This paper addresses the problem of image-based surface reconstruction. The main contribution is the computation of the exact derivative of the reprojection error functional. This allows its rigorous minimization via gradient descent surface evolution. The main difficulty has been to correctly take into account the visibility changes that occur when the surface moves. A geometric and analytical study of these changes is presented and used for the computation of derivative. Our analysis shows the strong influence that the movement of the contour generators has on the reprojection error. As a consequence, during the proper minimization of the reprojection error, the contour generators of the surface are automatically moved to their correct location in the images. Therefore, current methods adding additional silhouettes or apparent contour constraints to ensure this alignment can now be understood and justified by a single criterion: the reprojection error.