This question is for people who know the Aitchison Geometry - I'm working on a (more mathematical and not statistical) paper on the Aitchison Geometry and I try to understand how ellipses (or any other geometrical objects) are defined in the Aitchison Geometry and how to present them. I've seen in many papers that for any $\mathbf{x} \in \mathcal{S}^{3}$, which is the part simplex, a possible pair of coordinates is given by $$ \frac{2}{\sqrt{6}} \ln \frac{x_{1}}{\left(x_{2} x_{3}\right)^{1 / 2}}, \quad \frac{1}{\sqrt{2}} \ln \frac{x_{2}}{x_{3}}. $$ What does this exactly mean? and how to set up graphics like this one (can also be by using R): Ellipses in the space of coordinates (left) and in the simplex (right), where discontinuous lines represent orthogonal axes.
Thanks in advance!
Let us give maybe the most natural way to consider simplex $S_3$ (with coordinates $(x_1,x_2,x_3)$), i.e., as a 3D object ; it is indeed the equilateral triangle obtained as the intersection of the plane with equation:
$$x_1+x_2+x_3=1$$
with the coordinate planes $x_1Ox_2, x_2Ox_3, x_3Ox_1$.
This the convention used in this excellent Question here.
Fig. 1.: The simplex triangle in a 3D setting.
In a second step, let us understand the genesis of the Aitchinson coordinates in this 3D representation. Let $c_1$ and $c_2$ be some positive constants ; let us consider the isolevel surfaces with the following equations (please note that we have got rid of the logarithms):
$$\dfrac{x_1}{\sqrt{x_2x_3}}=c_1 \ \ \iff \ \ x_1^2=c_1^2x_2x_3\tag{1}$$
which, when $c_1$ varies, is a variable cone (see explanation below) and
$$\dfrac{x_2}{x_3}=c_2 \ \ \iff \ \ x_2=c_2x_3 \tag{2}$$
which is a plane, rotating about the $x_1$ axis when $c_2$ varies.
The trace of the (increasingly squeezed) cones on the triangle will be (increasingly thin) ellipses sharing two common points.
The traces of the planes are straight lines passing through point $A$ (with coordinates $(x_1=1,x_2=0,x_3=0)$) .
These curves correspond to what is shown in your reference document. They can indeed be used as a system giving a new system of coordinates through coefficients $c_1$ and $c_2$ given in (1) and (2).
Fig. 2: Simplex triangle (red) with its plane (light blue). Plane $x_1Ox_2$ in grey. Variable plane with equation (2) (lighter grey). Variable cone (purple) always containing $x_2$ and $x_3$ axes.
Fig. 3: When $c_1$ and $c_2$ are varying... one gets curvilinear coordinates inside the triangle. It is to be noted that not all of the curves are ellipses. The leftmost are branches of hyperbolas.
Remark: coefficients $2/\sqrt{6},\ 1/\sqrt{2}$ are surely there for the benefit of some normalization. To be checked.
Explanation about the fact that (1) is the equation of a cone: the reason is homogeneity ; if a point $(x_1,x_2,x_3)$ belongs to the surface defined by (1), then any multiple $(ax_1,ax_2,ax_3)$ belongs to it as well.