\(G^1\) curves are visually smooth and aesthetic curves. They play an important role in the domain of computer-aided design (CAD) and 3D modeling. Interpolatory geometric subdivision schemes is a methodological approach for attaining these curves while mitigating undesirable artifacts. In this paper, we introduce a novel geometric interpolatory subdivision scheme, named the angle-based 6-point geometric scheme, designed for curves on surfaces of constant curvature (Euclidean, spherical, and hyperbolic). This proposed scheme incorporates a tension parameter. We show that the scheme converges if the tension parameter belongs to a specific interval, and yields \(G^1\) -continuous limit curves. This allows us to manipulate a whole family of curve subdivision schemes on constant curvature surfaces according to our needs. Numerical tests indicate the possibility of selecting the parameter within a well-defined range to achieve \(G^2\) -continuity across all three surface models. Experimental examples are presented to illustrate the convergence and \(G^1\) property of this scheme, accompanied by substantial applications showcasing its effectiveness.
