Extraction of tori from minimal point sets

•Sharp upper bound for the number of tori that interpolate three points, two of them being equipped with a normal direction.•Efficient algorithm for computing all tori that interpolates three points, two of them being equipped with a normal vector.•Analysis of the 3D circles that interpolate two 3D points, both equipped with a normal vector. A new algebraic method for extracting tori from a minimal point set, made of two oriented points and a simple point, is proposed. We prove a degree bound on the number of such tori; this bound is reached on examples, even when we restrict to smooth tori. Our method is based on pre-computed closed formulae well suited for numerical computations with approximate input data.