Data.VectorSpace

Vector space type relation and basic instances.

Documentation

class (Eq a, Floating a) => VectorSpace v a | v -> a where

Vector space type relation.

A vector space is a set (type) closed under addition and multiplication by a scalar. The type of the scalar is the field of the vector space, and it is said that v is a vector space over a.

The encoding uses a type class |VectorSpace| v a, where v represents the type of the vectors and a represents the types of the scalars.

Minimal complete definition

zeroVector, (*^), (^+^), dot

Methods

zeroVector :: v

Vector with no magnitude (unit for addition).

(*^) :: a -> v -> v infixr 9

Multiplication by a scalar.

(^/) :: v -> a -> v infixl 9

Division by a scalar.

(^+^) :: v -> v -> v infixl 6

Vector addition

(^-^) :: v -> v -> v infixl 6

Vector subtraction

negateVector :: v -> v

Vector negation. Addition with a negated vector should be same as subtraction.

dot :: v -> v -> a infix 7

Dot product (also known as scalar or inner product).

For two vectors, mathematically represented as a = a1,a2,...,an and b = b1,b2,...,bn, the dot product is a . b = a1*b1 + a2*b2 + ... + an*bn.

Some properties are derived from this. The dot product of a vector with itself is the square of its magnitude (norm), and the dot product of two orthogonal vectors is zero.

norm :: v -> a

Vector's norm (also known as magnitude).

For a vector represented mathematically as a = a1,a2,...,an, the norm is the square root of a1^2 + a2^2 + ... + an^2.

normalize :: v -> v

Return a vector with the same origin and orientation (angle), but such that the norm is one (the unit for multiplication by a scalar).

Instances

VectorSpace Double Double

Vector space instance for Doubles, with Double scalars.

Defined in Data.VectorSpace

Methods

zeroVector :: Double

(*^) :: Double -> Double -> Double

(^/) :: Double -> Double -> Double

(^+^) :: Double -> Double -> Double

(^-^) :: Double -> Double -> Double

negateVector :: Double -> Double

dot :: Double -> Double -> Double

norm :: Double -> Double

normalize :: Double -> Double

VectorSpace Float Float

Vector space instance for Floats, with Float scalars.

Defined in Data.VectorSpace

Methods

zeroVector :: Float

(*^) :: Float -> Float -> Float

(^/) :: Float -> Float -> Float

(^+^) :: Float -> Float -> Float

(^-^) :: Float -> Float -> Float

negateVector :: Float -> Float

dot :: Float -> Float -> Float

norm :: Float -> Float

normalize :: Float -> Float

RealFloat a => VectorSpace (Vector3 a) a

Defined in Data.Vector3

Methods

zeroVector :: Vector3 a

(*^) :: a -> Vector3 a -> Vector3 a

(^/) :: Vector3 a -> a -> Vector3 a

(^+^) :: Vector3 a -> Vector3 a -> Vector3 a

(^-^) :: Vector3 a -> Vector3 a -> Vector3 a

negateVector :: Vector3 a -> Vector3 a

dot :: Vector3 a -> Vector3 a -> a

norm :: Vector3 a -> a

normalize :: Vector3 a -> Vector3 a

RealFloat a => VectorSpace (Vector2 a) a

Defined in Data.Vector2

Methods

zeroVector :: Vector2 a

(*^) :: a -> Vector2 a -> Vector2 a

(^/) :: Vector2 a -> a -> Vector2 a

(^+^) :: Vector2 a -> Vector2 a -> Vector2 a

(^-^) :: Vector2 a -> Vector2 a -> Vector2 a

negateVector :: Vector2 a -> Vector2 a

dot :: Vector2 a -> Vector2 a -> a

norm :: Vector2 a -> a

normalize :: Vector2 a -> Vector2 a

(Eq a, Floating a) => VectorSpace (a, a) a

Vector space instance for pairs of Floating point numbers.

Defined in Data.VectorSpace

Methods

zeroVector :: (a, a)

(*^) :: a -> (a, a) -> (a, a)

(^/) :: (a, a) -> a -> (a, a)

(^+^) :: (a, a) -> (a, a) -> (a, a)

(^-^) :: (a, a) -> (a, a) -> (a, a)

negateVector :: (a, a) -> (a, a)

dot :: (a, a) -> (a, a) -> a

norm :: (a, a) -> a

normalize :: (a, a) -> (a, a)

(Eq a, Floating a) => VectorSpace (a, a, a) a

Vector space instance for triplets of Floating point numbers.

Defined in Data.VectorSpace

Methods

zeroVector :: (a, a, a)

(*^) :: a -> (a, a, a) -> (a, a, a)

(^/) :: (a, a, a) -> a -> (a, a, a)

(^+^) :: (a, a, a) -> (a, a, a) -> (a, a, a)

(^-^) :: (a, a, a) -> (a, a, a) -> (a, a, a)

negateVector :: (a, a, a) -> (a, a, a)

dot :: (a, a, a) -> (a, a, a) -> a

norm :: (a, a, a) -> a

normalize :: (a, a, a) -> (a, a, a)

(Eq a, Floating a) => VectorSpace (a, a, a, a) a

Vector space instance for tuples with four Floating point numbers.

Defined in Data.VectorSpace

Methods

zeroVector :: (a, a, a, a)

(*^) :: a -> (a, a, a, a) -> (a, a, a, a)

(^/) :: (a, a, a, a) -> a -> (a, a, a, a)

(^+^) :: (a, a, a, a) -> (a, a, a, a) -> (a, a, a, a)

(^-^) :: (a, a, a, a) -> (a, a, a, a) -> (a, a, a, a)

negateVector :: (a, a, a, a) -> (a, a, a, a)

dot :: (a, a, a, a) -> (a, a, a, a) -> a

norm :: (a, a, a, a) -> a

normalize :: (a, a, a, a) -> (a, a, a, a)

(Eq a, Floating a) => VectorSpace (a, a, a, a, a) a

Vector space instance for tuples with five Floating point numbers.

Defined in Data.VectorSpace

Methods

zeroVector :: (a, a, a, a, a)

(*^) :: a -> (a, a, a, a, a) -> (a, a, a, a, a)

(^/) :: (a, a, a, a, a) -> a -> (a, a, a, a, a)

(^+^) :: (a, a, a, a, a) -> (a, a, a, a, a) -> (a, a, a, a, a)

(^-^) :: (a, a, a, a, a) -> (a, a, a, a, a) -> (a, a, a, a, a)

negateVector :: (a, a, a, a, a) -> (a, a, a, a, a)

dot :: (a, a, a, a, a) -> (a, a, a, a, a) -> a

norm :: (a, a, a, a, a) -> a

normalize :: (a, a, a, a, a) -> (a, a, a, a, a)