in this study, a generalization of the theory of involute-evolute curves is presented for ruledsurfaces based on line geometry. using lines instead of points, two ruled surfaces which are offset in the senseof involute-evolute are defined. moreover, the found results are clarified using computer-aided examples

We study the geometry of bifurcation sets generic unfoldings $$D_4^\pm $$ -functions. Taking blow-ups, we show each -functions admits a parametrization as surface in $${\varvec{R}}^3$$ . Using this parametrization, investigate behavior Gaussian curvature and principal curvatures. Furthermore, number ridge curves subparabolic near their singular point.