There have been recent interests in defining hyperbolic versions of the mean cur vature flow and other curvature flows for hypersurfaces, which are typically parabolic equations.An important feature of these curvature equations is the invariant prop erty under rigid motions.In generalizing these equations to the hyperbolic case, it is thus highly desirable to retain this feature.In this talk, we start with the non parametric curve flow case and apply the prolongation formula to investigate which corresponding PDEs are invariant under the Euclidean group.We then move on to look at convex hypersurfaces.A rather satisfactory family of equations was proposed which can be naturally interpreted from the point of view of support functions.In ad dition, short time existence is established and finite time singularity is characterized for these equations.