A hybrid symbolic-numerical approach for ruled surface detection and parameterization

Ruled surfaces play a crucial role in geometric modeling and computer-aided design (CAD) due to their structural simplicity and broad practical applications. In this paper, we present a novel hybrid symbolic-numerical approach for determining whether an implicitly or parametrically defined surface is ruled. Unlike previous methods, which are either purely symbolic or require complex numerical optimization techniques, our approach extends symbolic algorithms to numerical inputs given in an approximate form. This is a significant advancement, as previous symbolic methods required exact algebraic information. Our algorithm, in contrast, is designed to handle approximate data while preserving efficiency and robustness. For implicitly defined surfaces, our method relies on computing two planar curve parameterizations. If these curves are rational, the surface is ruled. If they are only approximately rational, we introduce the concept of ϵ-ruled surfaces, allowing us to classify surfaces that are 'almost ruled' within a given numerical tolerance. For parametrically defined surfaces, we analyze whether an existing rational parameterization can be rewritten in the standard ruled form: One of the key advantages of our method is its efficiency. Rather than relying on complex implicitization or high-dimensional computations, it only requires the parameterization of two curves, making it computationally efficient and practical for real-world applications Our results show that this hybrid symbolic-numeric framework significantly outperforms previous methods, particularly in handling geometric input data that is not perfectly exact. This breakthrough provides a powerful tool for geometric modeling, enabling more accurate and efficient ruled surface detection in industrial and scientific applications.