Math.Manifold.Real.Coordinates

Documentation

type Coordinate m = forall q. CoordinateIsh q m => q

A coordinate is a function that can be used both to determine the position of a point on a manifold along the one of some family of (possibly curved) axes on which it lies, and for moving the point along that axis. Basically, this is a Lens and can indeed be used with the ^., .~ and %~ operators.

Coordinate m ~ Lens' m 'ℝ'

In addition, each type may also have a way of identifying particular coordinate axes. This is done with CoordinateIdentifier, which is what should be used for defining given coordinate axes.

coordinate :: CoordinateIdentifier m -> Coordinate m

class HasCoordinates m where

To give a custom type coordinate axes, first define an instance of this class.

Minimal complete definition

coordinateAsLens

Associated Types

data CoordinateIdentifier m :: *

A unique description of a coordinate axis.

Methods

coordinateAsLens :: CoordinateIdentifier m -> Lens' m ℝ

How to use a coordinate axis for points in the containing space. This is what coordinate calls under the hood.

validCoordinateRange :: CoordinateIdentifier m -> m -> (ℝ, ℝ)

Delimiters for the possible values one may choose for a given coordinate, around a point on the manifold. For example, in spherical coordinates, the azimuth generally has a range of (-pi, pi), except at the poles where it's (0,0).

Instances

HasCoordinates S¹

Associated Types

data CoordinateIdentifier S¹ :: *

Methods

coordinateAsLens :: CoordinateIdentifier S¹ -> Lens' S¹ ℝ

validCoordinateRange :: CoordinateIdentifier S¹ -> S¹ -> (ℝ, ℝ)

HasCoordinates S²

Associated Types

data CoordinateIdentifier S² :: *

Methods

coordinateAsLens :: CoordinateIdentifier S² -> Lens' S² ℝ

validCoordinateRange :: CoordinateIdentifier S² -> S² -> (ℝ, ℝ)

HasCoordinates ℝ

Associated Types

data CoordinateIdentifier ℝ :: *

Methods

coordinateAsLens :: CoordinateIdentifier ℝ -> Lens' ℝ ℝ

validCoordinateRange :: CoordinateIdentifier ℝ -> ℝ -> (ℝ, ℝ)

HasCoordinates ℝ⁰

Associated Types

data CoordinateIdentifier ℝ⁰ :: *

Methods

coordinateAsLens :: CoordinateIdentifier ℝ⁰ -> Lens' ℝ⁰ ℝ

validCoordinateRange :: CoordinateIdentifier ℝ⁰ -> ℝ⁰ -> (ℝ, ℝ)

HasCoordinates ℝ³

Associated Types

data CoordinateIdentifier ℝ³ :: *

Methods

coordinateAsLens :: CoordinateIdentifier ℝ³ -> Lens' ℝ³ ℝ

validCoordinateRange :: CoordinateIdentifier ℝ³ -> ℝ³ -> (ℝ, ℝ)

HasCoordinates ℝ²

Associated Types

data CoordinateIdentifier ℝ² :: *

Methods

coordinateAsLens :: CoordinateIdentifier ℝ² -> Lens' ℝ² ℝ

validCoordinateRange :: CoordinateIdentifier ℝ² -> ℝ² -> (ℝ, ℝ)

(HasCoordinates a, HasCoordinates b) => HasCoordinates (a, b)

Associated Types

data CoordinateIdentifier (a, b) :: *

Methods

coordinateAsLens :: CoordinateIdentifier (a, b) -> Lens' (a, b) ℝ

validCoordinateRange :: CoordinateIdentifier (a, b) -> (a, b) -> (ℝ, ℝ)

(HasCoordinates b, HasCoordinates f) => HasCoordinates (FibreBundle b f)

Associated Types

data CoordinateIdentifier (FibreBundle b f) :: *

Methods

coordinateAsLens :: CoordinateIdentifier (FibreBundle b f) -> Lens' (FibreBundle b f) ℝ

validCoordinateRange :: CoordinateIdentifier (FibreBundle b f) -> FibreBundle b f -> (ℝ, ℝ)

Vector space axes

class HasCoordinates m => HasXCoord m where

Minimal complete definition

xCoord

Methods

xCoord :: Coordinate m

Instances

HasXCoord ℝ

Methods

xCoord :: Coordinate ℝ

HasXCoord ℝ³

Methods

xCoord :: Coordinate ℝ³

HasXCoord ℝ²

Methods

xCoord :: Coordinate ℝ²

(HasXCoord v, HasCoordinates w) => HasXCoord (v, w)

Methods

xCoord :: Coordinate (v, w)

class HasYCoord m where

Minimal complete definition

yCoord

Methods

yCoord :: Coordinate m

Instances

HasYCoord ℝ³

Methods

yCoord :: Coordinate ℝ³

HasYCoord ℝ²

Methods

yCoord :: Coordinate ℝ²

HasCoordinates w => HasYCoord ((ℝ, ℝ), w)

Methods

yCoord :: Coordinate ((ℝ, ℝ), w)

HasXCoord w => HasYCoord (ℝ, w)

Methods

yCoord :: Coordinate (ℝ, w)

class HasZCoord m where

Minimal complete definition

zCoord

Methods

zCoord :: Coordinate m

Instances

HasZCoord ℝ³

Methods

zCoord :: Coordinate ℝ³

HasXCoord w => HasZCoord ((ℝ, ℝ), w)

Methods

zCoord :: Coordinate ((ℝ, ℝ), w)

HasYCoord w => HasZCoord (ℝ, w)

Methods

zCoord :: Coordinate (ℝ, w)

Fibre bundle / tangent space diffs

location's :: (HasCoordinates b, Interior b ~ b, HasCoordinates f) => CoordinateIdentifier b -> Coordinate (FibreBundle b f)

class HasCoordinates m => CoordDifferential m where

Minimal complete definition

delta

Methods

delta :: CoordinateIdentifier m -> Coordinate (TangentBundle m)

Observe local, small variations (in the tangent space) of a coordinate. The idea is that ((p & coord+~δc) − p) ^. delta coord ≈ δc, thus the name “delta”. Note however that this only holds exactly for flat spaces; in most manifolds it can (by design) only be understood in an asymptotic sense, i.e. used for evaluating directional derivatives of some function. In particular, delta azimuth is unstable near the poles of a sphere, because it has to compensate for the sensitive rotation of the eφ unit vector.

Instances

CoordDifferential S¹

Methods

delta :: CoordinateIdentifier S¹ -> Coordinate (TangentBundle S¹)

CoordDifferential S²

Methods

delta :: CoordinateIdentifier S² -> Coordinate (TangentBundle S²)

CoordDifferential ℝ

Methods

delta :: CoordinateIdentifier ℝ -> Coordinate (TangentBundle ℝ)

CoordDifferential ℝ³

Methods

delta :: CoordinateIdentifier ℝ³ -> Coordinate (TangentBundle ℝ³)

CoordDifferential ℝ²

Methods

delta :: CoordinateIdentifier ℝ² -> Coordinate (TangentBundle ℝ²)

(CoordDifferential a, CoordDifferential b) => CoordDifferential (a, b)

Methods

delta :: CoordinateIdentifier (a, b) -> Coordinate (TangentBundle (a, b))

Spherical coordinates

class HasAzimuth m where

Minimal complete definition

azimuth

Methods

azimuth :: Coordinate m

Instances

HasAzimuth S¹

Methods

azimuth :: Coordinate S¹

HasAzimuth S²

Methods

azimuth :: Coordinate S²

class HasZenithDistance m where

Minimal complete definition

zenithAngle

Methods

zenithAngle :: Coordinate m

Instances

HasZenithDistance S²

Methods

zenithAngle :: Coordinate S²