linear-1.9.0.1: Linear Algebra

Portability non-portable experimental Edward Kmett Trustworthy

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

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) 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

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

 Eq LinePass Show LinePass Generic LinePass

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 `Line`s (or a `Line` and a `Ray`) checks that the two lines coincide in 3D space. Directed equality, between two `Ray`s, 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

 Eq (Coincides a)

# Basis elements

p01 :: Lens' (Plucker a) aSource

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) aSource

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) aSource

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) aSource

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) aSource

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) aSource

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
```