In this article, we unify several recently developed analytic algorithms for spiral cone-beam computed tomography (CT), including both the filtered-backprojection algorithm and the backprojected-filtration algorithms in the cases of standard spiral, nonstandard spiral, and more general scanning loci. Using Tuy's inversion scheme, we give concise proofs of these reconstruction formulas for cone-beam CT. While a similar proof of the Katsevich algorithm was previously reported, our proof of the Zou-Pan algorithm is new. More importantly, our formulation is generally valid for nonstandard spiral loci and other curves, in agreement with another paper from our group. Furthermore, two sets of simulation results are presented, showing both filtered-backprojection reconstruction using asymmetric filtering lines and backprojected-filtration reconstruction using a saddle curve.