A Space is a continuous set of numbers. Continuous here means that the set has an upper and lower bound, and an element that is between these two bounds is a member of the Space.

a `union` b == b `union` a
a `intersection` b == b `intersection` a
(a `union` b) `intersection` c == (a `intersection` b) `union` (a `intersection` c)
(a `intersection` b) `union` c == (a `union` b) `intersection` (a `union` c)
norm (norm a) = norm a
a |>| b == b |<| a
a |.| singleton a

a convex hull

https://en.wikipedia.org/wiki/Intersection_(set_theory)

a space that can be divided neatly

unsafeSpace1 (grid OuterPos s g) == s
getUnion (sconcat (Union <$> (gridSpace s g))) == s

middle element of the space

interpolate a space

interpolate s x == project s (zero ... one) x

project an element from one space to another, preserving relative position.

project o n (lower o) = lower n
project o n (upper o) = upper n
project o n (mid o) = mid n
project a a x = x

Pos suggests where points should be placed in forming a grid across a field space.

Maybe containing space of a traversable.

the containing space of a non-empty Traversable.

partial function.

all $ unsafeSpace1 a `contains` <$> a

supply a random element within a Space

>>> randomS (one :: Range Double) g
(0.43085240252163404,StdGen {unStdGen = SMGen 4530528345362647137 13679457532755275413})

StatefulGen version of randomS

>>> import Control.Monad
>>> runStateGen_ g (randomSM (one :: Range Double))
0.43085240252163404

list of n random elements within a Space

>>> let g = mkStdGen 42
>>> fst (randomSs 3 (one :: Range Double) g)
[0.43085240252163404,-6.472345419562497e-2,0.3854692674681801]

>>> fst (randomSs 3 (Rect 0 10 0 10 :: Rect Int) g)
[Point 0 7,Point 0 2,Point 1 7]

is an element contained within a space

is a space contained within another?

(a `union` b) `contains` a
(a `union` b) `contains` b

are two spaces disjoint?

distance between boundaries

create a space centered on a plus or minus b

lift a monotone function (increasing or decreasing) over a given space

a small space

widen a space

widen by a small amount

Scale a Space. (scalar multiplication)

Move a Space. (scalar addition)

linear transform + translate of a point-like number

(x, y) -> (ax + by + c, dx + ey + d)

or

\[ \begin{pmatrix} a & b & c\\ d & e & f\\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x\\ y\\ 1 \end{pmatrix} \]

Calculate the inverse of a transformation.

An Affinity is something that can be subjected to an affine transformation in 2-dimensional space, where affine means a linear matrix operation or a translation (+).

https://en.wikipedia.org/wiki/Affine_transformation

Apply a Transform to an Affinity

Rotate an Affinity (counter-clockwise)