| Copyright | (c) Justus Sagemüller 2015 |
|---|---|
| License | GPL v3 |
| Maintainer | (@) jsag $ hvl.no |
| Stability | experimental |
| Portability | portable |
| Safe Haskell | Safe-Inferred |
| Language | Haskell2010 |
Data.Manifold.Riemannian
Description
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 × S¹S¹ is
simply treated as the unit disk, etc..)
Synopsis
- class SemimanifoldWithBoundary x => Geodesic x where
- geodesicBetween :: x -> x -> Maybe (D¹ -> x)
- middleBetween :: x -> x -> Maybe x
- interpolate :: (Geodesic x, IntervalLike i) => x -> x -> Maybe (i -> x)
- class WithField ℝ PseudoAffine (Interior i) => IntervalLike i where
- toClosedInterval :: i -> D¹
- class Geodesic m => Riemannian m where
- pointsBarycenter :: Geodesic m => NonEmpty m -> Maybe m
- type FlatSpace x = (AffineManifold x, Geodesic x, SimpleSpace x)
Documentation
class SemimanifoldWithBoundary x => Geodesic x where Source #
Minimal complete definition
Methods
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 |
middleBetween :: x -> x -> Maybe x Source #
Instances
interpolate :: (Geodesic x, IntervalLike i) => x -> x -> Maybe (i -> x) Source #
class WithField ℝ PseudoAffine (Interior i) => IntervalLike i where Source #
One-dimensional manifolds, whose closure is homeomorpic to the unit interval.
Methods
toClosedInterval :: i -> D¹ Source #
Instances
| IntervalLike D¹ Source # | |
Defined in Data.Manifold.Riemannian Methods toClosedInterval :: D¹ -> D¹ Source # | |
| IntervalLike ℝ Source # | |
Defined in Data.Manifold.Riemannian Methods toClosedInterval :: ℝ -> D¹ Source # | |
class Geodesic m => Riemannian m where Source #
Instances
type FlatSpace x = (AffineManifold x, Geodesic x, SimpleSpace x) Source #