Copyright	(C) 2012-2013 Edward Kmett,
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	experimental
Portability	non-portable
Safe Haskell	Trustworthy
Language	Haskell98

Linear.Plucker

Contents

Operations on lines
Basis elements

Description

Plücker coordinates for lines in 3d homogeneous space.

Synopsis

Documentation

data Plucker a Source

Plücker coordinates for lines in a 3-dimensional space.

Constructors

Plucker !a !a !a !a !a !a

Instances

Monad Plucker
Functor Plucker
MonadFix Plucker
Applicative Plucker
Foldable Plucker
Traversable Plucker
Generic1 Plucker
Distributive Plucker
Representable Plucker
MonadZip Plucker
Traversable1 Plucker
Foldable1 Plucker
Apply Plucker
Bind Plucker
Additive Plucker
Metric Plucker
Trace Plucker
Affine Plucker
Unbox a => Vector Vector (Plucker a)
Unbox a => MVector MVector (Plucker a)
Eq a => Eq (Plucker a)
Fractional a => Fractional (Plucker a)
Num a => Num (Plucker a)
Ord a => Ord (Plucker a)
Read a => Read (Plucker a)
Show a => Show (Plucker a)
Ix a => Ix (Plucker a)
Generic (Plucker a)
Storable a => Storable (Plucker a)
NFData a => NFData (Plucker a)
Hashable a => Hashable (Plucker a)
Unbox a => Unbox (Plucker a)
Ixed (Plucker a)
Epsilon a => Epsilon (Plucker a)
FunctorWithIndex (E Plucker) Plucker
FoldableWithIndex (E Plucker) Plucker
TraversableWithIndex (E Plucker) Plucker
Each (Plucker a) (Plucker b) a b
type Rep1 Plucker
type Rep Plucker = E Plucker
type Diff Plucker = Plucker
data MVector s (Plucker a) = MV_Plucker !Int (MVector s a)
type Rep (Plucker a)
data Vector (Plucker a) = V_Plucker !Int (Vector a)
type Index (Plucker a) = E Plucker
type IxValue (Plucker a) = a

squaredError :: (Eq a, Num a) => Plucker a -> a Source

Valid Plücker coordinates p will have squaredError p == 0

That said, floating point makes a mockery of this claim, so you may want to use nearZero.

isotropic :: Epsilon a => Plucker a -> Bool Source

Checks if the line is near-isotropic (isotropic vectors in this quadratic space represent lines in real 3d space).

(><) :: Num a => Plucker a -> Plucker a -> a infixl 5 Source

This isn't th actual metric because this bilinear form gives rise to an isotropic quadratic space

plucker :: Num a => V4 a -> V4 a -> Plucker a Source

Given a pair of points represented by homogeneous coordinates generate Plücker coordinates for the line through them, directed from the second towards the first.

plucker3D :: Num a => V3 a -> V3 a -> Plucker a Source

Given a pair of 3D points, generate Plücker coordinates for the line through them, directed from the second towards the first.

Operations on lines

parallel :: Epsilon a => Plucker a -> Plucker a -> Bool Source

Checks if two lines are parallel.

intersects :: (Epsilon a, Ord a) => Plucker a -> Plucker a -> Bool Source

Checks if two lines intersect (or nearly intersect).

data LinePass Source

Describe how two lines pass each other.

Constructors

Coplanar	The lines are coplanar (parallel or intersecting).
Clockwise	The lines pass each other clockwise (right-handed screw)
Counterclockwise	The lines pass each other counterclockwise (left-handed screw).

Instances

Eq LinePass
Show LinePass
Generic LinePass
type Rep LinePass

passes :: (Epsilon a, Num a, Ord a) => Plucker a -> Plucker a -> LinePass Source

Check how two lines pass each other. passes l1 l2 describes l2 when looking down l1.

quadranceToOrigin :: Fractional a => Plucker a -> a Source

The minimum squared distance of a line from the origin.

closestToOrigin :: Fractional a => Plucker a -> V3 a Source

The point where a line is closest to the origin.

isLine :: Epsilon a => Plucker a -> Bool Source

Not all 6-dimensional points correspond to a line in 3D. This predicate tests that a Plücker coordinate lies on the Grassmann manifold, and does indeed represent a 3D line.

coincides :: (Epsilon a, Fractional a) => Plucker a -> Plucker a -> Bool Source

Checks if two lines coincide in space. In other words, undirected equality.

coincides' :: (Epsilon a, Fractional a, Ord a) => Plucker a -> Plucker a -> Bool Source

Checks if two lines coincide in space, and have the same orientation.

Basis elements

p01 :: Lens' (Plucker a) a Source

These elements form a basis for the Plücker space, or the Grassmanian manifold Gr(2,V4).

p01 :: Lens' (Plucker a) a
p02 :: Lens' (Plucker a) a
p03 :: Lens' (Plucker a) a
p23 :: Lens' (Plucker a) a
p31 :: Lens' (Plucker a) a
p12 :: Lens' (Plucker a) a

p02 :: Lens' (Plucker a) a Source

These elements form a basis for the Plücker space, or the Grassmanian manifold Gr(2,V4).

p01 :: Lens' (Plucker a) a
p02 :: Lens' (Plucker a) a
p03 :: Lens' (Plucker a) a
p23 :: Lens' (Plucker a) a
p31 :: Lens' (Plucker a) a
p12 :: Lens' (Plucker a) a

p03 :: Lens' (Plucker a) a Source

These elements form a basis for the Plücker space, or the Grassmanian manifold Gr(2,V4).

p01 :: Lens' (Plucker a) a
p02 :: Lens' (Plucker a) a
p03 :: Lens' (Plucker a) a
p23 :: Lens' (Plucker a) a
p31 :: Lens' (Plucker a) a
p12 :: Lens' (Plucker a) a

p10 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source