Spiral folding of flat and planar membranes with finite thickness is of relevant interest to develop the spin-type deployable membrane structures for space environments and for consumer applications. Examples involve the design and development of origami-based structures, airbags, antenna design, wrapping of food by thin membranes, wheel design, and membrane deployment for medical applications. In this paper, we propose the governing equations to fold planar membranes with finite thickness by using curved creases, whose governing equations render fold patterns whose radius of curvature tends to increase linearly by accommodating membrane thickness. The consideration of curvature along in the crease patterns is relevant and potential to balance the tension of outer layers with the compression of inner layers, and to distribute the out-of plane and localized bending near the creases and vertices. We present the mathematical formulations that consider the curved creases and describe folding examples of a planar membrane with a defined thickness. Our computational experiments have shown (1) the versatility to model a plural number of curvature profiles, and (2) the feasibility of global deployment by using the compliant and explicit numerical simulations. From viewpoints of configuration and deployment performance, the curved crease patterns are potential to extend the versatility and smoothness of spiral folding mechanisms.