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 |

Plücker coordinates for lines in 3d homogeneous space.

- data Plucker a = Plucker !a !a !a !a !a !a
- squaredError :: 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
- passes :: (Epsilon 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

Plücker coordinates for lines in a 3-dimensional space.

Plucker !a !a !a !a !a !a |

squaredError :: Num a => Plucker a -> a #

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 #

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 #

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 #

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 #

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

# Operations on lines

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

Checks if two lines intersect (or nearly intersect).

Describe how two lines pass each other.

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). |

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

Check how two lines pass each other. `passes l1 l2`

describes
`l2`

when looking down `l1`

.

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

The minimum squared distance of a line from the origin.

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

The point where a line is closest to the origin.

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

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 #

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

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

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

# Basis elements

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

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

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

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) #

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) #

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

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

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

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

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