Physics.Learn.Position

A module for working with the idea of position and coordinate systems.

Documentation

data Position

A type for position. Position is not a vector because it makes no sense to add positions.

Instances

Show Position

Methods

showsPrec :: Int -> Position -> ShowS

show :: Position -> String

showList :: [Position] -> ShowS

StateSpace Position

Position is not a vector, but displacement (difference in position) is a vector.

Associated Types

type Diff Position :: *

Methods

(.-.) :: Position -> Position -> Diff Position

(.+^) :: Position -> Diff Position -> Position

type Diff Position

type Diff Position = Vec

type Displacement = Vec

A displacement is a vector.

type ScalarField = Position -> Double

A scalar field associates a number with each position in space.

type VectorField = Position -> Vec

A vector field associates a vector with each position in space.

type Field v = Position -> v

Sometimes we want to be able to talk about a field without saying whether it is a scalar field or a vector field.

type CoordinateSystem = (Double, Double, Double) -> Position

A coordinate system is a function from three parameters to space.

cartesian :: CoordinateSystem

The Cartesian coordinate system. Coordinates are (x,y,z).

cylindrical :: CoordinateSystem

The cylindrical coordinate system. Coordinates are (s,phi,z), where s is the distance from the z axis and phi is the angle with the x axis.

spherical :: CoordinateSystem

The spherical coordinate system. Coordinates are (r,theta,phi), where r is the distance from the origin, theta is the angle with the z axis, and phi is the azimuthal angle.

cart

Arguments

:: Double

x coordinate

-> Double

y coordinate

-> Double

z coordinate

-> Position

A helping function to take three numbers x, y, and z and form the appropriate position using Cartesian coordinates.

cyl

Arguments

:: Double

s coordinate

-> Double

phi coordinate

-> Double

z coordinate

-> Position

A helping function to take three numbers s, phi, and z and form the appropriate position using cylindrical coordinates.

sph

Arguments

:: Double

r coordinate

-> Double

theta coordinate

-> Double

phi coordinate

-> Position

A helping function to take three numbers r, theta, and phi and form the appropriate position using spherical coordinates.

cartesianCoordinates :: Position -> (Double, Double, Double)

Returns the three Cartesian coordinates as a triple from a position.

cylindricalCoordinates :: Position -> (Double, Double, Double)

Returns the three cylindrical coordinates as a triple from a position.

sphericalCoordinates :: Position -> (Double, Double, Double)

Returns the three spherical coordinates as a triple from a position.

displacement

Arguments

:: Position

source position

-> Position

target position

-> Displacement

Displacement from source position to target position.

shiftPosition :: Displacement -> Position -> Position

Shift a position by a displacement.

shiftObject :: Displacement -> (a -> Position) -> a -> Position

An object is a map into Position.

shiftField :: Displacement -> (Position -> v) -> Position -> v

A field is a map from Position.

addFields :: AdditiveGroup v => [Field v] -> Field v

Add two scalar fields or two vector fields.

rHat :: VectorField

The vector field in which each point in space is associated with a unit vector in the direction of increasing spherical coordinate r, while spherical coordinates theta and phi are held constant. Defined everywhere except at the origin. The unit vector rHat points in different directions at different points in space. It is therefore better interpreted as a vector field, rather than a vector.

thetaHat :: VectorField

The vector field in which each point in space is associated with a unit vector in the direction of increasing spherical coordinate theta, while spherical coordinates r and phi are held constant. Defined everywhere except on the z axis.

phiHat :: VectorField

The vector field in which each point in space is associated with a unit vector in the direction of increasing (cylindrical or spherical) coordinate phi, while cylindrical coordinates s and z (or spherical coordinates r and theta) are held constant. Defined everywhere except on the z axis.

sHat :: VectorField

The vector field in which each point in space is associated with a unit vector in the direction of increasing cylindrical coordinate s, while cylindrical coordinates phi and z are held constant. Defined everywhere except on the z axis.

xHat :: VectorField

The vector field in which each point in space is associated with a unit vector in the direction of increasing Cartesian coordinate x, while Cartesian coordinates y and z are held constant. Defined everywhere.

yHat :: VectorField

The vector field in which each point in space is associated with a unit vector in the direction of increasing Cartesian coordinate y, while Cartesian coordinates x and z are held constant. Defined everywhere.

zHat :: VectorField

The vector field in which each point in space is associated with a unit vector in the direction of increasing Cartesian coordinate z, while Cartesian coordinates x and y are held constant. Defined everywhere.