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

middleBetween :: x -> x -> Maybe x

Instances

Geodesic ℝ

Geodesic S⁰

Geodesic S¹

Geodesic ℝ⁴

Geodesic ℝ³

Geodesic ℝ²

Geodesic ℝ¹

Geodesic (V0 ℝ)

Geodesic (ZeroDim s)

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

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

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

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

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

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

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

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

Methods

toClosedInterval :: i -> D¹

Instances

IntervalLike ℝ

IntervalLike D¹

class Geodesic m => Riemannian m where

Methods

rieMetric :: RieMetric m

Instances

Riemannian ℝ

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

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