Parametrization of a spherical triangle?

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I want to calculate an integral in the curve defined by a spherical triangle with vertices $\left(a,0,0\right),\left(0,a,0\right),\left(0,0,a\right)$ over the sphere of center $\left(0,0,0\right)$, and I would like to have a parametrization in 1 variable in order to do it, and I don't know how since the sphere itself has a parametrization in 2 variables. Any ideas?