Estimating Error Bounds and Subdivision Depths for Loop Subdivision Surfaces Zhangjin Huang Scbool of Electronic Engineering and Computer Science, Peking University,Beijing 100871,China 0ct,2007 Abstract We derive a bound on the maximal distanoe betwreen a Loop sub- division surfnce patch and its control mesh in terms of the maximum norm of the tnixed second dilferenoes of the control points and a con- stant that pemb only on the lenx of the pateh.A sulxlivisim dopth formmla is also proposed Keywords:Loop subdivision surface,control mesh.error bound,sub- division depth 1 Introduction A subdivision surfaoc is defined as the limit of a finer and finer control mesh by subdividing the mesh recursively.The Loop subdivision surface geteral izes the quartic three-directional box spline surface to triangular meshes of arbitrary topology [1]. Previons erroe estimation techniques for Loop suhdivision surfnces can be clasified into two classes. One is the vertex hnsed method 2,3,which measures the distance between the vertloes and their limit positions.Lanquetin et al.derived a wrong exponential bound and consequently a wrong subdivision depth formula [2].Wang et al.propoeed a proper exponential bound with an inefficient subdivision depth estimation technique 3 As pointed out in 3, the vertex basei bounds all sulfer Irom one proble:they iay be sanaller than the nctnnl distance in some enses. The other is the patch based method [4.5].which estimates the para- metric distanoe between a Loop surfnce patch and its control mesh.Wu ct aL.presented an accurate error measure [4,5].But their estimate is de- pendent on recursive subdivision,thus can not be used to pre-compute the Estimating Error Bounds and Subdivision Depths for Loop Subdivision Surfaces Zhangjin Huang School of Electronic Engineering and Computer Science, Peking University, Beijing 100871, China Oct., 2007 Abstract We derive a bound on the maximal distance between a Loop sub￾division surface patch and its control mesh in terms of the maximum norm of the mixed second differences of the control points and a con￾stant that depends only on the valence of the patch. A subdivision depth formula is also proposed. Keywords: Loop subdivision surface, control mesh, error bound, sub￾division depth 1 Introduction A subdivision surface is defined as the limit of a finer and finer control mesh by subdividing the mesh recursively. The Loop subdivision surface general￾izes the quartic three-directional box spline surface to triangular meshes of arbitrary topology [1]. Previous error estimation techniques for Loop subdivision surfaces can be classified into two classes. One is the vertex based method [2, 3], which measures the distance between the vertices and their limit positions. Lanquetin et al. derived a wrong exponential bound and consequently a wrong subdivision depth formula [2]. Wang et al. proposed a proper exponential bound with an inefficient subdivision depth estimation technique [3]. As pointed out in [3], the vertex based bounds all suffer from one problem: they may be smaller than the actual distance in some cases. The other is the patch based method [4, 5], which estimates the para￾metric distance between a Loop surface patch and its control mesh. Wu et al. presented an accurate error measure [4, 5]. But their estimate is de￾pendent on recursive subdivision, thus can not be used to pre-compute the 1