This Mathematica notebook implements a symbolic and geometric software framework for constructing and interpolating curves intrinsically embedded in the four-dimensional hypersphere (S³) using stereographic projection and its inverse. The framework addresses the challenge of preserving hyperspherical constraints during interpolation by projecting curve data to the three-dimensional hyperplane, performing classical Lagrange interpolation in Euclidean space, and lifting the result back to S³ through an exact inverse mapping. This approach ensures that interpolated curves remain entirely intrinsic to the hypersphere, avoiding deviations that occur in direct 4D Euclidean interpolation. Key features include: - Exact symbolic implementations of 4D stereographic projection and inverse projection - Lagrange interpolation routines for curves and tensor-product surfaces - Visualization tools based on fixed immersions from 4D to 3D - Modular architecture allowing replacement of interpolation schemes - Validation of hyperspherical constraints at every step The software provides explicit symbolic implementations for projection, inverse projection, and interpolation procedures, along with visualization tools suitable for high-dimensional geometric modeling, theoretical physics, and computational differential geometry applications. All computations yield exact analytic expressions, enabling further theoretical analysis or adaptation to other manifolds. Keywords: stereographic projection, hypersphere, 4D geometry, Lagrange interpolation, symbolic computation, Mathematica, geometric modeling, computational geometry, differential geometry, higher-dimensional interpolation License: MIT