linear-1.20.3: Linear Algebra

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

- data Plucker a = Plucker !a !a !a !a !a !a
- squaredError :: (Eq a, Num a) => Plucker a -> a
- isotropic :: Epsilon a => Plucker a -> Bool
- (><) :: Num a => Plucker a -> Plucker a -> a
- plucker :: Num a => V4 a -> V4 a -> Plucker a
- plucker3D :: Num a => V3 a -> V3 a -> Plucker a
- parallel :: Epsilon a => Plucker a -> Plucker a -> Bool
- intersects :: (Epsilon a, Ord a) => Plucker a -> Plucker a -> Bool
- data LinePass
  - = Coplanar
  - | Clockwise
  - | Counterclockwise
- passes :: (Epsilon a, Num a, Ord a) => Plucker a -> Plucker a -> LinePass
- quadranceToOrigin :: Fractional a => Plucker a -> a
- closestToOrigin :: Fractional a => Plucker a -> V3 a
- isLine :: Epsilon a => Plucker a -> Bool
- coincides :: (Epsilon a, Fractional a) => Plucker a -> Plucker a -> Bool
- coincides' :: (Epsilon a, Fractional a, Ord a) => Plucker a -> Plucker a -> Bool
- p01 :: Lens' (Plucker a) a
- p02 :: Lens' (Plucker a) a
- p03 :: Lens' (Plucker a) a
- p10 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- p12 :: Lens' (Plucker a) a
- p13 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- p20 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- p21 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- p23 :: Lens' (Plucker a) a
- p30 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- p31 :: Lens' (Plucker a) a
- p32 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- e01 :: E Plucker
- e02 :: E Plucker
- e03 :: E Plucker
- e12 :: E Plucker
- e31 :: E Plucker
- e23 :: E Plucker

# 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

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

    p10 :: Num a => Lens' (Plucker a) a
    p20 :: Num a => Lens' (Plucker a) a
    p30 :: Num a => Lens' (Plucker a) a
    p32 :: Num a => Lens' (Plucker a) a
    p13 :: Num a => Lens' (Plucker a) a
    p21 :: Num a => Lens' (Plucker a) a

p12 :: 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

p13 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source

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

    p10 :: Num a => Lens' (Plucker a) a
    p20 :: Num a => Lens' (Plucker a) a
    p30 :: Num a => Lens' (Plucker a) a
    p32 :: Num a => Lens' (Plucker a) a
    p13 :: Num a => Lens' (Plucker a) a
    p21 :: Num a => Lens' (Plucker a) a

p20 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source

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

    p10 :: Num a => Lens' (Plucker a) a
    p20 :: Num a => Lens' (Plucker a) a
    p30 :: Num a => Lens' (Plucker a) a
    p32 :: Num a => Lens' (Plucker a) a
    p13 :: Num a => Lens' (Plucker a) a
    p21 :: Num a => Lens' (Plucker a) a

p21 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source

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

    p10 :: Num a => Lens' (Plucker a) a
    p20 :: Num a => Lens' (Plucker a) a
    p30 :: Num a => Lens' (Plucker a) a
    p32 :: Num a => Lens' (Plucker a) a
    p13 :: Num a => Lens' (Plucker a) a
    p21 :: Num a => Lens' (Plucker a) a

p23 :: 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

p30 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source

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

    p10 :: Num a => Lens' (Plucker a) a
    p20 :: Num a => Lens' (Plucker a) a
    p30 :: Num a => Lens' (Plucker a) a
    p32 :: Num a => Lens' (Plucker a) a
    p13 :: Num a => Lens' (Plucker a) a
    p21 :: Num a => Lens' (Plucker a) a

p31 :: 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

p32 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source

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

    p10 :: Num a => Lens' (Plucker a) a
    p20 :: Num a => Lens' (Plucker a) a
    p30 :: Num a => Lens' (Plucker a) a
    p32 :: Num a => Lens' (Plucker a) a
    p13 :: Num a => Lens' (Plucker a) a
    p21 :: Num a => Lens' (Plucker a) a

e01 :: E Plucker Source

e02 :: E Plucker Source

e03 :: E Plucker Source

e12 :: E Plucker Source

e31 :: E Plucker Source

e23 :: E Plucker Source