Data.Manifold.Web

The web data type

data PointsWeb :: * -> * -> *

A PointsWeb is almost, but not quite a mesh. It is a stongly connected† directed graph, backed by a tree for fast nearest-neighbour lookup of points.

†In general, there can be disconnected components, but every connected component is strongly connected.

Instances

Functor (PointsWeb a)

Foldable (PointsWeb a)

Traversable (PointsWeb a)

Foldable (PointsWeb x) (->) (->)

Traversable (PointsWeb x) (PointsWeb x) (->) (->)

Generic (PointsWeb a b)

(NFData x, NFData (Metric x), NFData (Needle' x), NFData y) => NFData (PointsWeb x y)

type TraversalObject (->) (PointsWeb x) b = ()

type Rep (PointsWeb a b)

Construction

fromWebNodes :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => MetricChoice x -> [(x, y)] -> PointsWeb x y

fromShadeTree_auto :: forall x. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => ShadeTree x -> PointsWeb x ()

fromShadeTree :: forall x. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => (Shade x -> Metric x) -> ShadeTree x -> PointsWeb x ()

fromShaded

Arguments

:: (WithField ℝ Manifold x, SimpleSpace (Needle x))

=> MetricChoice x

Local scalar-product generator. You can always use recipMetric . _shadeExpanse (but this may give distortions compared to an actual Riemannian metric).

-> (x `Shaded` y)

Source tree.

-> PointsWeb x y

Lookup

nearestNeighbour :: (WithField ℝ Manifold x, SimpleSpace (Needle x)) => PointsWeb x y -> x -> Maybe (x, y)

indexWeb :: (WithField ℝ Manifold x, SimpleSpace (Needle x)) => PointsWeb x y -> WebNodeId -> Maybe (x, y)

webEdges :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => PointsWeb x y -> [((x, y), (x, y))]

toGraph :: (WithField ℝ Manifold x, SimpleSpace (Needle x)) => PointsWeb x y -> (Graph, Vertex -> (x, y))

Decomposition

sliceWeb_lin :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x), Geodesic x, Geodesic y) => PointsWeb x y -> Cutplane x -> [(x, y)]

Fetch a point between any two neighbouring web nodes on opposite sides of the plane, and linearly interpolate the values onto the cut plane.

sampleWeb_2Dcartesian_lin :: (x ~ ℝ, y ~ ℝ, Geodesic z) => PointsWeb (x, y) z -> ((x, x), Int) -> ((y, y), Int) -> [(y, [(x, Maybe z)])]

sampleEntireWeb_2Dcartesian_lin :: (x ~ ℝ, y ~ ℝ, Geodesic z) => PointsWeb (x, y) z -> Int -> Int -> [(y, [(x, Maybe z)])]

Local environments

localFocusWeb :: WithField ℝ Manifold x => PointsWeb x y -> PointsWeb x ((x, y), [(Needle x, y)])

Uncertain functions

differentiateUncertainWebFunction :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x), WithField ℝ Manifold y, SimpleSpace (Needle y), Refinable y) => PointsWeb x (Shade' y) -> PointsWeb x (Shade' (LocalLinear x y))

Differential equations

Fixed resolution

filterDEqnSolution_static :: forall x y m. (WithField ℝ Manifold x, SimpleSpace (Needle x), Refinable y, Geodesic (Interior y)) => InconsistencyStrategy m x (Shade' y) -> DifferentialEqn x y -> PointsWeb x (Shade' y) -> m (PointsWeb x (Shade' y))

iterateFilterDEqn_static :: (WithField ℝ Manifold x, SimpleSpace (Needle x), Refinable y, Geodesic (Interior y), Applicative m) => InconsistencyStrategy m x (Shade' y) -> DifferentialEqn x y -> PointsWeb x (Shade' y) -> [PointsWeb x (Shade' y)]

Automatic resolution

filterDEqnSolutions_adaptive

Arguments

:: (WithField ℝ Manifold x, SimpleSpace (Needle x), WithField ℝ AffineManifold y, Refinable y, Geodesic y, badness ~ ℝ, Monad m)

=> MetricChoice x

Scalar product on the domain, for regularising the web.

-> InconsistencyStrategy m x (Shade' y)

-> DifferentialEqn x y

-> (x -> Shade' y -> badness)

-> PointsWeb x (SolverNodeState x y)

-> m (PointsWeb x (SolverNodeState x y))

iterateFilterDEqn_adaptive

Arguments

:: (WithField ℝ Manifold x, SimpleSpace (Needle x), WithField ℝ AffineManifold y, Refinable y, Geodesic y, Monad m)

=> MetricChoice x

Scalar product on the domain, for regularising the web.

-> InconsistencyStrategy m x (Shade' y)

-> DifferentialEqn x y

-> (x -> Shade' y -> ℝ)

Badness function for local results.

-> PointsWeb x (Shade' y)

-> [PointsWeb x (Shade' y)]

Configuration

data InconsistencyStrategy m x y where

Constructors

AbortOnInconsistency :: InconsistencyStrategy Maybe x y

IgnoreInconsistencies :: InconsistencyStrategy Identity x y

HighlightInconsistencies :: y -> InconsistencyStrategy Identity x y

Instances

Functor (InconsistencyStrategy m x)

Misc

data ConvexSet x

Constructors

EmptyConvex

ConvexSet

Fields

convexSetHull :: Shade' x

If p is in all intersectors, it must also be in the hull.

convexSetIntersectors :: [Shade' x]

Instances

(WithField ℝ Manifold x, SimpleSpace (Needle x), Show (Interior x), Show (Needle' x)) => Show (ConvexSet x)

Refinable x => Semigroup (ConvexSet x)

Under intersection.

ellipsoid :: Shade' x -> ConvexSet x

coerceWebDomain :: forall a b y. (Manifold a, Manifold b, LocallyCoercible a b) => PointsWeb a y -> PointsWeb b y