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Parametric scaling representations of the asymptotic equation of state, correlation lengths, and singular part of the Helmholtz free energy in the critical region of a system with a scalar order parameter are considered with the aim of fitting all 15 available independent universal amplitude ratios and describing naturally van der Waals loops. Defects in previous linear and cubic models and their extensions are described: they cannot represent analytically connected van der Waals loops or approximate satisfactorily all the known amplitude ratios. However, novel trigonometric parametric models for the free energy ͑and correlations͒ meet all the desiderata as demonstrated by the explicit numerical data presented.

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Parametric scaling representations are obtained and studied for the asymptotic behavior of interfacial tensions in the full neighborhood of a fluid (or Ising-type) critical endpoint, i.e., as a function both of temperature and of density/order parameter or chemical potential/ordering field. Accurate nonclassical critical exponents and reliable estimates for the universal amplitude ratios are included naturally on the basis of the "extended de Gennes-Fisher" local-functional theory. Serious defects in previous scaling treatments are rectified and complete wetting behavior is represented; however, quantitatively small, but unphysical residual nonanalyticities on the wetting side of the critical isotherm are smoothed out "manually." Comparisons with the limited available observations are presented elsewhere but the theory invites new, searching experiments and simulations, e.g., for the vapor-liquid interfacial tension on the two sides of the critical endpoint isotherm for which an amplitude ratio -3.25+/-0.05 is predicted.