Data.Manifold.PseudoAffine

This is the second prototype of a manifold class. It appears to give considerable advantages over Manifold, so that class will probably soon be replaced with the one we define here (though PseudoAffine does not follow the standard notion of a manifold very closely, it should work quite equivalently for pretty much all Haskell types that qualify as manifolds).

Manifolds are interesting as objects of various categories, from continuous to diffeomorphic. At the moment, we mainly focus on region-wise differentiable functions, which are a promising compromise between flexibility of definition and provability of analytic properties. In particular, they are well-suited for visualisation purposes.

The classes in this module are mostly aimed at manifolds without boundary. Manifolds with boundary (which we call MWBound, never manifold!) are more or less treated as a disjoint sum of the interior and the boundary. To understand how this module works, best first forget about boundaries – in this case, Interior x ~ x, fromInterior and toInterior are trivial, and .+~|, |-~. and betweenBounds are irrelevant. The manifold structure of the boundary itself is not considered at all here.

Manifold class

class (PseudoAffine m, LinearManifold (Needle m), Interior m ~ m) => Manifold m

See Semimanifold and PseudoAffine for the methods.

Instances

(PseudoAffine m, LinearManifold (Needle m), (~) * (Interior m) m) => Manifold m

class (AdditiveGroup (Needle x), Interior (Interior x) ~ Interior x) => Semimanifold x where

Minimal complete definition

((.+~^) | fromInterior), toInterior, translateP

Associated Types

type Needle x :: *

The space of “natural” ways starting from some reference point and going to some particular target point. Hence, the name: like a compass needle, but also with an actual length. For affine spaces, Needle is simply the space of line segments (aka vectors) between two points, i.e. the same as Diff. The AffineManifold constraint makes that requirement explicit.

This space should be isomorphic to the tangent space (and is in fact used somewhat synonymously).

type Interior x :: *

Manifolds with boundary are a bit tricky. We support such manifolds, but carry out most calculations only in “the fleshy part” – the interior, which is an “infinite space”, so you can arbitrarily scale paths.

The default implementation is Interior x = x, which corresponds to a manifold that has no boundary to begin with.

Methods

(.+~^) :: Interior x -> Needle x -> x infixl 6

Generalised translation operation. Note that the result will always also be in the interior; scaling up the needle can only get you ever closer to a boundary.

fromInterior :: Interior x -> x

id sans boundary.

toInterior :: x -> Option (Interior x)

translateP :: Tagged x (Interior x -> Needle x -> Interior x)

The signature of .+~^ should really be Interior x -> Needle x -> Interior x, only, this is not possible because it only consists of non-injective type families. The solution is this tagged signature, which is of course rather unwieldy. That's why .+~^ has the stronger, but easier usable signature. Without boundary, these functions should be equivalent, i.e. translateP = Tagged (.+~^).

(.-~^) :: Interior x -> Needle x -> x infixl 6

Shorthand for \p v -> p .+~^ negateV v, which should obey the asymptotic law

p .-~^ v .+~^ v ≅ p

Meaning: if v is scaled down with sufficiently small factors η, then the difference (p.-~^v.+~^v) .-~. p should scale down even faster: as O (η²). For large vectors, it will however behave differently, except in flat spaces (where all this should be equivalent to the AffineSpace instance).

Instances

Semimanifold Double

Semimanifold Rational

Semimanifold D¹

Semimanifold ℝP²

Semimanifold S²

Semimanifold S¹

Semimanifold S⁰

Semimanifold (ZeroDim k)

AffineManifold x => Semimanifold (ShadeTree x)

Experimental. There might be a more powerful instance possible.

AffineManifold x => Semimanifold (Shade' x)

AffineManifold x => Semimanifold (Shade x)

(Semimanifold a, Semimanifold b) => Semimanifold (a, b)

(HasMetric a, FiniteDimensional b, (~) * (Scalar a) (Scalar b)) => Semimanifold ((:-*) a b)

Semimanifold x => Semimanifold (WithAny x y)

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

(HasMetric a, FiniteDimensional b, (~) * (Scalar a) s, (~) * (Scalar b) s) => Semimanifold (Linear s a b)

type Needle' x = DualSpace (Needle x)

A co-needle can be understood as a “paper stack”, with which you can measure the length that a needle reaches in a given direction by counting the number of holes punched through them.

class (Semimanifold x, Semimanifold (Interior x), Needle (Interior x) ~ Needle x, Interior (Interior x) ~ Interior x) => PseudoAffine x where

This is the class underlying manifolds. (Manifold only precludes boundaries and adds an extra constraint that would be circular if it was in a single class. You can always just use Manifold as a constraint in your signatures, but you must define only PseudoAffine for manifold types – the Manifold instance follows universally from this, if 'Interior x ~ x.)

The interface is (boundaries aside) almost identical to the better-known AffineSpace class, but we don't require associativity of .+~^ with ^+^ – except in an asymptotic sense for small vectors.

That innocent-looking change makes the class applicable to vastly more general types: while an affine space is basically nothing but a vector space without particularly designated origin, a pseudo-affine space can have nontrivial topology on the global scale, and yet be used in practically the same way as an affine space. At least the usual spheres and tori make good instances, perhaps the class is in fact equivalent to manifolds in their usual maths definition (with an atlas of charts: a family of overlapping regions of the topological space, each homeomorphic to the Needle vector space or some simply-connected subset thereof).

Minimal complete definition

(.-~.) | (.-~!)

Methods

(.-~.) :: x -> Interior x -> Option (Needle x) infix 6

The path reaching from one point to another. Should only yield Nothing if

• The points are on disjoint segments of a non–path-connected space.
• Either of the points is on the boundary. Use |-~. to deal with this.

On manifolds, the identity

p .+~^ (q.-~.p) ≡ q

should hold, at least save for floating-point precision limits etc..

.-~. and .+~^ only really work in manifolds without boundary. If you consider the path between two points, one of which lies on the boundary, it can't really be possible to scale this path any longer – it would have to reach “out of the manifold”. To adress this problem, these functions basically consider only the interior of the space.

(.-~!) :: x -> Interior x -> Needle x

Unsafe version of .-~.. If the two points lie in disjoint regions, the behaviour is undefined.

Instances

PseudoAffine Double

PseudoAffine Rational

PseudoAffine D¹

PseudoAffine ℝP²

PseudoAffine S²

PseudoAffine S¹

PseudoAffine S⁰

PseudoAffine (ZeroDim k)

(PseudoAffine a, PseudoAffine b) => PseudoAffine (a, b)

(HasMetric a, FiniteDimensional b, (~) * (Scalar a) (Scalar b)) => PseudoAffine ((:-*) a b)

PseudoAffine x => PseudoAffine (WithAny x y)

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

(HasMetric a, FiniteDimensional b, (~) * (Scalar a) s, (~) * (Scalar b) s) => PseudoAffine (Linear s a b)

Type definitions

Metrics

type Metric x = HerMetric (Needle x)

The word “metric” is used in the sense as in general relativity. Cf. HerMetric.

type Metric' x = HerMetric' (Needle x)

euclideanMetric :: EuclidSpace x => proxy x -> Metric x

type RieMetric x = x -> Metric x

A Riemannian metric assigns each point on a manifold a scalar product on the tangent space. Note that this association is not continuous, because the charts/tangent spaces in the bundle are a priori disjoint. However, for a proper Riemannian metric, all arising expressions of scalar products from needles between points on the manifold ought to be differentiable.

type RieMetric' x = x -> Metric' x

Constraints

type RealDimension r = (PseudoAffine r, Interior r ~ r, Needle r ~ r, HasMetric r, DualSpace r ~ r, Scalar r ~ r, RealFloat r, r ~ ℝ)

The RealFloat class plus manifold constraints.

type AffineManifold m = (PseudoAffine m, Interior m ~ m, AffineSpace m, Needle m ~ Diff m, LinearManifold' (Diff m))

The AffineSpace class plus manifold constraints.

type LinearManifold x = (AffineManifold x, Needle x ~ x, HasMetric x)

Basically just an “updated” version of the VectorSpace class. Every vector space is a manifold, this constraint makes it explicit.

(Actually, LinearManifold is stronger than VectorSpace at the moment, since HasMetric requires FiniteDimensional. This might be lifted in the future.)

type WithField s c x = (c x, s ~ Scalar (Needle x))

Require some constraint on a manifold, and also fix the type of the manifold's underlying field. For example, WithField ℝ HilbertSpace v constrains v to be a real (i.e., Double-) Hilbert space. Note that for this to compile, you will in general need the -XLiberalTypeSynonyms extension (except if the constraint is an actual type class (like Manifold): only those can always be partially applied, for type constraints this is by default not allowed).

type HilbertSpace x = (LinearManifold x, InnerSpace x, Interior x ~ x, Needle x ~ x, DualSpace x ~ x, Floating (Scalar x))

A Hilbert space is a complete inner product space. Being a vector space, it is also a manifold.

(Stricly speaking, that doesn't have much to do with the completeness criterion; but since Manifolds are at the moment confined to finite dimension, they are in fact (trivially) complete.)

type EuclidSpace x = (AffineManifold x, InnerSpace (Diff x), DualSpace (Diff x) ~ Diff x, Floating (Scalar (Diff x)))

An euclidean space is a real affine space whose tangent space is a Hilbert space.

type LocallyScalable s x = (PseudoAffine x, HasMetric (Needle x), s ~ Scalar (Needle x))

Local functions

type LocalLinear x y = Linear (Scalar (Needle x)) (Needle x) (Needle y)

type LocalAffine x y = (Needle y, LocalLinear x y)

Misc

alerpB :: forall x. (AffineSpace x, VectorSpace (Diff x), Scalar (Diff x) ~ ℝ) => x -> x -> D¹ -> x

Like alerp, but actually restricted to the interval between the points.

palerp :: forall x. Manifold x => Interior x -> Interior x -> Option (Scalar (Needle x) -> x)

Interpolate between points, approximately linearly. For points that aren't close neighbours (i.e. lie in an almost flat region), the pathway is basically undefined – save for its end points.

A proper, really well-defined (on global scales) interpolation only makes sense on a Riemannian manifold, as Geodesic.

palerpB :: forall x. WithField ℝ Manifold x => Interior x -> Interior x -> Option (D¹ -> x)

Like palerp, but actually restricted to the interval between the points, with a signature like geodesicBetween rather than alerp.

class (PseudoAffine x, PseudoAffine ξ, Scalar (Needle x) ~ Scalar (Needle ξ)) => LocallyCoercible x ξ where

Instances of this class must be diffeomorphic manifolds, and even have canonically isomorphic tangent spaces, so that fromPackedVector . asPackedVector :: Needle x -> Needle ξ defines a meaningful “representational identity“ between these spaces.

Methods

locallyTrivialDiffeomorphism :: x -> ξ

Must be compatible with the canonical isomorphism on the tangent spaces, i.e. locallyTrivialDiffeomorphism (p .+~^ fromPackedVector v) ≡ locallyTrivialDiffeomorphism p .+~^ fromPackedVector v

Instances

LocallyCoercible ℝ ℝ

(PseudoAffine a, PseudoAffine b, PseudoAffine c) => LocallyCoercible ((a, b), c) (a, (b, c))

LocallyCoercible ((ℝ, ℝ), ℝ) ((ℝ, ℝ), ℝ)

(PseudoAffine a, PseudoAffine b, PseudoAffine c) => LocallyCoercible (a, (b, c)) ((a, b), c)

LocallyCoercible (ℝ, (ℝ, ℝ)) (ℝ, (ℝ, ℝ))

LocallyCoercible (ℝ, ℝ) (ℝ, ℝ)

(PseudoAffine a, PseudoAffine b, PseudoAffine c) => LocallyCoercible ((a, b), c) (a, b, c)

(PseudoAffine a, PseudoAffine b, PseudoAffine c) => LocallyCoercible (a, (b, c)) (a, b, c)

(PseudoAffine a, PseudoAffine b, PseudoAffine c) => LocallyCoercible (a, b, c) (a, (b, c))

(PseudoAffine a, PseudoAffine b, PseudoAffine c) => LocallyCoercible (a, b, c) ((a, b), c)

class ImpliesMetric s where

Minimal complete definition

inferMetric | inferMetric'

Associated Types

type MetricRequirement s x :: Constraint

Methods

inferMetric :: (MetricRequirement s x, HasMetric (Needle x)) => s x -> Option (Metric x)

inferMetric' :: (MetricRequirement s x, HasMetric (Needle x)) => s x -> Option (Metric' x)

Instances

ImpliesMetric HerMetric'

ImpliesMetric HerMetric

ImpliesMetric ShadeTree

ImpliesMetric Shade'

ImpliesMetric Shade