y) -> [(Twig x y, TwigEnviron x y)] Source #

Example: https://nbviewer.jupyter.org/github/leftaroundabout/manifolds/blob/master/test/Trees-and-Webs.ipynb#pseudorandomCloudTree

type Twig x y = (Int, x `Shaded` y) Source #

type TwigEnviron x y = [Twig x y] Source #

seekPotentialNeighbours :: forall x y. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => (x `Shaded` y) -> x `Shaded` (y, [Int]) Source #

completeTopShading :: forall x y. (WithField ℝ PseudoAffine x, WithField ℝ PseudoAffine y, SimpleSpace (Needle x), SimpleSpace (Needle y)) => (x `Shaded` y) -> [Shade' (x, y)] Source #

flexTwigsShading :: forall x y f. (WithField ℝ Manifold x, WithField ℝ Manifold y, SimpleSpace (Needle x), SimpleSpace (Needle y), Applicative f) => (Shade' (x, y) -> f (x, (Shade' y, LocalLinear x y))) -> (x `Shaded` y) -> f (x `Shaded` y) Source #

traverseTrunkBranchChoices :: Applicative f => ((Int, x `Shaded` y) -> (x `Shaded` y) -> f (x `Shaded` z)) -> (x `Shaded` y) -> f (x `Shaded` z) Source #

data Shaded x y Source #

Constructors

PlainLeaves [(x, y)]
DisjointBranches !LeafCount (NonEmpty (x `Shaded` y))
OverlappingBranches !LeafCount !(Shade x) (NonEmpty (DBranch x y))

Instances

Functor (Shaded x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods fmap :: (a -> b) -> Shaded x a -> Shaded x b # (<$) :: a -> Shaded x b -> Shaded x a #
Foldable (Shaded x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods fold :: Monoid m => Shaded x m -> m # foldMap :: Monoid m => (a -> m) -> Shaded x a -> m # foldr :: (a -> b -> b) -> b -> Shaded x a -> b # foldr' :: (a -> b -> b) -> b -> Shaded x a -> b # foldl :: (b -> a -> b) -> b -> Shaded x a -> b # foldl' :: (b -> a -> b) -> b -> Shaded x a -> b # foldr1 :: (a -> a -> a) -> Shaded x a -> a # foldl1 :: (a -> a -> a) -> Shaded x a -> a # toList :: Shaded x a -> [a] # null :: Shaded x a -> Bool # length :: Shaded x a -> Int # elem :: Eq a => a -> Shaded x a -> Bool # maximum :: Ord a => Shaded x a -> a # minimum :: Ord a => Shaded x a -> a # sum :: Num a => Shaded x a -> a # product :: Num a => Shaded x a -> a #
Traversable (Shaded x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods traverse :: Applicative f => (a -> f b) -> Shaded x a -> f (Shaded x b) # sequenceA :: Applicative f => Shaded x (f a) -> f (Shaded x a) # mapM :: Monad m => (a -> m b) -> Shaded x a -> m (Shaded x b) # sequence :: Monad m => Shaded x (m a) -> m (Shaded x a) #
(WithField ℝ Manifold x, SimpleSpace (Needle x)) => Semigroup (ShadeTree x) Source #	WRT union.
Instance details Defined in Data.Manifold.TreeCover Methods (<>) :: ShadeTree x -> ShadeTree x -> ShadeTree x # sconcat :: NonEmpty (ShadeTree x) -> ShadeTree x # stimes :: Integral b => b -> ShadeTree x -> ShadeTree x #
(WithField ℝ Manifold x, SimpleSpace (Needle x)) => Monoid (ShadeTree x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods mempty :: ShadeTree x # mappend :: ShadeTree x -> ShadeTree x -> ShadeTree x # mconcat :: [ShadeTree x] -> ShadeTree x #
(WithField ℝ PseudoAffine x, Show x, Show (Interior x), Show (Needle' x), Show (Metric' x)) => Show (Shaded x ()) Source #
Instance details Defined in Data.Manifold.TreeCover Methods showsPrec :: Int -> Shaded x () -> ShowS # show :: Shaded x () -> String # showList :: [Shaded x ()] -> ShowS #
Generic (Shaded x y) Source #
Instance details Defined in Data.Manifold.TreeCover Associated Types type Rep (Shaded x y) :: Type -> Type # Methods from :: Shaded x y -> Rep (Shaded x y) x0 # to :: Rep (Shaded x y) x0 -> Shaded x y #
(NFData x, NFData (Needle' x), NFData y) => NFData (DBranch x y) Source #
Instance details Defined in Data.Manifold.TreeCover Methods rnf :: DBranch x y -> () #
(NFData x, NFData (Needle' x), NFData y) => NFData (Shaded x y) Source #
Instance details Defined in Data.Manifold.TreeCover Methods rnf :: Shaded x y -> () #
type Rep (Shaded x y) Source #
Instance details Defined in Data.Manifold.TreeCover type Rep (Shaded x y)

fmapShaded :: (Semimanifold x, SimpleSpace (Needle x)) => (y -> υ) -> (x `Shaded` y) -> x `Shaded` υ Source #

constShaded :: y -> (x `Shaded` y₀) -> x `Shaded` y Source #

zipTreeWithList :: (x `Shaded` w) -> NonEmpty y -> x `Shaded` (w, y) Source #

stiAsIntervalMapping :: (x ~ ℝ, y ~ ℝ) => (x `Shaded` y) -> [(x, ((y, Diff y), LinearMap ℝ x y))] Source #

spanShading :: forall x y. (WithField ℝ Manifold x, WithField ℝ Manifold y, SimpleSpace (Needle x), SimpleSpace (Needle y)) => (Shade x -> Shade y) -> ShadeTree x -> x `Shaded` y Source #

type DBranch x y = DBranch' x (x `Shaded` y) Source #

data DBranch' x c Source #

Constructors

DBranch
Fields boughDirection :: !(Needle' x) boughContents :: !(Hourglass c)

Instances

Functor (DBranch' x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods fmap :: (a -> b) -> DBranch' x a -> DBranch' x b # (<$) :: a -> DBranch' x b -> DBranch' x a #
Foldable (DBranch' x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods fold :: Monoid m => DBranch' x m -> m # foldMap :: Monoid m => (a -> m) -> DBranch' x a -> m # foldr :: (a -> b -> b) -> b -> DBranch' x a -> b # foldr' :: (a -> b -> b) -> b -> DBranch' x a -> b # foldl :: (b -> a -> b) -> b -> DBranch' x a -> b # foldl' :: (b -> a -> b) -> b -> DBranch' x a -> b # foldr1 :: (a -> a -> a) -> DBranch' x a -> a # foldl1 :: (a -> a -> a) -> DBranch' x a -> a # toList :: DBranch' x a -> [a] # null :: DBranch' x a -> Bool # length :: DBranch' x a -> Int # elem :: Eq a => a -> DBranch' x a -> Bool # maximum :: Ord a => DBranch' x a -> a # minimum :: Ord a => DBranch' x a -> a # sum :: Num a => DBranch' x a -> a # product :: Num a => DBranch' x a -> a #
Traversable (DBranch' x) Source #
Instance details Defined in Data.Manifold.TreeCover Methods traverse :: Applicative f => (a -> f b) -> DBranch' x a -> f (DBranch' x b) # sequenceA :: Applicative f => DBranch' x (f a) -> f (DBranch' x a) # mapM :: Monad m => (a -> m b) -> DBranch' x a -> m (DBranch' x b) # sequence :: Monad m => DBranch' x (m a) -> m (DBranch' x a) #
(WithField ℝ PseudoAffine x, Show (Needle' x), Show c) => Show (DBranch' x c) Source #
Instance details Defined in Data.Manifold.TreeCover Methods showsPrec :: Int -> DBranch' x c -> ShowS # show :: DBranch' x c -> String # showList :: [DBranch' x c] -> ShowS #
Generic (DBranch' x c) Source #
Instance details Defined in Data.Manifold.TreeCover Associated Types type Rep (DBranch' x c) :: Type -> Type # Methods from :: DBranch' x c -> Rep (DBranch' x c) x0 # to :: Rep (DBranch' x c) x0 -> DBranch' x c #
(NFData x, NFData (Needle' x), NFData y) => NFData (DBranch x y) Source #
Instance details Defined in Data.Manifold.TreeCover Methods rnf :: DBranch x y -> () #
type Rep (DBranch' x c) Source #
Instance details Defined in Data.Manifold.TreeCover type Rep (DBranch' x c) = D1 (MetaData "DBranch'" "Data.Manifold.TreeCover" "manifolds-0.5.1.0-1Kt2fUEXtjvw812WS2MXA" False) (C1 (MetaCons "DBranch" PrefixI True) (S1 (MetaSel (Just "boughDirection") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 (Needle' x)) :*: S1 (MetaSel (Just "boughContents") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 (Hourglass c))))

data Hourglass s Source #

Hourglass as the geometric shape (two opposing ~conical volumes, sharing only a single point in the middle); has nothing to do with time.

Constructors

Hourglass
Fields upperBulb, lowerBulb :: !s

Instances

Functor Hourglass Source #
Instance details Defined in Data.Manifold.TreeCover Methods fmap :: (a -> b) -> Hourglass a -> Hourglass b # (<$) :: a -> Hourglass b -> Hourglass a #
Applicative Hourglass Source #
Instance details Defined in Data.Manifold.TreeCover Methods pure :: a -> Hourglass a # (<>) :: Hourglass (a -> b) -> Hourglass a -> Hourglass b # liftA2 :: (a -> b -> c) -> Hourglass a -> Hourglass b -> Hourglass c # (>) :: Hourglass a -> Hourglass b -> Hourglass b # (<*) :: Hourglass a -> Hourglass b -> Hourglass a #
Foldable Hourglass Source #
Instance details Defined in Data.Manifold.TreeCover Methods fold :: Monoid m => Hourglass m -> m # foldMap :: Monoid m => (a -> m) -> Hourglass a -> m # foldr :: (a -> b -> b) -> b -> Hourglass a -> b # foldr' :: (a -> b -> b) -> b -> Hourglass a -> b # foldl :: (b -> a -> b) -> b -> Hourglass a -> b # foldl' :: (b -> a -> b) -> b -> Hourglass a -> b # foldr1 :: (a -> a -> a) -> Hourglass a -> a # foldl1 :: (a -> a -> a) -> Hourglass a -> a # toList :: Hourglass a -> [a] # null :: Hourglass a -> Bool # length :: Hourglass a -> Int # elem :: Eq a => a -> Hourglass a -> Bool # maximum :: Ord a => Hourglass a -> a # minimum :: Ord a => Hourglass a -> a # sum :: Num a => Hourglass a -> a # product :: Num a => Hourglass a -> a #
Traversable Hourglass Source #
Instance details Defined in Data.Manifold.TreeCover Methods traverse :: Applicative f => (a -> f b) -> Hourglass a -> f (Hourglass b) # sequenceA :: Applicative f => Hourglass (f a) -> f (Hourglass a) # mapM :: Monad m => (a -> m b) -> Hourglass a -> m (Hourglass b) # sequence :: Monad m => Hourglass (m a) -> m (Hourglass a) #
Foldable Hourglass ((->) :: Type -> Type -> Type) ((->) :: Type -> Type -> Type) Source #
Instance details Defined in Data.Manifold.TreeCover Methods ffoldl :: (ObjectPair (->) a b, ObjectPair (->) a (Hourglass b)) => ((a, b) -> a) -> (a, Hourglass b) -> a # foldMap :: (Object (->) a, Object (->) (Hourglass a), Semigroup m, Monoid m, Object (->) m, Object (->) m) => (a -> m) -> Hourglass a -> m #
Show s => Show (Hourglass s) Source #
Instance details Defined in Data.Manifold.TreeCover Methods showsPrec :: Int -> Hourglass s -> ShowS # show :: Hourglass s -> String # showList :: [Hourglass s] -> ShowS #
Generic (Hourglass s) Source #
Instance details Defined in Data.Manifold.TreeCover Associated Types type Rep (Hourglass s) :: Type -> Type # Methods from :: Hourglass s -> Rep (Hourglass s) x # to :: Rep (Hourglass s) x -> Hourglass s #
Semigroup s => Semigroup (Hourglass s) Source #
Instance details Defined in Data.Manifold.TreeCover Methods (<>) :: Hourglass s -> Hourglass s -> Hourglass s # sconcat :: NonEmpty (Hourglass s) -> Hourglass s # stimes :: Integral b => b -> Hourglass s -> Hourglass s #
(Monoid s, Semigroup s) => Monoid (Hourglass s) Source #
Instance details Defined in Data.Manifold.TreeCover Methods mempty :: Hourglass s # mappend :: Hourglass s -> Hourglass s -> Hourglass s # mconcat :: [Hourglass s] -> Hourglass s #
NFData s => NFData (Hourglass s) Source #
Instance details Defined in Data.Manifold.TreeCover Methods rnf :: Hourglass s -> () #
type Rep (Hourglass s) Source #
Instance details Defined in Data.Manifold.TreeCover type Rep (Hourglass s) = D1 (MetaData "Hourglass" "Data.Manifold.TreeCover" "manifolds-0.5.1.0-1Kt2fUEXtjvw812WS2MXA" False) (C1 (MetaCons "Hourglass" PrefixI True) (S1 (MetaSel (Just "upperBulb") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 s) :*: S1 (MetaSel (Just "lowerBulb") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 s)))

unsafeFmapTree :: (NonEmpty (x, y) -> NonEmpty (ξ, υ)) -> (Needle' x -> Needle' ξ) -> (Shade x -> Shade ξ) -> (x `Shaded` y) -> ξ `Shaded` υ Source #

External

type AffineManifold m = (Atlas m, Manifold m, AffineSpace m, Needle m ~ Diff m, HasTrie (ChartIndex m)) Source #

The AffineSpace class plus manifold constraints.

euclideanMetric :: EuclidSpace x => proxy x -> Metric x Source #