Optimizing best-fit algorithms for complex cross-vault geometries in HBIM generation using point cloud data

Builders of the past naturally adjusted geometries to fit existing surfaces. Today, replicating these forms during the 3D digitization of historical elements poses a significant challenge for BIM operators. Achieving a precise fit for the geometry of a cross-vault facilitates the implementation of the Scan-to-BIM approach for repetitive objects with significant variations in their geometry. This paper introduces a descriptive mathematical model that provides BIM experts with a foundation for creating multiple geometric replicas. The approach employs clustering algorithms, optimization techniques, frequency analysis via Fourier transform, and ordinary Kriging interpolation. Two parametric BIM models are developed: one simple model defined by five variables and another more complex model defined by nine geometric variables. Both models are validated against the segmented point cloud. The results indicate interpolated standard deviations of ±0.0085 m for the simple vault and ± 0.0066 m for the complex vault. The difference between using the simple and complex vault models is ±0.0082 m, representing a variation of 0.01 % in the values of the five optimized parameters.