Copyright	(C) 2012-2015 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 Source
Functor Plucker Source
MonadFix Plucker Source
Applicative Plucker Source
Foldable Plucker Source
Traversable Plucker Source
Generic1 Plucker Source
Distributive Plucker Source
Representable Plucker Source
MonadZip Plucker Source
Serial1 Plucker Source
Traversable1 Plucker Source
Apply Plucker Source
Bind Plucker Source
Foldable1 Plucker Source
Eq1 Plucker Source
Ord1 Plucker Source
Read1 Plucker Source
Show1 Plucker Source
Additive Plucker Source
Metric Plucker Source
Trace Plucker Source
Affine Plucker Source
Unbox a => Vector Vector (Plucker a) Source
Unbox a => MVector MVector (Plucker a) Source
Eq a => Eq (Plucker a) Source
Floating a => Floating (Plucker a) Source
Fractional a => Fractional (Plucker a) Source
Num a => Num (Plucker a) Source
Ord a => Ord (Plucker a) Source
Read a => Read (Plucker a) Source
Show a => Show (Plucker a) Source
Ix a => Ix (Plucker a) Source
Generic (Plucker a) Source
Storable a => Storable (Plucker a) Source
Binary a => Binary (Plucker a) Source
Serial a => Serial (Plucker a) Source
Serialize a => Serialize (Plucker a) Source
NFData a => NFData (Plucker a) Source
Hashable a => Hashable (Plucker a) Source
Unbox a => Unbox (Plucker a) Source
Ixed (Plucker a) Source
Epsilon a => Epsilon (Plucker a) Source
FunctorWithIndex (E Plucker) Plucker Source
FoldableWithIndex (E Plucker) Plucker Source
TraversableWithIndex (E Plucker) Plucker Source
Each (Plucker a) (Plucker b) a b Source
type Rep1 Plucker Source
type Rep Plucker = E Plucker Source
type Diff Plucker = Plucker Source
data MVector s (Plucker a) = MV_Plucker !Int (MVector s a) Source
type Rep (Plucker a) Source
data Vector (Plucker a) = V_Plucker !Int (Vector a) Source
type Index (Plucker a) = E Plucker Source
type IxValue (Plucker a) = a Source

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 Source
Show LinePass Source
Generic LinePass Source
type Rep LinePass Source

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