Clif.Algebra

This module provides the type Clif for representing the elements of a Clifford algebra along with some standard operations.

See The inner products of geometric algebra by Leo Dorst for a concise explanation of the different inner products.

The Clif type

data Clif b a

A data type representing a Clif (multivector) composed of direct sum of scaled blades

Instances

Functor (Clif b)

Methods

fmap :: (a -> b) -> Clif b a -> Clif b b

(<$) :: a -> Clif b b -> Clif b a

(Eq b, Eq a, Basis b a) => Eq (Clif b a)

The Eq instance calculates the canonical forms of the compared Clifs before comparison.

Methods

(==) :: Clif b a -> Clif b a -> Bool

(/=) :: Clif b a -> Clif b a -> Bool

(Eq a, Basis b a, Fractional a) => Fractional (Clif b a)

Inverse elements only exist for Clifs c for which c times c is scalar. For others, recip does not terminate.

Methods

(/) :: Clif b a -> Clif b a -> Clif b a

recip :: Clif b a -> Clif b a

fromRational :: Rational -> Clif b a

(Eq a, Basis b a) => Num (Clif b a)

Note that abs and signum are only well-defined on the scalar component of each Clif, and zero otherwise.

Methods

(+) :: Clif b a -> Clif b a -> Clif b a

(-) :: Clif b a -> Clif b a -> Clif b a

(*) :: Clif b a -> Clif b a -> Clif b a

negate :: Clif b a -> Clif b a

abs :: Clif b a -> Clif b a

signum :: Clif b a -> Clif b a

fromInteger :: Integer -> Clif b a

(Show b, Show a) => Show (Clif b a)

Show instance just shows the underlying Map for now

Methods

showsPrec :: Int -> Clif b a -> ShowS

show :: Clif b a -> String

showList :: [Clif b a] -> ShowS

Constructing and deconstructing Clifs

blade :: (Eq a, Basis b a) => [b] -> a -> Clif b a

Constructor for a blade

(*:) :: (Eq a, Basis b a) => a -> [b] -> Clif b a infix 9

Infix synonym for flip blade

vec :: (Eq a, Basis b a) => b -> a -> Clif b a

Constructor for basis vector values. Note that vec a s is equivalent to blade [a] s.

fromList :: (Eq a, Basis b a) => [([b], a)] -> Clif b a

Constructs a Clif from a list of blades and their multipliers in canonical form.

>>> fromList [([], 42), ([E 1, E 2], 1)]
42 *: [] + 1 *: [E 1,E 2]

Geometric algebra operations

grade :: (Eq a, Basis b a) => Int -> Clif b a -> Clif b a

Grade projection on the given grade. For negative values, returns zero.

Note that this always calculates the canonical form of a Clif before projecting it.

rev :: Ord b => Clif b a -> Clif b a

Reverse of a Clif, i.e. the reverse of all its component blades.

Outer product

wedge :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a

Wedge product

(/\) :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a infixl 8

Infix synonym for wedge

Inner products

(<\) :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a infixl 8

Infix synonym for lContract

(/>) :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a infixl 8

Infix synonym for rContract

(.|.) :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a infixl 8

Infix synonym for scalarProd

(<.>) :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a infixl 8

Infix synonym for dot

lContract :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a

Left contraction

rContract :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a

Right contraction

scalarProd :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a

Scalar product (0-grade components of the blade products)

dot :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a

Dot product

hestenes :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a

Hestenes dot product

Hodge duality

hodge :: (Eq a, Basis b a) => [b] -> Clif b a -> Clif b a

Hodge dual of a Clif in a Clifford algebra specified by a given pseudoscalar (volume element):

hodge (E <$> "abc") $ blade [E 'b'] 1 == blade (E <$> "ac") 1

Projections

proj :: (Eq a, Basis b a, Fractional a) => Clif b a -> Clif b a -> Clif b a

Projection of Clif x in the direction of Clif y, defined as

proj x y == (x <\ recip y) <\ y