This paper studies Newton-type methods for minimization of partly smooth convex functions. Sequential Newton methods are provided using local parameterizations obtained from U-Lagrangian theory and from Riemannian geometry. The Hessian based on the U-Lagrangian depends on the selection of a dual parameter g; by revealing the connection to Riemannian geometry, a natural choice of g emerges for which the two Newton directions become identical. This choice of g is also shown to be related to the least-squares multiplier estimate from a sequential quadratic programming (SQP) approach, and with this multiplier, SQP gives the same search direction as the Newton methods.