linear-1.3: Linear Algebra

Portabilitynon-portable
Stabilityexperimental
MaintainerEdward Kmett <ekmett@gmail.com>
Safe HaskellTrustworthy

Linear.Plucker

Contents

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 

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

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 -> BoolSource

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 -> aSource

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 aSource

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 aSource

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 -> BoolSource

Checks if two lines are parallel.

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

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

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

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

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

The minimum squared distance of a line from the origin.

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

The point where a line is closest to the origin.

isLine :: Epsilon a => Plucker a -> BoolSource

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.

data Coincides a whereSource

When lines are represented as Plücker coordinates, we have the ability to check for both directed and undirected equality. Undirected equality between Lines (or a Line and a Ray) checks that the two lines coincide in 3D space. Directed equality, between two Rays, checks that two lines coincide in 3D, and have the same direction. To accomodate these two notions of equality, we use an Eq instance on the Coincides data type.

For example, to check the directed equality between two lines, p1 and p2, we write, Ray p1 == Ray p2.

Constructors

Line :: (Epsilon a, Fractional a) => Plucker a -> Coincides a 
Ray :: (Epsilon a, Fractional a, Ord a) => Plucker a -> Coincides a 

Instances

Basis elements

p01 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker 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 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker 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 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker 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 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker 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 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker 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 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker 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