Support Vector Machine (SVM) has become a very effective method in statistical machine learning and it has proved that training SMV is to solve Nearest Point pair Problem (NPP) between two disjoint closed convex sets. Later Keerthi pointed out that it is difficult to apply classical excellent geometric algorithms direcly to SVM and so designed a new geometric algorithm for SVM. In this article, a new algorithm for geometrically solving SVM, Kernel Projection Algorithm, is presented based on the theorem on fixed-points of projection mapping. This new algorithm makes it easy to apply classical geometric algorithms to solving SVM and is more understandable than Keerthi’s. Experiments show that the new algorithm can also handle large-scale SVM problems. Geometric algorithms for SVM, such as Keerthi’s algorithm, require that two closed convex sets be disjoint and otherwise the algorithms are meaningless. In this article, this requirement will be guaranteed in theory by using the theoretic result on universal kernel functions.