<p>Shape features of geological boundaries exert a primary control on the spatial distribution of mineralization by defining fluid pathways, depositional traps, and loci of metal precipitation. Three-dimensional (3D) shape analysis provides a mean to encode these model features quantitatively and to evaluate the association with mineralization. Existing approaches, however, are often task-dependent, scale-specific, and poorly adapted to the complex geometries and topologies typical of geological boundaries, such as natural faults and contact interfaces. We present here a unified and scale-controllable 3D shape analysis framework based on a shape-adaptive, closed-form local implicit function representation of geological boundaries. Local implicit functions are constructed via moving least squares (MLS), wherein the local surface of a geological boundary is approximated by localized polynomials, thereby inherently adapting to complex shapes and topologies. A key advantage of the proposed framework is analytical differentiability, which enables closed-form computation of first- and higher-order derivatives as fundamental components for a spectrum of shape descriptors that characterize boundary features. Moreover, owing to the adjustable support radius in MLS, both the local implicit function and the derived shape descriptors are scale-controllable. Together these attributes yield a single framework that is analytically differentiable for 3D shape analysis of geological boundaries, which enables both shape-adaptive and scale-controllable extraction of shape descriptors. As such, a comprehensive spectrum of shape descriptors is derived, including local orientations (strike and dip) from first-order derivatives, rates of change in orientation from second-order derivatives, and curvature metrics (mean and Gaussian curvatures) from a combination of closed-form first- and second-order derivatives. We applied the method to the Zhaoping fault at the structurally controlled Xiadian gold deposit (Eastern China) to quantify associations between fault geometry and gold mineralization. Extracted shape descriptors showed significant and scale-dependent correlations with mineralization. When incorporated as predictor variables into 3D mineral prospectivity mapping, the descriptors improved model fit to known mineralization patterns across multiple modeling approaches. The framework therefore provides an effective and comprehensive tool for revealing shape controls of geological models and enhancing performance of 3D mineral prospectivity modeling in future exploration.</p>