<p>Spherical parameterization serves as a critical technique in computer graphics, with existing methods primarily focusing on minimizing isometric distortion. Despite significant advancements, current approaches often struggle to simultaneously achieve both bijectivity and low isometric distortion. This paper presents a novel method for computing continuous spherical parameterizations that addresses these challenging constraints. Our approach combines inverse stereographic projection with a bijective mapping that maps mesh to the extended complex plane. Unlike traditional methods that map triangles to geodesic spherical triangles, our technique continuously maps input mesh triangles to stereographic triangles, resulting in a strictly one-to-one and onto mapping. We also identify a critical limitation in existing approaches: vanilla triangles (i.e., triangles with straight edges) in the extended complex plane can induce significant distortions near poles during stereographic projection. To mitigate this issue, we propose utilizing Bézier triangles instead of traditional vanilla triangles, which notably reduce the isometric distortion. Through extensive experimentation and comparisons with state-of-the-art techniques, we demonstrate the effectiveness and robustness of our method across various input scenarios.</p>