I am interested in scaling isotropic functions. The functions depend on the geodesic distance $d(x,y)$ on a unit sphere in $\mathbb{R}^d$.
I was interested in functions with the argument $\varepsilon d(x,y)$ and my idea was that the point $\epsilon x$ and $\epsilon y$ on the sphere of radius $\varepsilon$ should satisfy $$ \varepsilon d(x,y)=d_{\varepsilon}(\varepsilon x,\varepsilon y).$$ Is this wrong, allready?
In the next step I would aim to embedd this sphere of radius $\varepsilon$ in a sphere in $\mathbb{R}^{d+1}$ with additional coefficient $x_{d+1}=\sqrt{1-\varepsilon^2}$ and still the distance between the points should be preserved.
But this would lead to $\varepsilon \arccos (x^Ty)= \arccos(\varepsilon^2x^Ty-(1-\epsilon^2))$. Which is clearly wrong.
The first part is correct i.e. $d_{\epsilon}\cdot (\epsilon x, \epsilon y) = \epsilon \cdot d_1(x,y)$.
The second is not. Your embedding is not isometric. It embeds (in the special case of $d=2$) a circle into a unit sphere as "a circle of latitude". Route of an airplane flying from (say) Germany to US is not close to a given circle of latitude, but resembles the great circle of Earth over Iceland or even Greenland.