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At fixed radial comoving distance and fixed emission scale factor, open spatial geometry has transverse radius $\sinh (\kappa D_C)/\kappa >D_C$. Hence its angular diameter distance exceeds the flat value, and a standard object of fixed proper size hasThe same comparison can be expressed at fixed proper radial distance on the emission slice. A comparison at fixed redshift must additionally account for the different expansion histories in $D_C(z)$; curvature and expansion effects must not be silently conflated.

Known-size cosmological standard rulers with measured redshifts provide $D_A(z)$ through their angular sizes. The function depends on the spatial-curvature parameter directly and on matter, radiation or dark energy through the expansion history. For a comoving ruler such as a calibrated baryon acoustic oscillation scale, the transverse angular measurement determines $(1+z)D_A$. Angular acoustic scales in the cosmic microwave background similarly probe the ratio of a calibrated early-time length to a cosmological distance. A range of redshifts constrains parameter-dependent shapes and the overall distance scale, but one angular measurement cannot uniquely separate all density parameters. Ruler calibration, source evolution and lensing must be modeled, and independent expansion or distance information helps break the degeneracies.

For the transverse baryon acoustic oscillation ruler, $d_A=a\chi$ and the physical ruler length is $a{\chi }_d$. Its observed angle is thereforewhereThe unknown cosmological standard ruler length and $H_0$ occur only through the common amplitude $H_0{\chi }_d$. Measurements at two distinct redshifts givewhich generically determines the one shape parameter ${\Omega }_{m,0}$. Thus two redshifts suffice; additional redshifts overdetermine the model and improve precision.