Linear.Metric

class Metric :: (Type -> Type) -> Constraintclass (Additive f) <= Metric f  where

A metric space with an inner product.

Laws:

• dot v v >= 0 (positive semi-definite)
• dot v v = 0 implies v = zero
• dot u v = dot v u (symmetry)
• dot (a *^ u) v = a * dot u v (linearity)

Members

• dot :: forall a. Semiring a => f a -> f a -> a

  The inner (dot) product of two vectors.

  dot (V2 1.0 2.0) (V2 3.0 4.0) = 11.0  -- 1*3 + 2*4
  

• quadrance :: forall a. Semiring a => f a -> a

  The squared norm of a vector (quadrance from rational trigonometry).

  This is more efficient than norm when you only need to compare magnitudes.

  quadrance (V2 3.0 4.0) = 25.0  -- 3² + 4²
  

• qd :: forall a. Ring a => f a -> f a -> a

  The squared distance between two vectors.

  qd (V2 0.0 0.0) (V2 3.0 4.0) = 25.0
  

• distance :: f Number -> f Number -> Number

  The Euclidean distance between two vectors.

  distance (V2 0.0 0.0) (V2 3.0 4.0) = 5.0
  

• norm :: f Number -> Number

  The Euclidean norm (magnitude) of a vector.

  norm (V2 3.0 4.0) = 5.0
  

• signorm :: f Number -> f Number

  Convert a non-zero vector to a unit vector (signorm = "sign of norm").

  Returns the zero vector when given the zero vector.

  signorm (V2 3.0 4.0) = V2 0.6 0.8
  signorm (V2 0.0 0.0) = V2 0.0 0.0
  

normalize :: forall f. Metric f => Epsilon Number => f Number -> f Number

Normalize a vector to unit length.

Unlike signorm, this function handles the zero vector by returning zero rather than NaN.

normalize (V2 3.0 4.0) = V2 0.6 0.8
normalize (V2 0.0 0.0) = V2 0.0 0.0

project :: forall f. Metric f => f Number -> f Number -> f Number

Project the first vector onto the second.

project (V2 1.0 0.0) (V2 1.0 1.0) = V2 0.5 0.5