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We describe singularities of height functions on singular surfaces in R³ parameterized by smooth map-germs A-equivalent to one of Sₖ, Bₖ, Cₖ and F₄ singularities in terms of extended geometric language via finite succession of blowing-ups. We investigate singularities of dual surfaces of such singular surfaces.
Fukui et al. (Thu,) studied this question.