Pointwise regularity of parameterized affine zipper fractal curves

We study the pointwise regularity of zipper fractal curves generated by affine mappings. Under the assumption of dominated splitting of index-1, we calculate the Hausdorff dimension of the level sets of the pointwise H\"older exponent for a subinterval of the spectrum. We give an equivalent characterization for the existence of regular pointwise H\"older exponent for Lebesgue almost every point. In this case, we extend the multifractal analysis to the full spectrum. In particular, our results apply for de Rham's curve.