Goal.Geometry.Manifold

This module provides the core mathematical definitions used by the rest of Goal. In Goal, all mathematical structures are Manifolds, even when they are not especially complicated ones; Manifolds may indicate highly articulated structures, but may also indicate simpler concepts such as (vector) spaces.

Manifolds are sets of points which can be described locally as Euclidean spaces. In geometry, a point is typically a member of the actual Manifold. However, arbitrary types of points will often be difficult to represent directly, and so points in Goal are always represented in terms of their Coordinates in terms of a given chart.

Charts are in turn represented by phantom types. Mathematically, charts are maps between the Manifold and the relevant Cartesian coordinate system. However, since we do not represent the points of a Manifold explicility, we also cannot represent Charts explicitly. As such, Atlases merely index a point so as to indicate how to interpret its particular Coordinates.

Manifolds

class Eq m => Manifold m where

A geometric object with a certain dimension. We assume that a Manifold somehow represents all the geometric, coordinate independent structure under consideration. Manifolds should satisfy

dimension m = length $ coordinates (Point m cs)

Methods

dimension :: m -> Int

Instances

Manifold Continuum

Manifold Euclidean

Manifold m => Manifold (Replicated m)

(Manifold m, Manifold n) => Manifold (m, n)

(Manifold m, Manifold n) => Manifold (Tensor n m)

(Manifold m, Manifold n) => Manifold (Affine m n)

Manifold m => Manifold (Bundle c m)

Manifold m => Manifold (Tangent c m)

(Manifold m, Manifold n, Manifold o) => Manifold (m, n, o)

class Transition c d m where

A transition involves taking a point represented by the chart c, and re-representing in terms of the chart d. This will usually require recomputation of the Coordinates. Transitions should satisfy the law

transition $ transition p = p

Methods

transition :: (c :#: m) -> d :#: m

Instances

Transition c c m

Transition Polar Cartesian Euclidean

Transition Cartesian Polar Euclidean

Sets

data Embedded m c

Constructors

Embedded

Fields

disembed :: m

Instances

Eq m => Eq (Embedded m c)

Read m => Read (Embedded m c)

Show m => Show (Embedded m c)

Manifold m => Set (Embedded m c)

type Element (Embedded m c) = (:#:) c m

Points

type Coordinates = Vector Double

Elements of Euclidean spaces are referred to as Coordinates.

data c :#: m infixr 1

A point is an element of a Manifold m in terms of a particular chart c.

Instances

Eq m => Eq ((:#:) c m)

Read m => Read ((:#:) c m)

Show m => Show ((:#:) c m)

coordinate :: Int -> (c :#: m) -> Double

chart :: Manifold m => c -> (c :#: m) -> c :#: m

chart allows one to specify the Atlas of a new point. This is often necessary when typeclass methods are used to generate points under a variety of coordinate systems.

breakChart :: Manifold m => (c :#: m) -> d :#: m

alterChart :: Manifold m => d -> (c :#: m) -> d :#: m

Combines breakChart and chart.

listCoordinates :: (c :#: m) -> [Double]

Returns the Coordinates of the point in list form.

alterCoordinates :: Manifold m => (Double -> Double) -> (c :#: m) -> c :#: m

alterCoordinates allows one to map a function over the coordinates of a point without changing the chart.

toPair :: (c :#: m) -> (Double, Double)

Charts

data Cartesian

The Cartesian coordinate system.

Constructors

Cartesian

Instances

Primal Cartesian

Riemannian Cartesian Continuum

Riemannian Cartesian Euclidean

Transition Polar Cartesian Euclidean

Transition Cartesian Polar Euclidean

type Dual Cartesian = Cartesian

data Polar

The Polar coordinate system.

Constructors

Polar

Instances

Transition Polar Cartesian Euclidean

Transition Cartesian Polar Euclidean

Constructors

fromList :: Manifold m => m -> [Double] -> c :#: m

fromList builds points without the need to work with vectors.

fromCoordinates :: Manifold m => m -> Coordinates -> c :#: m

euclideanPoint :: [Double] -> Cartesian :#: Euclidean

A convenience function for building Euclidean vectors.

realNumber :: Double -> Cartesian :#: Continuum

A convenience function for building elements of a Continuum.

Direct Sums

Replicated

mapReplicated :: Manifold m => ((c :#: m) -> x) -> (c :#: Replicated m) -> [x]

A function to map functions over a point on a Replicated Manifold.

joinReplicated :: Manifold m => [c :#: m] -> c :#: Replicated m

Joins a list of distributions into a Replicated Manifold. Be advised that this function assumes that the families of the individual distributions are equal.

concatReplicated :: (c :#: Replicated m) -> (c :#: Replicated m) -> c :#: Replicated m

Joins two Replicated Manifolds.

DirectSum

joinPair :: (Manifold m, Manifold n) => (c :#: m) -> (d :#: n) -> (c, d) :#: (m, n)

Joins a pair of Points into a Point on the the direct sum of the underlying Charts and Manifolds.

splitPair :: (Manifold m, Manifold n) => ((c, d) :#: (m, n)) -> (c :#: m, d :#: n)

Splits a direct sum pair.

joinPair' :: (Manifold m, Manifold n) => (c :#: m) -> (c :#: n) -> c :#: (m, n)

Alternative version where we assume that the Charts are shared.

splitPair' :: (Manifold m, Manifold n) => (c :#: (m, n)) -> (c :#: m, c :#: n)

Alternative version where we assume that the Charts are shared.

joinTriple :: (Manifold m, Manifold n, Manifold o) => (c :#: m) -> (d :#: n) -> (e :#: o) -> (c, d, e) :#: (m, n, o)

Joins a triple of Points into a Point on the the direct sum of the underlying Charts and Manifolds.

splitTriple :: (Manifold m, Manifold n, Manifold o) => ((c, d, e) :#: (m, n, o)) -> (c :#: m, d :#: n, e :#: o)

Splits a direct sum triple.

joinTriple' :: (Manifold m, Manifold n, Manifold o) => (c :#: m) -> (c :#: n) -> (c :#: o) -> c :#: (m, n, o)

Alternative version where we assume that the Charts are shared.

splitTriple' :: (Manifold m, Manifold n, Manifold o) => (c :#: (m, n, o)) -> (c :#: m, c :#: n, c :#: o)

Alternative version where we assume that the Charts are shared.