Software framework for intrinsic curve interpolation in the 4D hypersphere using stereographic projection

We present a symbolic and geometric software framework, implemented in Mathematica, for constructing and interpolating curves intrinsically embedded in the four-dimensional hypersphere (S3⊂E4) 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 W=0, performing classical Lagrange interpolation in E3, 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 E4 interpolation. The software provides explicit symbolic implementations for the projection, inverse projection, and interpolation procedures, along with visualization tools based on fixed immersions from E4 to E3. Illustrative examples demonstrate the framework’s accuracy and its ability to handle both curve and surface constructions, highlighting its potential for applications in high-dimensional geometric modeling, theoretical physics, and computational differential geometry. The full source code and examples are available in a public repository for reproducibility and further development.