There has been an increasing interest in design and construction of deployable mechanisms (DMs) with multiple degrees of freedom (DOFs). This paper summarizes a family of deployable mechanisms that approximates a series of curves and surfaces using the polygonal approximation technique. These mechanisms are obtained by linking the two- and three-dimensional deployable units, which are constitutive of Sarrus and scissor linkages. Multiple unit mechanisms with varying sizes are assembled and alter their shape within a different family of parameterized curves and surfaces. A systematic methodology for polygonal approximation method is presented. Quadratic, semi-cubic, cubic, quartic and sextic curve boundaries, and quadric surfaces are approximated and controlled. Computer-aided design (CAD) models and kinematic simulations elucidate the mechanism’s ability to approximate a set of curves and surfaces.