Viviani4D: A Python framework for exact NURBS representation and interactive visualization of 4D algebraic hypersurfaces

We present Viviani4D, an open-source Python framework for the exact construction, manipulation, and visualization of four-dimensional geometric objects using Non-Uniform Rational B-Splines (NURBS). The software provides a computational pipeline that transforms a rational parameterization of a 4D algebraic hypersurface into an exact NURBS representation, enabling real-time exploration through interactive orthogonal projections into R 3 . Unlike general-purpose visualization tools, Viviani4D maintains mathematical exactness while offering an intuitive interface for generating continuous families of 3D surfaces from a single 4D model. The framework is built on a modular architecture with core classes for B-spline basis evaluation, NURBS curve/surface manipulation, and specialized 4D geometric objects. It supports symbolic validation of geometric properties, high-quality static and interactive visualizations, and export capabilities for further analysis. Designed for researchers and educators in computational geometry, CAD, and scientific visualization, Viviani4D bridges the gap between theoretical higher-dimensional geometry and practical implementation. The complete code, documentation, and reproducible examples are publicly available to encourage verification, extension, and adoption in both academic and applied contexts.