We present several non-linear 4-point interpolatory schemes, derived from the "classical" linear 4-point scheme. These new schemes have variable tension parameter instead of the fixed tension parameter in the linear 4-point scheme. The tension parameter is adapted locally according to the geometry of the control polygon within the 4-point stencil. This allows the schemes to remain local and in the same time to achieve two important shape-preserving properties - artifacts elimination and convexity-preservation. The proposed schemes are robust and have special features such as "double-knot" edges corresponding to continuity without geometrical smoothness and inflection edges support for convexity-preservation. Convergence proof is given and experimental smoothness analysis is done in detail, which indicates that the limit curves are C^1.