create a space centered on a plus or minus b

is a space contained within another?

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

are two spaces disjoint?

a small space

interpolate a space

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

Calculate the inverse of a transformation.

is an element contained within a space

middle element of the space

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

Move a Space. (scalar addition)

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

Rotate an Affinity (counter-clockwise)

Scale a Space. (scalar multiplication)

Maybe containing space of a traversable.

the containing space of a non-empty Traversable.

partial function.

all $ unsafeSpace1 a `contains` <$> a

widen a space

widen by a small amount

distance between boundaries

Apply a Transform to an Affinity

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

the underlying element in the space

a space that can be divided neatly

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

the type that divides or quotients the space

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

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

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

linear transform + translate of a point-like number

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

or

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

a convex hull