Nonlinear spectral unmixing for hyperspectral remote sensing images can overcome the shortage of linear unmixing methods that failing in explaining the nonlinear mixing effect in more complex scenarios. Meanwhile, bilinear mixture models and their corresponding algorithms are the hot topic of related researches. A nonlinear spectral unmixing algorithm based on the geometric characteristics of bilinear mixture models was proposed. By representing the models’ nonlinear mixing terms as the linear contribution of one extra vertex concentrating the common nonlinear mixing effect, solving the complex bilinear mixture models was converted to do the simple linear spectral unmixing. Further, a traditional linear spectral unmixing algorithm was adopted to estimate the abundances directly in an iterative way. Experimental results on simulated and real hyperspectral images indicate that the proposed algorithm can overcome the collinearity effect and the adverse impact caused by fitting too many parameters, and improve both unmixing accuracy and computational speed.