Data.Manifold.Riemannian

Riemannian manifolds are manifolds equipped with a Metric at each point. That means, these manifolds aren't merely topological objects anymore, but have a geometry as well. This gives, in particular, a notion of distance and shortest paths (geodesics) along which you can interpolate.

Keep in mind that the types in this library are generally defined in an abstract-mathematical spirit, which may not always match the intuition if you think about manifolds as embedded in ℝ³. (For instance, the torus inherits its geometry from the decomposition as 'S¹' × 'S¹', not from the “doughnut” embedding; the cone over S¹ is simply treated as the unit disk, etc..)

Documentation

data GeodesicWitness x where

Constructors

GeodesicWitness :: Geodesic (Interior x) => SemimanifoldWitness x -> GeodesicWitness x

class Semimanifold x => Geodesic x where

Minimal complete definition

geodesicBetween

Methods

geodesicBetween

Arguments

:: x

Starting point; the interpolation will yield this at -1.

-> x

End point, for +1.

If the two points are actually connected by a path...

-> Maybe (D¹ -> x)

...then this is the interpolation function. Attention: the type will change to Differentiable in the future.

geodesicWitness :: GeodesicWitness x

geodesicWitness :: Geodesic (Interior x) => GeodesicWitness x

middleBetween :: x -> x -> Maybe x

Instances

Geodesic S⁰

Methods

geodesicBetween :: S⁰ -> S⁰ -> Maybe (D¹ -> S⁰)

geodesicWitness :: GeodesicWitness S⁰

middleBetween :: S⁰ -> S⁰ -> Maybe S⁰

Geodesic S¹

Methods

geodesicBetween :: S¹ -> S¹ -> Maybe (D¹ -> S¹)

geodesicWitness :: GeodesicWitness S¹

middleBetween :: S¹ -> S¹ -> Maybe S¹

Geodesic ℝ

Methods

geodesicBetween :: ℝ -> ℝ -> Maybe (D¹ -> ℝ)

geodesicWitness :: GeodesicWitness ℝ

middleBetween :: ℝ -> ℝ -> Maybe ℝ

Geodesic ℝ⁴

Methods

geodesicBetween :: ℝ⁴ -> ℝ⁴ -> Maybe (D¹ -> ℝ⁴)

geodesicWitness :: GeodesicWitness ℝ⁴

middleBetween :: ℝ⁴ -> ℝ⁴ -> Maybe ℝ⁴

Geodesic ℝ³

Methods

geodesicBetween :: ℝ³ -> ℝ³ -> Maybe (D¹ -> ℝ³)

geodesicWitness :: GeodesicWitness ℝ³

middleBetween :: ℝ³ -> ℝ³ -> Maybe ℝ³

Geodesic ℝ²

Methods

geodesicBetween :: ℝ² -> ℝ² -> Maybe (D¹ -> ℝ²)

geodesicWitness :: GeodesicWitness ℝ²

middleBetween :: ℝ² -> ℝ² -> Maybe ℝ²

Geodesic ℝ¹

Methods

geodesicBetween :: ℝ¹ -> ℝ¹ -> Maybe (D¹ -> ℝ¹)

geodesicWitness :: GeodesicWitness ℝ¹

middleBetween :: ℝ¹ -> ℝ¹ -> Maybe ℝ¹

Geodesic (V0 ℝ)

Methods

geodesicBetween :: V0 ℝ -> V0 ℝ -> Maybe (D¹ -> V0 ℝ)

geodesicWitness :: GeodesicWitness (V0 ℝ)

middleBetween :: V0 ℝ -> V0 ℝ -> Maybe (V0 ℝ)

Geodesic (ZeroDim s)

Methods

geodesicBetween :: ZeroDim s -> ZeroDim s -> Maybe (D¹ -> ZeroDim s)

geodesicWitness :: GeodesicWitness (ZeroDim s)

middleBetween :: ZeroDim s -> ZeroDim s -> Maybe (ZeroDim s)

(Geodesic v, FiniteFreeSpace v, FiniteFreeSpace (DualVector v), LinearSpace v, (~) * (Scalar v) ℝ, Geodesic (DualVector v), InnerSpace (DualVector v)) => Geodesic (Stiefel1 v)

Methods

geodesicBetween :: Stiefel1 v -> Stiefel1 v -> Maybe (D¹ -> Stiefel1 v)

geodesicWitness :: GeodesicWitness (Stiefel1 v)

middleBetween :: Stiefel1 v -> Stiefel1 v -> Maybe (Stiefel1 v)

(WithField ℝ AffineManifold x, Geodesic x, SimpleSpace (Needle x)) => Geodesic (Shade' x)

Methods

geodesicBetween :: Shade' x -> Shade' x -> Maybe (D¹ -> Shade' x)

geodesicWitness :: GeodesicWitness (Shade' x)

middleBetween :: Shade' x -> Shade' x -> Maybe (Shade' x)

(WithField ℝ PseudoAffine x, Geodesic (Interior x), SimpleSpace (Needle x)) => Geodesic (Shade x)

Methods

geodesicBetween :: Shade x -> Shade x -> Maybe (D¹ -> Shade x)

geodesicWitness :: GeodesicWitness (Shade x)

middleBetween :: Shade x -> Shade x -> Maybe (Shade x)

(Geodesic a, Geodesic b) => Geodesic (a, b)

Methods

geodesicBetween :: (a, b) -> (a, b) -> Maybe (D¹ -> (a, b))

geodesicWitness :: GeodesicWitness (a, b)

middleBetween :: (a, b) -> (a, b) -> Maybe (a, b)

(Geodesic a, Geodesic b, Geodesic c) => Geodesic (a, b, c)

Methods

geodesicBetween :: (a, b, c) -> (a, b, c) -> Maybe (D¹ -> (a, b, c))

geodesicWitness :: GeodesicWitness (a, b, c)

middleBetween :: (a, b, c) -> (a, b, c) -> Maybe (a, b, c)

(LinearSpace v, (~) * (Scalar v) ℝ, TensorSpace w, (~) * (Scalar w) ℝ) => Geodesic (LinearMap ℝ v w)

Methods

geodesicBetween :: LinearMap ℝ v w -> LinearMap ℝ v w -> Maybe (D¹ -> LinearMap ℝ v w)

geodesicWitness :: GeodesicWitness (LinearMap ℝ v w)

middleBetween :: LinearMap ℝ v w -> LinearMap ℝ v w -> Maybe (LinearMap ℝ v w)

(TensorSpace v, (~) * (Scalar v) ℝ, TensorSpace w, (~) * (Scalar w) ℝ) => Geodesic (Tensor ℝ v w)

Methods

geodesicBetween :: Tensor ℝ v w -> Tensor ℝ v w -> Maybe (D¹ -> Tensor ℝ v w)

geodesicWitness :: GeodesicWitness (Tensor ℝ v w)

middleBetween :: Tensor ℝ v w -> Tensor ℝ v w -> Maybe (Tensor ℝ v w)

(TensorSpace v, (~) * (Scalar v) ℝ, TensorSpace w, (~) * (Scalar w) ℝ) => Geodesic (LinearFunction ℝ v w)

Methods

geodesicBetween :: LinearFunction ℝ v w -> LinearFunction ℝ v w -> Maybe (D¹ -> LinearFunction ℝ v w)

geodesicWitness :: GeodesicWitness (LinearFunction ℝ v w)

middleBetween :: LinearFunction ℝ v w -> LinearFunction ℝ v w -> Maybe (LinearFunction ℝ v w)

interpolate :: (Geodesic x, IntervalLike i) => x -> x -> Maybe (i -> x)

class WithField ℝ PseudoAffine i => IntervalLike i where

One-dimensional manifolds, whose closure is homeomorpic to the unit interval.

Minimal complete definition

toClosedInterval

Methods

toClosedInterval :: i -> D¹

Instances

IntervalLike D¹

Methods

toClosedInterval :: D¹ -> D¹

IntervalLike ℝ

Methods

toClosedInterval :: ℝ -> D¹

class Geodesic m => Riemannian m where

Minimal complete definition

rieMetric

Methods

rieMetric :: RieMetric m

Instances

Riemannian ℝ

Methods

rieMetric :: RieMetric ℝ

pointsBarycenter :: Geodesic m => NonEmpty m -> Maybe m

type FlatSpace x = (AffineManifold x, Geodesic x, SimpleSpace x)