Triangulated coordinates

2D plane

Rectangular coordinates consist of two distances (x, y), and polar coordinates consist of a distance and an angle (r, θ). Perhaps there could be a coordinate system consisting of two angles (α, β):

A diagram of 2D triangulated coordinates

There are two arbitrarily chosen points: the origin, which is (0, 0) in rectangular coordinates, and the ascended origin, which I arbitrarily choose to be (1, 0). The axis that the origin and ascended origin are on is the cool axis (or x-axis in this case). A point (α, β) is the intersection between a line that forms the angle α with the cool axis from the origin and a line that forms the angle β with the cool axis from the ascended origin.

For example, the triangulated point (45°, 90°) would be (1, 1) in rectangular, and (5, 4) in rectangular would be (tan⁻¹0.8, 45°).

Like polar coordinates, there are many ways to state a point. (45°, 90°) can also be (225°, 90°), (-135°, 270°), etc.

A limitation of this system is that specific points on the cool axis can't be expressed; maybe a third, noncollinear origin and angle could be used. For this reason, neither α nor β can be kπ, where k is an integer (e.g. 0, π, etc.).

Rectangular to triangulated

(x, y) = (tan⁻¹(y / x), tan⁻¹(y / (x − 1)))

Rectangular x:

Rectangular y:

Triangulated:

Triangulated to rectangular

(α, β) = (tanβ / (tanβ - tanα), (tanα · tanβ) / (tanβ - tanα))

Triangulated α: °

Triangulated β: °

Rectangular:

Graphs can be made with triangulated coordinates as well:

α = 45° produces the triangulated equivalent of y = x (α = 0 could also work despite points on the cool axis not being expressible because if they were, they would have an α of 0).
α = β produces nothing because parallel lines can't intersect.
However, α = 2β produces a circle of radius 1 with centre (0, 0).
α = 3β produces a weird loopity-loop thing with an asymptote at x = 3/2
α = 4β and α = 5βare rather sophisticated.

...I think?