The optimization of cam mechanisms with translating flat-face followers, upon consideration of curvature constraints, is discussed here. In fact, when minimizing the size of the cam disk, cam profiles with cusps arising from discontinuities in the curvature of the profile, can appear. On the other hand, cusps are undesirable because they give rise to inadmissibly large contact stresses and hence, should be prevented from occurring. Conditions under which these discontinuities appear, although available in the literature, are not readily applicable, for they are highly problem-dependent. However, if both the follower-displacement program and the profile are determined based on the normal spline, a design tool introduced elsewhere, then readily applicable design formulae, leading to unconstrained optimum designs, are derived. As well, design charts are derived, which produce the cusp-free, minimum-size profile for a given displacement program. The same problem for oscillating flat-face followers is addressed in an accompanying paper.