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 ( S 3 ⊂ E 4 ) 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 E 3 , and lifting the result back to S 3 through an exact inverse mapping. This approach ensures that interpolated curves remain entirely intrinsic to the hypersphere, avoiding deviations that occur in direct E 4 interpolation. The software provides explicit symbolic implementations for the projection, inverse projection, and interpolation procedures, along with visualization tools based on fixed immersions from E 4 to E 3 . 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.