A class of $$C^1$$C1 rational interpolation splines in one and two dimensions with region control

In this work, we use a kind of $$C^1$$ rational interpolation splines in one and two dimensions to generate curves and surfaces with region control. Simple data-dependent sufficient constraints are derived on the local control parameters to generate $$C^1$$ interpolation curves lying strictly between two given piecewise linear curves and $$C^1$$ interpolation surfaces lying strictly between two given piecewise bi-cubic blending linear interpolation surfaces.