We have derived a quartic equation for computing the direction of an internuclear vector from residual dipolar
couplings (RDCs) measured in two aligning media, and two simple trigonometric equations for computing the
backbone phi-psi angles from two backbone vectors in consecutive peptide planes. These equations make it possible
to compute, exactly and in constant time, the backbone phi-psi angles for a residue from RDCs in two media on any
single backbone vector type. Building upon these exact solutions we have designed a novel algorithm for determining
a protein backbone substructure consisting of alpha-helices and beta-sheets. Our algorithm employs a systematic
search technique to refine the conformation of both alpha-helices and beta-sheets and to determine their orientations using
exclusively the angular restraints from RDCs. The algorithm computes the backbone substructure employing very
sparse distance restraints between pairs of alpha-helices and beta-sheets refined by the systematic search. The algorithm
has been demonstrated on the protein human ubiquitin using only backboneNH RDCs, plus twelve hydrogen bonds
and four NOE distance restraints. Further, our results show that both the global orientations and the conformations
of alpha-helices and beta-strands can be determined with high accuracy using only two RDCs per residue. The algorithm
requires, as its input, backbone resonance assignments, the identification of alpha-helices and beta-sheets as well as sparse
NOE distance and hydrogen bond restraints.