  The relationship between $L_p$ affine surface area and curvature measures is
investigated. As a result, a new representation of the existing notion of $L_p$
affine surface area depending only on curvature measures is derived. Direct
proofs of the equivalence between this new representation and those previously
known are provided. The proofs show that the new representation is, in a sense,
"polar" to that of Lutwak's and "dual" to that of Sch\"utt & Werner's.
