Construction of Navau and Garcia

Basic steps Construction has two parameters: smoothness k and n > k, defining how closely the surface follows the control mesh We consider the case k = 2 (C2 surfaces), and n = 3 The number of charts grows with n, so smaller is better for most applications Steps subdivide using Catmull-Clark 3 times define 1 quad chart for each regular vertex not adjacent to irregular, 2 quad charts for some adjacent, K quad charts for vertices of valence K define 1 basis function per chart assign a 1 control point per vertex

Step 1: Subdivide Subdivide 3 times (use Catmull-Clark) charts will (more or less) correspond to 2- neighborhoods of vertices in the refined mesh subdivide enough so that if charts overlap they can contain only one irregular vertex

Step 2: Define charts If 2-neighborhood has no irregular vertices simply map part of the mesh to a square

Step 2: Define charts Regular vertices close to irregular (within 2 ring) make a curved star mapping 4-nbhd of irregular vertex to the plane; irregular vertex splits into K. sweep the outer curve of the curved 2-nbhd to get a chart

Step 2: Define charts Regular vertices diagonal from irregular and irregular for these vertices, the outer boundary has multiple segments; use multiple charts

Step 3: basis functions One control point assigned per vertex several charts may share a control point similar to splines, weighted blend of control points Basis functions are tensor product splines of degree k+1, remapped to the charts Quad charts make the definition easy

Step 4: transition maps Transition maps are affine for charts containing irregular vertices (these charts always share the same star) affine for regular to regular chart polynomial for regular to irregular

Summary of properties smoothness: C k flexibility: unknown at irregular charts shape: all quads charts are associated with vertices, multiple charts for some vertices number of charts: large; approx. 4^(subdiv steps) * # of vertices for C 2, 3 subdiv. steps, i.e. 64* # of vertices flexibility: unknown embedding functions: constants (1 control point/chart) blending functions: tensor-product B-splines remapped transition functions: linear or polynomial of degree k+1 reduce to splines of degree k+1 for regular control meshes

Construction of Ying and Zorin

Charts and Transition Maps Building local geometry for one-ring neighborhood is easy How to blend them together smoothly ? The transition map must be smooth ? Smooth

Charts and Transition Maps Charts One for each vertex Map only depends on valence Transition maps Conformal, thus Easy to compute Alternatives transition map smooth

Local Geometry Maps Needs to be smooth and 3-flexible at center vertex Use monomials as basis for local geometry High order splines can also be used Needs to provide good visual quality Make it close to an existing surface with good quality Use Catmull-Clark subdivision surface

Local Geometry Maps Constraints: 3d positions Positions in chart Basis Monomials in chart Unknowns Monomial coefficients Least square fitting Catmull-Clark subdivision Pseudo-inverse can be pre-computed Dyadic points Fitting

Blending Functions Need to be a partition of unity Need to be smooth in each chart tensor product rotate and copy map

Evaluation Procedure Find correspondent positions in each chart Evaluate local geometry by local chart position Compute blending function in each chart Blend local geometry to get surface position

Local Parameterization Local parameterization around extraordinary vertex Magnitude of derivatives Our surface Catmull-Clark 1 st order2 nd order3 rd order

High-order smoothness Reflection lines Curvature behavior Catmull-Clark Our surface

Surface Examples

Summary of Features charts: star-shaped, depend on vertex valence one chart per vertex embedding functions: polynomials or splines blending functions: compositions of conformal and polynomial transition functions: conformal C 1 or C k smooth for a prescribed k At least 3-flexible at vertices Explicit local parameterizations exist for any point