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We present an analytical calculation of the theoretical minimum emittance in storage rings with arbitrary but nonreversing bending magnets. Our derivation is based on a dipole with a short segment of constant bending radius and linear ramps at the ends, which has been shown to be very close to the optimal bending profile. The analytical results confirm and extend the previous ones obtained by numerical optimizations (except for a minor uncertainty on the profile for minimum effective emittance). Simple approximate formulas are given for calculating the theoretical minimum emittance and the parameters of the required bending profile and optics functions, which are of practical value for storage-ring designs. To facilitate designs of linear optics using the optimal bending profiles, we derive a closed expression for the transfer matrix of a linear-ramp dipole. Besides that, the minimum emittance theory is further refined, especially with more rigorous proof of the theory and parameter ranges.