the nearest point map of a real algebraic variety with respect to euclidean distance is an algebraic function. the euclidean distance degree is the number of critical points for this optimization problem. we focus on projective varieties seen in engineering applications, and we discuss tools for exact computation. our running example is the eckart-young theorem which relates the nearest point map for low rank matrices with the singular value decomposition. this is joint work with jan draisma, emil horobet, giorgio ottaviani, rekha thomas.