Matrix-Driven Quartic Overhauser (QOVR) Surfaces Structural Framework: Continuity Limitations, Computer Graphics Algorithms, and Software Implementation

Abstract

This study introduces the spatial and analytical construction of the Quartic Overhauser (QOVR) surface generation framework designed to resolve boundary alignment and localized shape modification constraints. This framework implements a variable parameter fourth degree novel architecture to achieve exact parameter isolation across orthogonal coordinate axes. The analytical pipeline integrates directional spline blending functions with symmetric spatial control matrices, ensuring that internal knot vector variations allow localized surface adjustments while preserving the absolute positional invariance of global edge boundaries. Computational verification confirms that while the current formulation satisfies explicit C^0 positional closure and C^1 tangent continuity conditions across the internal and boundary interfaces without triggering global curvature propagation, edge joint separation, or wave-like artifacts, C^2 curvature continuity is not maintained. As demonstrated by three-dimensional mesh models and colormap visualizations, the spatial intensity fields condense strictly within the immediate neighborhood of the modified element, verifying that the displacement effect decreases exponentially as the distance from the perturbed control points increases. This explicit decoupling preserves structural symmetry and boundary invariance across adjacent geometric patches, satisfying manufacturing and reverse engineering sealing criteria. Keywords: Quartic Overhauser surface; computer aided geometric design (CAGD); local shape control; geometric continuity; symmetric tensor products; computer graphics algorithms; software implementation framework.