Many effective automatic surface simplification algorithms have been developed in order to produce approximations from a complex polygonal model. However, previous algorithms often have difficulty identifying and preserving important geometric features such as high-curvature regions when the model is simplified to a very low level of detail. In this paper, we define a new feature constraint quadric error metric and propose a user-controlled geometric feature preserving simplification that aims to improve this weakness. We apply quadric fitting method to estimate the parameters and the local geometry of the surface; then we present adaptive weighting of the quadric which is associated with each vertex, relying on the discrete differential-geometry properties of the vertex. Before the simplification process is executed, our algorithm requires users to specify several global parameters to control the quality of approximations and overall processing time, and then our algorithm can adaptively control the relative importance of different surface regions to implicitly reorder the edge collapses sequence. We also propose additional feature constraint quadrics to control the selection of new vertices' optimal positions for preserving geometric features and properties of the surface. This paper shows, with very little effort, our algorithm can preserve various geometric features of the original model and produce approximations with smaller differential-geometry errors than previous algorithms.