Origami-inspired metasurfaces have attracted growing interest for their ability to achieve real-time, programmable shape morphing, giving rise to a new interdisciplinary field in science and engineering that explores the transformation of two-dimensional planar sheets into complex three-dimensional structures through controlled folding [1]. In the present study, we have proposed programmable origami curvature that realises shape-reconfigurable metasurfaces for antenna design. The proposed metasurfaces are derived from the higher-order origami derivatives, and their rigid foldability and the motion dynamics are thoroughly examined through computational simulation [2]. An efficient mathematical approach is developed based on idealising the origami derivative as a spherical linkage involving Denavit-Hartenberg matrix notations. This formulation enables the generation of metasurfaces that realise pre-defined shapes with single-degree-of-freedom programmable mechanics. The proposed geometries of origami-based metasurfaces offer dynamic control over surface geometry, orientation, and operational parameters, including frequency, radiation pattern and polarisation characteristics. The seamless transition between a two-dimensional sheet and a three-dimensional configuration highlights the potential of programmable origami curvature as an innovative approach for next-generation space applications.