Programmable 3D shape morphing of hot-drawn polymeric sheets has been demonstrated using photothermal local shrinkage of patterned hinges. However, the hinge designs have been limited to simple linear hinges used to generate in-plane local folding or global curvature. Herein, we report an unprecedented design strategy to realize localized curvature engineering in 3D structures employing radial hinges and stress-releasing facets on 2D polymeric sheets. The shape and height of the 3D structures are readily controlled by varying the number of radial patterns. Moreover, they are numerically predictable by finite elemental modeling simulation with consideration of the spatio-temporal stress distribution, as well as of stress competition effects. Localized curvature engineering provides programming capabilities for various designs including soft-turtle-shell, sea-shell shapes, and saddle architectures with the desired chirality. The results of local curvilinear actuation with quantifiable stress implies options to advance the applicability of self-folded architectures embodying coexisting curved and linear geometric surfaces.