Research about morphing of continuum bodies represents a frontier in form-finding and shape optimization for lightweight shell structures, offering a novel approach to manipulate shapes towards desired targets. This study presents a parametric analysis method based on the metric properties of surfaces. The method enables the generation of different shell forms, starting from a flat reference domain, through a shape control process that involves local metric and curvature changes. In this work, we explore how to achieve configurations with constant Gaussian curvature throughout the shape by assigning non-uniform but isotropic local metrics to a flat disk. Exploiting a non-linear shell model already presented in [1], the parametric approach facilitates the exploration of different distortion fields by defining the target first and second fundamental forms. The resulting shape’s elastic energy is separated into membrane and bending contributions, enabling independent study to determine the ideal structural behavior parameters. This work advances the field of morphing approaches for shell structures by providing new knowledge on shape optimization and manipulation using curvature control and parametric analysis, facilitating new efficient and sustainable design solutions in architectural and engineering applications.