# Subdivision Surfaces

### Introduction

The most common way to model complex smooth surfaces is by using a patchwork of bicubic patches such as BSplines or NURBS.

However, while they do provide a reliable smooth limit surface definition, bi-cubic patch surfaces are limited to 2-dimensional topologies, which only describe a very small fraction of real-world shapes. This fundamental parametric limitation requires authoring tools to implement at least the following functionalities:

• smooth trimming
• seams stitching

Both trimming and stitching need to guarantee the smoothness of the model both spatially and temporally as the model is animated. Attempting to meet these requirements introduces a lot of expensive computations and complexity.

Subdivision surfaces on the other hand can represent arbitrary topologies, and therefore are not constrained by these difficulties.

### Arbitrary Topology

A subdivision surface, like a parametric surface, is described by its control mesh of points. The surface itself can approximate or interpolate this control mesh while being piecewise smooth. But where polygonal surfaces require large numbers of data points to approximate being smooth, a subdivision surface is smooth - meaning that polygonal artifacts are never present, no matter how the surface animates or how closely it is viewed.

Ordinary cubic B-spline surfaces are rectangular grids of tensor-product patches. Subdivision surfaces generalize these to control grids with arbitrary connectivity.

### Uniform Subdivision

Applies a uniform refinement scheme to the coarse faces of a mesh. The mesh converges closer to the limit surface with each iteration of the algorithm.

Applies a progressive refinement strategy to isolate irregular features. The resulting vertices can be assembled into bi-cubic patches defining the limit surface.

Feature adaptive refinement can be much more economical in terms of time and memory use, but the best method to use depends on application needs.

The following table identifies several factors to consider:

• Exponential geometry growth
• Geometry growth close to linear and occuring only in the neighborhood of isolated topological features
• Current implementation only produces bi-linear patches for uniform refinement
• Current implementation only produces bi-cubic patches for feature adaptive refinement
• All face-varying interpolation rules supported at refined vertex locations
• Currently, only bi-linear face-varying interpolation is supported for bi-cubic patches

Release Notes (3.0.0)

• Full support for bi-cubic face-varying interpolation is a significant feature which will be supported in future releases.
• Feature adaptive refinement for the Loop subdivision scheme is expected to be supported in future releases.

### Boundary Interpolation Rules

Boundary interpolation rules control how boundary edges and vertices are interpolated.

The following rule sets can be applied to vertex data interpolation:

Mode Behavior
VTX_BOUNDARY_NONE No boundary edge interpolation should occur; instead boundary faces are tagged as holes so that the boundary edge-chain continues to support the adjacent interior faces but is not considered to be part of the refined surface
VTX_BOUNDARY_EDGE_ONLY All the boundary edge-chains are sharp creases; boundary vertices are not affected
VTX_BOUNDARY_EDGE_AND_CORNER All the boundary edge-chains are sharp creases and boundary vertices with exactly one incident face are sharp corners

On a grid example:

### Face-Varying Interpolation Rules

Face-varying data is used when discontinuities are required in the data over the surface -- mostly commonly the seams between disjoint UV regions. Face-varying data can follow the same interpolation behavior as vertex data, or it can be constrained to interpolate linearly around selective features from corners, boundaries, or the entire interior of the mesh.

The following rules can be applied to face-varying data interpolation -- the ordering here applying progressively more linear constraints:

Mode Behavior
FVAR_LINEAR_NONE smooth everywhere the mesh is smooth
FVAR_LINEAR_CORNERS_ONLY sharpen (linearly interpolate) corners only
FVAR_LINEAR_CORNERS_PLUS1 CORNERS_ONLY + sharpening of junctions of 3 or more regions
FVAR_LINEAR_CORNERS_PLUS2 CORNERS_PLUS1 + sharpening of darts and concave corners
FVAR_LINEAR_BOUNDARIES linear interpolation along all boundary edges and corners
FVAR_LINEAR_ALL linear interpolation everywhere (boundaries and interior)

These rules cannot make the interpolation of the face-varying data smoother than that of the vertices. The presence of sharp features of the mesh created by sharpness values, boundary interpolation rules, or the subdivision scheme itself (e.g. Bilinear) take precedence.

All face-varying interpolation modes illustrated in UV space using the catmark_fvar_bound1 regression shape -- a simple 4x4 grid of quads segmented into three UV regions (their control point locations implied by interpolation in the FVAR_LINEAR_ALL case):

### Semi-Sharp Creases

It is possible to modify the subdivision rules to create piecewise smooth surfaces containing infinitely sharp features such as creases and corners. As a special case, surfaces can be made to interpolate their boundaries by tagging their boundary edges as sharp.

However, we've recognized that real world surfaces never really have infinitely sharp edges, especially when viewed sufficiently close. To this end, we've added the notion of semi-sharp creases, i.e. rounded creases of controllable sharpness. These allow you to create features that are more akin to fillets and blends. As you tag edges and edge chains as creases, you also supply a sharpness value that ranges from 0-10, with sharpness values >=10 treated as infinitely sharp.

It should be noted that infinitely sharp creases are really tangent discontinuities in the surface, implying that the geometric normals are also discontinuous there. Therefore, displacing along the normal will likely tear apart the surface along the crease. If you really want to displace a surface at a crease, it may be better to make the crease semi-sharp.

### Chaikin Rule

Chaikin's curve subdivision algorithm improves the appearance of multi-edge semi-sharp creases with varying weights. The Chaikin rule interpolates the sharpness of incident edges.

Mode Behavior
CREASE_UNIFORM Apply regular semi-sharp crease rules
CREASE_CHAIKIN Apply "Chaikin" semi-sharp crease rules

Example of contiguous semi-sharp creases interpolation:

### "Triangle Subdivision" Rule

The triangle subdivision rule is a rule added to the Catmull-Clark scheme that can be applied to all triangular faces; this rule was empirically determined to make triangles subdivide more smoothly. However, this rule breaks the nice property that two separate meshes can be joined seamlessly by overlapping their boundaries; i.e. when there are triangles at either boundary, it is impossible to join the meshes seamlessly

Mode Behavior
TRI_SUB_CATMARK Default Catmark scheme weights
TRI_SUB_SMOOTH "Smooth triangle" weights

Cylinder example :

### Manifold vs Non-Manifold Geometry

Continuous limit surfaces generally require that the topology be a two-dimensional manifold for the limit surface to be unambiguous. It is possible (and sometimes useful, if only temporarily) to model non-manifold geometry and so create surfaces whose limit is not as well-defined.

The following examples show typical cases of non-manifold topological configurations.

#### Non-Manifold Fan

This "fan" configuration shows an edge shared by 3 distinct faces.

With this configuration, it is unclear which face should contribute to the limit surface (assuming it is singular) as three of them share the same edge. Fan configurations are not limited to three incident faces: any configuration where an edge is shared by more than two faces incurs the same problem.

These and other regions involving non-manifold edges are dealt with by considering regions that are "locally manifold". Rather than a single limit surface through this problematic edge with its many incident faces, the edge locally partitions a single limit surface into more than one. So each of the three faces here will have their own (locally manifold) limit surface -- all of which meet at the shared edge.

#### Non-Manifold Disconnected Vertex

A vertex is disconnected from any edge and face.

This case is fairly trivial: there is a very clear limit surface for the four vertices and the face they define, but no possible way to exact a limit surface from the disconnected vertex.

While the vertex does not contribute to any limit surface, it may not be completely irrelevant though. Such vertices may be worth retaining during subdivision (if for no other reason than to preserve certain vertex ordering) and simply ignored when it comes time to consider the limit surface.