In hyperspectral image unmixing, a collection of pure spectra, the so-called endmembers, is identified and their abundance fractions are estimated at each pixel. While endmembers are often extracted using a geometric approach, the abundances are usually estimated by solving an inverse problem. In this paper, we bypass the problem of abundance estimation by using a geometric point of view. The proposed framework shows that a large number of endmember extraction techniques can be adapted to jointly estimate the abundance fractions, with no additional computational complexity. This is illustrated in this paper with the N-Findr, SGA, VCA, OSP, and ICE endmember extraction techniques. A nonlinear extension is also proposed, using non linear dime nsion reduction methods such as MDS, LLE and ISOMAP. These strategies maintain the geometric unmixing algorithms unchanged, for endmember extraction as well as abundance fraction estimation. The relevance of the proposed approach is illustrated through experiments on synthesized data and real hyperspectral image.