| Copyright | (c) Justus Sagemüller 2015 |
|---|---|
| License | GPL v3 |
| Maintainer | (@) sagemueller $ geo.uni-koeln.de |
| Stability | experimental |
| Portability | portable |
| Safe Haskell | None |
| Language | Haskell2010 |
Data.Manifold.Types
Contents
Description
Several commonly-used manifolds, represented in some simple way as Haskell
data types. All these are in the PseudoAffine class.
- type Real0 = ℝ⁰
- type Real1 = ℝ
- type RealPlus = ℝay
- type Real2 = ℝ²
- type Real3 = ℝ³
- type Sphere0 = S⁰
- type Sphere1 = S¹
- type Sphere2 = S²
- type Projective1 = ℝP¹
- type Projective2 = ℝP²
- type Disk1 = D¹
- type Disk2 = D²
- type Cone = CD¹
- type OpenCone = Cℝay
- data ZeroDim s :: * -> * = Origin
- type ℝ = Double
- type ℝ⁰ = ZeroDim ℝ
- type ℝ¹ = V1 ℝ
- type ℝ² = V2 ℝ
- type ℝ³ = V3 ℝ
- type ℝ⁴ = V4 ℝ
- newtype Stiefel1 v = Stiefel1 {
- getStiefel1N :: DualVector v
- stiefel1Project :: LinearManifold v => DualVector v -> Stiefel1 v
- stiefel1Embed :: (HilbertSpace v, RealFloat (Scalar v)) => Stiefel1 v -> v
- class (PseudoAffine v, InnerSpace v, NaturallyEmbedded (UnitSphere v) (DualVector v)) => HasUnitSphere v where
- type UnitSphere v :: *
- stiefel :: UnitSphere v -> Stiefel1 v
- unstiefel :: Stiefel1 v -> UnitSphere v
- data S⁰ :: *
- newtype S¹ :: * = S¹ {}
- data S² = S² {}
- type ℝP¹ = S¹
- data ℝP² = ℝP² {}
- newtype D¹ :: * = D¹ {}
- data D² = D² {}
- type ℝay = Cℝay ℝ⁰
- data CD¹ x = CD¹ {}
- data Cℝay x = Cℝay {
- hParamCℝay :: !Double
- pParamCℝay :: !x
- data Line x = Line {
- lineHandle :: x
- lineDirection :: Stiefel1 (Needle' x)
- lineAsPlaneIntersection :: forall x. (WithField ℝ Manifold x, FiniteDimensional (Needle' x)) => Line x -> [Cutplane x]
- data Cutplane x = Cutplane {}
- fathomCutDistance :: forall x. (WithField ℝ PseudoAffine x, LinearSpace (Needle x)) => Cutplane x -> Metric' x -> x -> Maybe ℝ
- sideOfCut :: (WithField ℝ PseudoAffine x, LinearSpace (Needle x)) => Cutplane x -> x -> Maybe S⁰
- cutPosBetween :: WithField ℝ Manifold x => Cutplane x -> (x, x) -> Maybe D¹
- data LinearMap s v w :: * -> * -> * -> *
- type LocalLinear x y = LinearMap (Scalar (Needle x)) (Needle x) (Needle y)
Index / ASCII names
type Projective1 = ℝP¹ Source
type Projective2 = ℝP² Source
Linear manifolds
data ZeroDim s :: * -> *
Constructors
| Origin |
Instances
| Refinable ℝ⁰ Source | |
| Monoid (ZeroDim s) | |
| AdditiveGroup (ZeroDim s) | |
| VectorSpace (ZeroDim s) | |
| HasBasis (ZeroDim s) | |
| AffineSpace (ZeroDim s) | |
| (Fractional' s, SemiInner s) => SemiInner (ZeroDim s) | |
| Num' s => FiniteDimensional (ZeroDim s) | |
| Num' s => TensorSpace (ZeroDim s) | |
| Num' s => LinearSpace (ZeroDim s) | |
| Semimanifold (ZeroDim k) | |
| PseudoAffine (ZeroDim k) | |
| Atlas (ZeroDim s) Source | |
| Geodesic (ZeroDim ℝ) Source | |
| NumberManifold s => LocallyCoercible (V0 s) (ZeroDim s) Source | |
| NumberManifold s => LocallyCoercible (ZeroDim s) (V0 s) Source | |
| NumberManifold s => LocallyCoercible (ZeroDim s) (ZeroDim s) Source | |
| type Scalar (ZeroDim s) = s | |
| type Basis (ZeroDim k) = Void | |
| type Diff (ZeroDim s) = ZeroDim s | |
| data SubBasis (ZeroDim s) = ZeroBasis | |
| type DualVector (ZeroDim s) = ZeroDim s | |
| type Interior (ZeroDim k) = ZeroDim k | |
| type Needle (ZeroDim k) = ZeroDim k | |
| type ChartIndex (ZeroDim s) = () Source | |
| type TensorProduct (ZeroDim s) v = ZeroDim s |
Hyperspheres
General form: Stiefel manifolds
Constructors
| Stiefel1 | |
Fields
| |
Arguments
| :: LinearManifold v | |
| => DualVector v | Must be nonzero. |
| -> Stiefel1 v |
stiefel1Embed :: (HilbertSpace v, RealFloat (Scalar v)) => Stiefel1 v -> v Source
Specific examples
class (PseudoAffine v, InnerSpace v, NaturallyEmbedded (UnitSphere v) (DualVector v)) => HasUnitSphere v where Source
Minimal complete definition
Nothing
Associated Types
type UnitSphere v :: * Source
Instances
data S⁰ :: *
The zero-dimensional sphere is actually just two points. Implementation might
therefore change to ℝ⁰ : the disjoint sum of two
single-point spaces.+ ℝ⁰
Constructors
| PositiveHalfSphere | |
| NegativeHalfSphere |
newtype S¹ :: *
The unit circle.
The ordinary unit sphere.
Projective spaces
The two-dimensional real projective space, implemented as a unit disk with opposing points on the rim glued together.
Intervals/disks/cones
newtype D¹ :: *
The “one-dimensional disk” – really just the line segment between the two points -1 and 1 of 'S⁰', i.e. this is simply a closed interval.
Instances
| Show D¹ | |
| Semimanifold D¹ | |
| PseudoAffine D¹ | |
| IntervalLike D¹ Source | |
| type Interior D¹ = ℝ | |
| type Needle D¹ = ℝ |
The standard, closed unit disk. Homeomorphic to the cone over 'S¹', but not in the the obvious, “flat” way. (And not at all, despite the identical ADT definition, to the projective space 'ℝP²'!)
Better known as ℝ⁺ (which is not a legal Haskell name), the ray of positive numbers (including zero, i.e. closed on one end).
A (closed) cone over a space x is the product of x with the closed interval 'D¹'
of “heights”,
except on its “tip”: here, x is smashed to a single point.
This construct becomes (homeomorphic-to-) an actual geometric cone (and to 'D²') in the
special case x = 'S¹'.
An open cone is homeomorphic to a closed cone without the “lid”,
i.e. without the “last copy” of x, at the far end of the height
interval. Since that means the height does not include its supremum, it is actually
more natural to express it as the entire real ray, hence the name.
Constructors
| Cℝay | |
Fields
| |
Affine subspaces
Lines
Constructors
| Line | |
Fields
| |
lineAsPlaneIntersection :: forall x. (WithField ℝ Manifold x, FiniteDimensional (Needle' x)) => Line x -> [Cutplane x] Source
Hyperplanes
Oriented hyperplanes, naïvely generalised to PseudoAffine manifolds:
Cutplane p wq such that
(q.-~.p) ^<.> w ≡ 0.
In vector spaces this is indeed a hyperplane; for general manifolds it should behave locally as a plane, globally as an (n−1)-dimensional submanifold.
Arguments
| :: (WithField ℝ PseudoAffine x, LinearSpace (Needle x)) | |
| => Cutplane x | Hyperplane to measure the distance from. |
| -> Metric' x | Metric to use for measuring that distance.
This can only be accurate if the metric
is valid both around the cut-plane's |
| -> x | Point to measure the distance to. |
| -> Maybe ℝ | A signed number, giving the distance from plane
to point with indication on which side the point lies.
|
sideOfCut :: (WithField ℝ PseudoAffine x, LinearSpace (Needle x)) => Cutplane x -> x -> Maybe S⁰ Source
Linear mappings
data LinearMap s v w :: * -> * -> * -> *
The tensor product between one space's dual space and another space is the space spanned by vector–dual-vector pairs, in bra-ket notation written as
m = ∑ |w⟩⟨v|
Any linear mapping can be written as such a (possibly infinite) sum. The
TensorProduct data structure only stores the linear independent parts
though; for simple finite-dimensional spaces this means e.g. LinearMap ℝ ℝ³ ℝ³|w⟩.
(The ⟨v| dual-vectors are then simply assumed to come from the canonical basis.)
For bigger spaces, the tensor product may be implemented in a more efficient
sparse structure; this can be defined in the TensorSpace instance.
Instances