manifolds- Coordinate-free hypersurfaces

Copyright(c) Justus Sagemüller 2015
LicenseGPL v3
Maintainer(@) sagemueller $
Safe HaskellNone







data Shade x 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 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.




_shadeCtr :: !(Interior x)
_shadeExpanse :: !(Metric' x)

pattern (:±) :: () => (WithField Manifold x, SimpleSpace (Needle x)) => Interior x -> [Needle x] -> Shade x 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.




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

(|±|) :: WithField EuclidSpace x => x -> [Needle x] -> Shade' x 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 where Source


shadeCtr :: Lens' (shade x) (Interior x) Source

Access the center of a Shade or a Shade'.

occlusion :: (PseudoAffine x, SimpleSpace (Needle x), s ~ Scalar (Needle x), RealDimension 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.

factoriseShade :: (Manifold x, SimpleSpace (Needle x), Manifold 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) => shade x -> shade y Source

linIsoTransformShade :: (LinearManifold x, LinearManifold y, SimpleSpace x, SimpleSpace y, Scalar x ~ Scalar y) => (x +> y) -> shade x -> shade y Source


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


pointsShades :: (WithField 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.

pointsCovers :: forall x. (WithField 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.



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

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

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

Minimal complete definition



subShade' :: 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' :: Shade' y -> Shade' y -> Maybe (Shade' y) Source

Intersection between two shades.

convolveShade' :: Shade' y -> Shade' (Needle y) -> Shade' y Source

mixShade's :: forall y. (WithField 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

data ShadeTree x Source


PlainLeaves [x] 
DisjointBranches !Int (NonEmpty (ShadeTree x)) 
OverlappingBranches !Int !(Shade x) (NonEmpty (DBranch x)) 

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

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


onlyLeaves :: WithField PseudoAffine x => ShadeTree x -> [x] Source

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

indexShadeTree :: forall x. WithField Manifold x => ShadeTree x -> Int -> Either Int ([ShadeTree x], x) 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.

positionIndex Source


:: (WithField 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.

-> ShadeTree x

The tree to index into

-> x

Position to look up

-> Maybe (Int, ([ShadeTree x], x))

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 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

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

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

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




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



class HasFlatView f where Source

Associated Types

type FlatView f x Source


flatView :: f x -> FlatView f x Source

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

shadesMerge Source


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

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]

mn 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.

smoothInterpolate :: forall x y. (WithField Manifold x, WithField LinearManifold y, SimpleSpace (Needle x)) => NonEmpty (x, y) -> x -> y Source

type Twig x = (Int, ShadeTree x) Source

type TwigEnviron x = [Twig x] Source

flexTwigsShading :: forall x y f. (WithField Manifold x, WithField 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

data WithAny x y Source

Essentially the same as (x,y), but not considered as a product topology. The Semimanifold etc. instances just copy the topology of x, ignoring y.




_untopological :: y
_topological :: !x

type Shaded x y = ShadeTree (x `WithAny` y) Source

This is to ShadeTree as Map is to Set.

fmapShaded :: (y -> υ) -> (x `Shaded` y) -> x `Shaded` υ Source

joinShaded :: ((x `WithAny` y) `Shaded` z) -> x `Shaded` (y, z) Source

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


type TriangBuild t n x = TriangT t (S n) x (State (Map (SimplexIT t n x) (Metric x, ISimplex (S n) x))) Source

doTriangBuild :: KnownNat n => (forall t. TriangBuild t n x ()) -> [Simplex (S n) x] Source

breakdownAutoTriang :: forall n n' x. (KnownNat n', n ~ S n') => AutoTriang n x -> [Simplex n x] Source