I want to clarify a bit. The vertices need to be the same distance a path that follows from (1,0,0), (0,1,0), and (0,0,1) and converges at ($\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$). The vertices are always equidistant from the two planes they start from, so point (1,0,0) is equidistant from xy and xz. Basically the triangle doesn't rotate on the surface, but rather just scales in size with the center at ($\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$). Here is an animated version starting at the midpoint going out
I created that gif in math3d.org using these three equations representing the three arcs forming the triangle: $$[sin(t),m\times sin(t+\frac{\pi}{4}),cos(t)]$$ $$[m\times sin(t+\frac{\pi}{4}),cos(t),sin(t)]$$ $$[cos(t),sin(t),m\times sin(t+\frac{\pi}{4})]$$
$$\{t \in [\left(1-s\right)^{\frac{4-\sqrt{2}}{2}}\cdot \frac{\pi }{4},\ \frac{\pi }{2}-\left(\left(1-s\right)^{\frac{4-\sqrt{2}}{2}}\cdot \frac{\pi }{4}\right)]\}$$
where $m = \frac{1}{\sqrt{2}}\left(1-s\right)$
I need to take that and then reparameterize it so that instead of just making three separate arcs, the path follows along each 0 to $\frac{2}{\pi3}$ to $\frac{4}{\pi3}$ and then to $2\pi$ for each arc no matter how long each arc is. The variable 's' is 1 when the spherical triangle is the largest with vertices perpendicular from each other from the origin to 0 where all three are at the midpoint ($\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$).
I attempted to figure it out in desmos and I got this, but it isn't quite right as the distance from the origin isn't always 1, but I'm not sure where to go from here especially with $x_2,y_3,z_1$:
$$ x_{1} = 0p ≤ t ≤ 1p : \left(\frac{1}{\sqrt{3}}\right)^{\left(1-s\right)}\cos\left(s\frac{3}{4}\left(t\right)\right)$$ $$ x_{2} = 1p ≤ t ≤ 2p : \frac{1}{\sqrt{3}}\left(1-\cos\left(\frac{\pi}{2}s-\frac{\pi}{2}\right)^{1.8}\right)$$ $$ x_{3} = 2p ≤ t ≤ 3p : \left(\frac{1}{\sqrt{3}}\right)^{\left(1-s\right)}\cos\left(s\frac{3}{4}\left(t-3p\right)\right)$$ $$ y_{1,2} = 0p ≤ t ≤ 2p : \left(\frac{1}{\sqrt{3}}\right)^{\left(1-s\right)}\cos\left(s\frac{3}{4}\left(t-p\right)\right)$$ $$ y_{3} = 2p ≤ t ≤ 3p : \frac{1}{\sqrt{3}}\left(1-\cos\left(\frac{\pi}{2}s-\frac{\pi}{2}\right)^{1.8}\right)$$ $$ z_{1} = 0p ≤ t ≤ 1p : \frac{1}{\sqrt{3}}\left(1-\cos\left(\frac{\pi}{2}s-\frac{\pi}{2}\right)^{1.8}\right)$$ $$ z_{2,3} = 1p ≤ t ≤ 3p : \left(\frac{1}{\sqrt{3}}\right)^{\left(1-s\right)}\cos\left(s\frac{3}{4}\left(t-2p\right)\right)$$
where $p = \frac{2pi}{3}$