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The Cartesian coordinates are the most widely used in Euclidean space of any dimension. However, there is another set of coordinate systems which can in some way be considered optimal.

Imagine hexagonal coordinates on the plane (see also my other question ):

enter image description here

They separate the plane into three sectors, while Cartesian coordinates - into four sectors. In this way it can be considered the most symmetrical coordinate system on a plane as well (if we apply rotation around the origin).

Also, hexagonal coordinate system can be used without negative coordinates involved.

I'm aware of the main disadvantage of such a system - it's not orthogonal.

The square of the Euclidean norm for this system can be written as follows:

$$||(a,b,c)||^2=a^2 + b^2+c^2-bc-ac-ab$$

However, there is a useful way to think of this coordinate system - as a projection of the three sides of 3D cube, in the direction of its main diagonal (see the picture above on the right).


We can extend this coordinate system to higher dimensions. For example, in 3D we have:

enter image description here

We can imagine the axes are pointing in the directions of tetrahedron vertices, or four vertices of a cube. The truncated octahedron is the only space filling Archimedean solid and represents the integer cell of this coordinate system (just like a hexagon on a plane).

These coordinates can also be imagined like a projection of four faces of a hypercube (they separate the whole space into four parts).

There is an obvious way to extend this coordinate system to any dimension $d$ with $d+1$ axes involved.


The question is - what are possible applications of this coordinate system in 3D and higher dimensions?

Any reference where this coordinate system is mentioned would be welcome as well.

I already know that hexagonal coordinates are useful, of course, but nothing about these higher dimensional cases.

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    $\begingroup$ (Demoting incorrect answer to a comment) It appears you're speaking of homogeneous or barycentric coordinates, which play a starring role in projective geometry. In the plane, they're used for, e.g., computer graphics and ternary plots. $\endgroup$ – Andrew D. Hwang Jul 17 '16 at 10:27
  • $\begingroup$ On further thought: You're right that they're not the same thing. Your coordinates are defined by orthogonal projection onto the hyperplane $H$ orthogonal to $e=(1,1,…,1)$, and are defined by where a line through a point of $H$ and parallel to $e$ hits the boundary of the positive orthant. $\endgroup$ – Andrew D. Hwang Jul 17 '16 at 10:28
  • $\begingroup$ @AndrewD.Hwang, thank you! $\endgroup$ – Yuriy S Jul 17 '16 at 10:33
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    $\begingroup$ Coda to my preceding comment: 1. Hexagonal coordinates can be viewed as living on the hyperplane $x_{1} + \dots + x_{n} = 0$, and are given by$$(x_{1}, \dots, x_{n}) - (\min x_{i}) (1, \dots, 1).$$2. The set of points having these Cartesian coordinates is the boundary of the non-negative orthant, which is not smooth along intersections of coordinate hyperplanes. That makes these coordinates vexing for applications requiring differentiability throughout $\mathbf{R}^{n-1}$, cf. homogeneous coordinates. $\endgroup$ – Andrew D. Hwang Jul 20 '16 at 11:47

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