The Viviani Surface in R4: A Rational Parametric Hypersurface for Symbolic Visualization and Geometric Modeling

Inspired by the classical Viviani curve, which arises as the intersection of a sphere and a cylinder tangent at a circle and passing through the poles, we introduce and study a four-dimensional analog—the Viviani surface. Defined as the intersection of a 3-sphere and a 3-cylinder in R4, this surface preserves the geometric essence of the original problem: determining the illuminated boundary of a hemisphere under sunlight incidence. We rigorously prove that the resulting set is a differentiable surface by constructing an atlas of Monge and geographic charts. We derive explicit parametric representations, study its orthogonal projections to three-dimensional subspaces, and compute the four-dimensional volume enclosed by the surface. Furthermore, we present a rational NURBS-based parametrization, allowing symbolic and numerical visualization using the Wolfram Language. Our approach bridges classical geometry with higher-dimensional modeling and contributes to the pedagogical and computational exploration of hypersurfaces in R4. This is, to our knowledge, the first symbolic and NURBS-compatible construction of such an intersection surface in dimension four.