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Copyright 2013 Pixar
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Licensed under the Apache License, Version 2.0 (the "Apache License")
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with the following modification; you may not use this file except in
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compliance with the Apache License and the following modification to it:
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Section 6. Trademarks. is deleted and replaced with:
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6. Trademarks. This License does not grant permission to use the trade
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names, trademarks, service marks, or product names of the Licensor
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and its affiliates, except as required to comply with Section 4(c) of
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the License and to reproduce the content of the NOTICE file.
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You may obtain a copy of the Apache License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the Apache License with the above modification is
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distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
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KIND, either express or implied. See the Apache License for the specific
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language governing permissions and limitations under the Apache License.
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Subdivision Surfaces
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--------------------
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.. contents::
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:local:
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:backlinks: none
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----
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Overview
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========
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Subdivision surfaces are a common modeling primitive that has gained popularity in
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animation and visual effects over the past decades.
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.. raw:: html
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<center>
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<p align="center">
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<IMG src="images/tetra.0.png" style="width: 20%;">
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<IMG src="images/tetra.1.png" style="width: 20%;">
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<IMG src="images/tetra.2.png" style="width: 20%;">
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<IMG src="images/tetra.3.png" style="width: 20%;">
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</p>
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</center>
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As the name suggests, subdivision surfaces are fundamentally *surfaces*.
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More specifically, subdivision surfaces are *piecewise parametric surfaces* defined over
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meshes of *arbitrary topology* -- both concepts that will be described in the sections
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that follow.
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Subdivision is both an operation that can be applied to a polygonal mesh to refine it, and
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a mathematical tool that defines the underlying smooth surface to which repeated subdivision
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of the mesh converges. Explicit subdivision is simple to apply some number of
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times to provide a smoother mesh, and that simplicity has historically lead to many tools
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representing the shape this way. In contrast, deriving the smooth surface that ultimately
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defines the shape -- its "limit surface" -- is considerably more complex but provides greater
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accuracy and flexibility. These differences have led to confusion in how some tools
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expose subdivision surfaces.
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The ultimate goal is to have all tools use subdivision surfaces as true surface primitives.
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The focus here is therefore less on subdivision and more on the nature
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of the surface that results from it. In addition to providing a consistent implementation of
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subdivision -- one that includes a number of widely used feature extensions -- a significant
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value of OpenSubdiv is that it makes the limit surface more accessible.
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Since its introduction, OpenSubdiv has received interest from users and developers
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with a wide variety of skills, interests and backgrounds. This document is
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intended to present subdivision surfaces from a perspective helpful in making use
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of OpenSubdiv. One purpose it serves is to provide a high level overview for those
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with less experience with the algorithms or mathematics of subdivision. The other
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is to provide an overview of the feature set available with OpenSubdiv, and to
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introduce those capabilities with the terminology used by OpenSubdiv (as much of it
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is overloaded).
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----
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Piecewise Parametric Surfaces
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=============================
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Piecewise parametric surfaces are arguably the most widely used geometric representation
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in industrial design, entertainment and many other areas. Many of the objects we deal
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with everyday -- cars, mobile phones, laptops -- were all designed and visualized first
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as piecewise parametric surfaces before those designs were approved and pursued.
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Piecewise parametric surfaces are ultimately just collections of simpler modeling primitives
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referred to as patches. Patches constitute the "pieces" of the larger surface in much the
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same way as a face or polygon constitutes a piece of a polygonal mesh.
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----
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Parametric Patches
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******************
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Patches are the building blocks of piecewise smooth surfaces, and many different kinds of
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patches have evolved to meet the needs of geometric modeling. Two of the more effective
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and common patches are illustrated below:
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+--------------------------------------+--------------------------------------+
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| .. image:: images/patch_bspline.jpg | .. image:: images/patch_bezier.jpg |
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| :align: center | :align: center |
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| :width: 95% | :width: 95% |
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| :target: images/patch_bspline.jpg | :target: images/patch_bezier.jpg |
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| Single bicubic B-Spline patch | Single bicubic Bezier patch |
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+--------------------------------------+--------------------------------------+
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Patches consist of a set of points or vertices that affect a rectangular piece of smooth
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surface (triangular patches also exist). That rectangle is "parameterized" in its two
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directions, transforming a simple 2D rectangle into the 3D surface:
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+--------------------------------------+--------------------------------------+
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| .. image:: images/param_uv.png | .. image:: images/param_uv2xyz.png |
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| :align: center | :align: center |
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| :width: 80% | :width: 80% |
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| :target: images/param_uv.png | :target: images/param_uv2xyz.png |
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| (u,v) 2D domain of a patch | Mapping from (u,v) to (x,y,z) |
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+--------------------------------------+--------------------------------------+
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The points that control the shape of the surface are usually referred to as control
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points or control vertices, and the collection of the entire set defining a patch as
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the control mesh, the control hull, the control cage or simply the hull, the cage,
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etc. For the sake of brevity we will frequently use the term "cage", which serves us
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more generally later.
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So a patch essentially consist of two entities: its control points and the surface
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affected by them.
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The way the control points affect the surface is what makes the different types of
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patches unique. Even patches defined by the same number of points can have different
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behavior. Note that all 16 points of the B-Spline patch above are relatively far from
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the surface they define compared to the similar Bezier patch. The two patches in
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that example actually represent exactly the same piece of surface -- each with a set
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of control points having different effects on it. In mathematical terms, each control
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point has a "basis function" associated with it that affects the surface in a particular
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way when only that point is moved. It is these basis functions that often give rise
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to the names of the different patches.
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There are pros and cons to these different properties of the control points of patches,
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which become more apparent as we assemble patches into piecewise surfaces.
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----
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Piecewise Surfaces
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******************
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Piecewise parametric surfaces are collections of patches.
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For rectangular patches, one of the simplest ways to construct a collection is to define
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a set of patches using a rectangular grid of control points:
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+----------------------------------------+----------------------------------------+
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| .. image:: images/surface_bspline.jpg | .. image:: images/surface_bezier.jpg |
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| :align: center | :align: center |
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| :width: 95% | :width: 95% |
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| :target: images/surface_bspline.jpg | :target: images/surface_bezier.jpg |
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| Piecewise B-Spline surface | Piecewise Bezier surface |
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+----------------------------------------+----------------------------------------+
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Note that we can overlap the points of adjacent B-spline patches. This overlapping
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means that moving one control point affects multiple patches -- but it also ensures
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that those patches always meet smoothly (this was a design intention and not true
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for other patch types). Adjacent Bezier patches only share points at their boundaries
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and coordinating the points across those boundaries to keep the surface smooth is
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possible, but awkward. This makes B-splines a more favorable surface representation
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for interactive modeling, but Bezier patches serve many other useful purposes.
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A more complicated B-spline surface:
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+------------------------------------------------+
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| .. image:: images/surface_bspline_complex.jpg |
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| :align: center |
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| :width: 95% |
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| :target: images/surface_bspline_complex.jpg |
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| |
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| Part of a more complicated B-Spline surface |
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+------------------------------------------------+
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Just as a patch consisted of a cage and a surface, the same is now true of the
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collection. The control cage is manipulated by a designer and the surface of each
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of the patches involved is displayed so they can assess its effect.
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----
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Arbitrary Topology
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==================
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Piecewise surfaces discussed thus far have been restricted to collections of patches
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over regular grids of control points. There is a certain simplicity with rectangular
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parametric surfaces that is appealing, but a surface representation that supports
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arbitrary topology has many other advantages.
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Rectangular parametric surfaces gained widespread adoption despite their topological
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limitations, and their popularity continues today in some areas. Complex objects often
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need many such surfaces to represent them and a variety of techniques have evolved to
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assemble them effectively, including "stitching" multiple surfaces together or cutting
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holes into them ("trimming"). These are complicated techniques, and while effective in
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some contexts (e.g. industrial design) they become cumbersome in others (e.g. animation
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and visual effects).
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A single polygonal mesh can represent shapes with far more complexity than a single
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rectangular piecewise surface, but its faceted nature eventually becomes a problem.
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Subdivision surfaces combine the topological flexibility of polygonal meshes with the
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underlying smoothness of piecewise parametric surfaces. Just as rectangular piecewise
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parametric surfaces have a collection of control points (its cage stored as a grid)
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and an underlying surface, subdivision surfaces also have a collection of control points
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(its cage stored as a mesh) and an underlying surface (often referred as its "limit
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surface").
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----
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Regular versus Irregular Features
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*********************************
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A mesh contains the vertices and faces that form the cage for the underlying
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surface, and the topology of that mesh can be arbitrarily complex.
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In areas where the faces and vertices of the mesh are connected to form rectangular
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grids, the limit surface becomes one of the rectangular piecewise parametric
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surfaces previously mentioned. These regions of the mesh are said to be "regular":
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they provide behavior familiar from the use of similar rectangular surfaces and
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their limit surface is relatively simple to deal with. All other areas are
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considered "irregular": they provide the desired topological flexibility and so
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are less familiar (and less predictable in some cases) and their limit surface
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can be much more complicated.
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Irregular features come in a number of forms. The most widely referred to is
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an extra-ordinary vertex, i.e. a vertex which, in the case of a quad subdivision
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scheme like Catmull-Clark, does not have four incident faces.
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+-------------------------------------+-------------------------------------+
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| .. image:: images/val6_cage.jpg | .. image:: images/val6_surface.jpg |
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| :align: center | :align: center |
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| :width: 95% | :width: 95% |
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| :target: images/val6_cage.jpg | :target: images/val6_surface.jpg |
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| Irregular vertex and incident | Regular and irregular regions of |
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| faces | the surface |
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+-------------------------------------+-------------------------------------+
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The presence of these irregular features makes the limit surface around them
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similarly irregular, i.e. it cannot be represented as simply as it can for regular
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regions.
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.. Explicit target to preserve external links:
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.. _feature-adaptive-subdivision:
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It's worth noting that irregular regions shrink in size and become more "isolated"
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as subdivision is applied. A face with a lot of extra-ordinary vertices around it
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makes for a very complicated surface, and isolating these features is a way to
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help deal with that complexity:
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+--------------------------------------+--------------------------------------+--------------------------------------+
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| .. image:: images/val5_iso_cage.jpg | .. image:: images/val5_iso_sub1.jpg | .. image:: images/val5_iso_sub2.jpg |
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| :align: center | :align: center | :align: center |
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| :width: 95% | :width: 95% | :width: 95% |
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| :target: images/val5_iso_cage.jpg | :target: images/val5_iso_sub1.jpg | :target: images/val5_iso_sub2.jpg |
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| Two valence-5 vertices nearby | Isolation subdivided once | Isolation subdivided twice |
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+--------------------------------------+--------------------------------------+--------------------------------------+
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It's generally necessary to perform some kind of local subdivision in these areas
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to break these pieces of surface into smaller, more manageable pieces, and the
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term "feature adaptive subdivision" has become popular in recent years to describe
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this process. Whether this is done explicitly or implicitly, globally or locally,
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what matters most is that there is an underlying piece of limit surface for each
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face -- albeit a potentially complicated one at an irregular feature -- that can
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be evaluated in much the same way as rectangular piecewise surfaces.
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+---------------------------------------+---------------------------------------+
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| .. image:: images/val6_regular.jpg | .. image:: images/val6_irregular.jpg |
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| :align: center | :align: center |
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| :width: 95% | :width: 95% |
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| :target: images/val6_regular.jpg | :target: images/val6_irregular.jpg |
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| Patches of the regular regions | Patches of the irregular region |
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+---------------------------------------+---------------------------------------+
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While supporting a smooth surface in these irregular areas is the main advantage
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of subdivision surfaces, both the complexity of the resulting surfaces and their
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quality are reasons to use them with care. When the topology is largely irregular,
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there is a higher cost associated with its surface, so minimizing irregularities
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is advantageous. And in some cases the surface quality, i.e. the perceived
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smoothness, of the irregular surfaces can lead to undesirable artefacts.
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An arbitrary polygonal mesh will often not make a good subdivision cage, regardless
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of how good that polygonal mesh appears.
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As with rectangular piecewise parametric surfaces, the cage should be shaped to
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affect the underlying surface it is intended to represent. See <Modeling Tips> for
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related recommendations.
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----
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Non-manifold Topology
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*********************
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Since the cage of a subdivision surface is stored in a mesh, and often
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manipulated in the same context as polygonal meshes, the topic of manifold
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versus non-manifold topology warrants some attention.
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There are many definitions or descriptions of what distinguishes a manifold
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mesh from one that is not.
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These range from concise but abstract mathematical definitions to sets of
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examples showing manifold and non-manifold meshes -- all have their value
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and an appropriate audience.
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The following is not a strict definition but serves
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well to illustrate most local topological configurations that cause a mesh
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to be non-manifold.
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Consider "standing" on the faces of a mesh and "walking" around each vertex
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in turn. Assuming a right-hand winding order of faces, stand on the side of
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the face in the positive normal direction. And when walking, step across each
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incident edge in a counter-clockwise direction to the next incident face.
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For an interior vertex:
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.. image:: images/walk_interior.png
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:align: center
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:width: 80%
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:target: images/walk_interior.png
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* start at the corner of any incident face
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* walk around the vertex across each incident edge to the next unvisited face; repeat
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* if you arrive back where you started and any incident faces or edges were not visited,
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the mesh is non-manifold
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Similarly, for a boundary vertex:
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.. image:: images/walk_boundary.png
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:align: center
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:width: 80%
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:target: images/walk_boundary.png
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* start at the corner of the face containing the leading boundary edge
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* walk around the vertex across each incident edge to the next unvisited face; repeat
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* if you arrive at another boundary edge and any incident faces or edges were not visited,
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the mesh is non-manifold
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If you can walk around all vertices this way and don't encounter any non-manifold
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features, the mesh is likely manifold.
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Obviously if a vertex has no faces,
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there is nothing to walk around and this test can't succeed, so it is again
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non-manifold. All of the faces around a vertex should also be in the same
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orientation, otherwise two adjacent faces have normals in opposite directions
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and the mesh will be considered non-manifold, so we should really include that
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constraint when stepping to the next face to be more strict.
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Consider walking around the indicated vertices of the following non-manifold meshes:
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+-----------------------------------------+-----------------------------------------+
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| .. image:: images/nonman_fan_cage.jpg | .. image:: images/nonman_vert_cage.jpg |
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| :align: center | :align: center |
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| :width: 95% | :width: 95% |
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| :target: images/nonman_fan_cage.jpg | :target: images/nonman_vert_cage.jpg |
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| Edges with > 2 incident faces | Faces sharing a vertex but no edges |
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+-----------------------------------------+-----------------------------------------+
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As mentioned earlier, many tools do not support non-manifold meshes, and in
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some contexts, e.g. 3D printing, they should be strictly avoided. Sometimes
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a manifold mesh may be desired and enforced as an end result, but the mesh
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may temporarily become non-manifold due to a particular sequence of modeling
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operations.
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Rather than supporting or advocating the use of non-manifold meshes, OpenSubdiv
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strives to be robust in the presence of non-manifold features to simplify the
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usage of its clients -- sparing them the need for topological analysis to
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determine when OpenSubdiv can or cannot be used. Although subdivision rules
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are not as well standardized in areas where the mesh is not manifold, OpenSubdiv
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provides simple rules and a reasonable limit surface in most cases.
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+--------------------------------------------+--------------------------------------------+
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| .. image:: images/nonman_fan_surface.jpg | .. image:: images/nonman_vert_surface.jpg |
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| :align: center | :align: center |
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| :width: 95% | :width: 95% |
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| :target: images/nonman_fan_surface.jpg | :target: images/nonman_vert_surface.jpg |
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| Surface around edges with > 2 incident | Surface for faces sharing a vertex but no |
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| faces | edges |
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+--------------------------------------------+--------------------------------------------+
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As with the case of regular versus irregular features, since every face has a
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corresponding piece of surface associated with it -- whether locally manifold or
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not -- the term "arbitrary topology" can be made to include non-manifold topology.
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----
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Subdivision versus Tessellation
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===============================
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The preceding sections illustrate subdivision surfaces as piecewise parametric surfaces of
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arbitrary topology. As piecewise parametric surfaces, they consist of a cage and the
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underlying surface defined by that cage.
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Two techniques used to display subdivision surfaces are subdivision and tessellation.
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Both have their legitimate uses, but there is an important distinction between them:
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* **subdivision** operates on a **cage** and produces a refined **cage**
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* **tessellation** operates on a **surface** and produces a discretization of that **surface**
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The existence and relative simplicity of the subdivision algorithm makes it easy to
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apply repeatedly to approximate the shape of the surface, but with the result being
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a refined cage, that approximation is not always very accurate. When compared to a
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cage refined to a different level, or a tessellation that uses points evaluated directly
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on the limit surface, the discrepancies can be confusing.
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Subdivision
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***********
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Subdivision is the process that gives "subdivision surfaces" their name, but it is not
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unique to them. Being piecewise parametric surfaces, let's first look at subdivision in
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the context of the simpler parametric patches that comprise them.
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Subdivision is a special case of *refinement*, which is key to the success of some of the
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most widely used types of parametric patches and their aggregate surfaces. A surface can
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be "refined" when an algorithm exists such that more control points can be introduced
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*while keeping the shape of the surface exactly the same*. For interactive and design
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purposes, this allows a designer to introduce more resolution for finer control without
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introducing undesired side effects in the shape. For more analytical purposes, it allows
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the surface to be broken into pieces, often adaptively, while being faithful to the
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original shape.
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One reason why both B-spline and Bezier patches are so widely used is that both of them
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can be refined. Uniform subdivision -- the process of splitting each of the patches
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in one or both of its directions -- is a special case of refinement that both of
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these patch types support:
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+---------------------------------------------+---------------------------------------------+---------------------------------------------+
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| .. image:: images/surface_bspline_cage.jpg | .. image:: images/surface_bspline_sub1.jpg | .. image:: images/surface_bspline_sub2.jpg |
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| :align: center | :align: center | :align: center |
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| :width: 95% | :width: 95% | :width: 95% |
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| :target: images/surface_bspline_cage.jpg | :target: images/surface_bspline_sub1.jpg | :target: images/surface_bspline_sub2.jpg |
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| | | |
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| B-Spline surface and its cage | Cage subdivided 1x | Cage subdivided 2x |
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+---------------------------------------------+---------------------------------------------+---------------------------------------------+
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In the cases illustrated above for B-Splines, the uniformly refined cages produce the same
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limit surface as the original (granted in more pieces). So it is fair to say that both
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uniform B-splines and Bezier surfaces are subdivision surfaces.
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The limit surface remains the same with the many more control points (roughly 4x with each
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iteration of subdivision), and those points are closer to (but not on) the surface. It
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may be tempting to use these new control points to represent the surface, but using the same
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number of points evaluated at corresponding uniformly spaced parametric locations on the
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|
surface is usually simpler and more effective.
|
|
|
|
Note also that points of the cage typically do not have any normal vectors associated with
|
|
them, though we can evaluate normals explicitly for arbitrary locations on the surface just
|
|
as we do for position. So if displaying a cage as a shaded surface, normal vectors at each
|
|
of the control points must be contrived. Both the positions and normals of the points on
|
|
the finer cage are therefore both approximations.
|
|
|
|
For more general subdivision surfaces, the same is true. Subdivision will refine a mesh of
|
|
arbitrary topology, but the resulting points will not lie on the limit surface and any normal
|
|
vectors contrived from and associated with these points will only be approximations to those
|
|
of the limit surface.
|
|
|
|
Tessellation
|
|
************
|
|
|
|
There is little need to use subdivision to approximate a parametric surface when it can be
|
|
computed directly, i.e. it can be tessellated. We can evaluate at arbitrary locations on the
|
|
surface and connect the resulting points to form a tessellation -- a discretization of the
|
|
limit surface -- that is far more flexible than the results achieved from uniform subdivision:
|
|
|
|
+----------------------------------------------+----------------------------------------------+
|
|
| .. image:: images/surface_bspline_tess1.jpg | .. image:: images/surface_bspline_tess2.jpg |
|
|
| :align: center | :align: center |
|
|
| :width: 95% | :width: 95% |
|
|
| :target: images/surface_bspline_tess1.jpg | :target: images/surface_bspline_tess2.jpg |
|
|
| | |
|
|
| Uniform tessellation of B-spline surface | Curvature-adaptive tessellation of B-spline |
|
|
| | surface |
|
|
+----------------------------------------------+----------------------------------------------+
|
|
|
|
For a simple parametric surface, the direct evaluation of the limit surface is also simple,
|
|
but for more complicated subdivision surfaces of arbitrary topology, this is less the case.
|
|
The lack of a clear understanding of the relationship between the limit surface and the
|
|
cage has historically lead to many applications avoiding tessellation.
|
|
|
|
It's worth mentioning that subdivision can be used to generate a tessellation even when the
|
|
limit surface is not available for direct evaluation. The recursive nature of subdivision
|
|
does give rise to formulae that allow a point on the limit surface to be computed that
|
|
corresponds to each point of the cage. This process is often referred to as "snapping"
|
|
or "pushing" the points of the cage onto the limit surface.
|
|
|
|
+--------------------------------------------+--------------------------------------------+
|
|
| .. image:: images/tess_snap1.jpg | .. image:: images/tess_snap2.jpg |
|
|
| :align: center | :align: center |
|
|
| :width: 95% | :width: 95% |
|
|
| :target: images/tess_snap1.jpg | :target: images/tess_snap2.jpg |
|
|
| | |
|
|
| Subdivided 1x and snapped to limit surface | Subdivided 2x and snapped to limit surface |
|
|
+--------------------------------------------+--------------------------------------------+
|
|
|
|
Since the end result is a
|
|
connected set of points on the limit surface, this forms a tessellation of the limit
|
|
surface, and we consider it a separate process to subdivision (though it does make use
|
|
of it). The fact that such a tessellation might have been achieved using subdivision is
|
|
indistinguishable from the final result -- the same tessellation might just as easily have
|
|
been generated by evaluating limit patches of the cage uniformly 2x, 4x, 8x, etc. along
|
|
each edge.
|
|
|
|
|
|
Which to Use?
|
|
*************
|
|
|
|
Subdivision is undeniably useful in creating finer cages to manipulate the surface,
|
|
but tessellation is preferred for displaying the surface when the patches are available
|
|
for direct evaluation. There was a time when global refinement was pursued in limited
|
|
circles as a way of rapidly evaluating parametric surfaces along isoparametric lines,
|
|
but patch evaluation, i.e. tessellation, generally prevails.
|
|
|
|
Considerable confusion has arisen due the way the two techniques have been employed and
|
|
presented when displaying the shape in end-user applications. One can argue that if an
|
|
application displays a representation of the surface that is satisfactory for its
|
|
purposes, then it is not necessary to burden the user with additional terminology and
|
|
choices. But when two representations of the same surface differ considerably between
|
|
two applications, the lack of any explanation or control leads to confusion.
|
|
|
|
As long as applications make different choices on how to display the surface, we seek a
|
|
balance between simplicity and control. Since subdivided points do not lie on the limit
|
|
surface, it is important to make it clear to users when subdivision is being used instead
|
|
of tessellation. This is particularly true in applications where the cage and the
|
|
surface are displayed in the same style as there is no visual cue for users to make that
|
|
distinction.
|
|
|
|
----
|
|
|
|
Mesh Data and Topology
|
|
======================
|
|
|
|
The ability of subdivision surfaces to support arbitrary topology leads to the use of
|
|
meshes to store both the topology of the cage and the data values associated with its
|
|
control points, i.e. its vertices. The shape of a mesh, or the subdivision surface
|
|
that results from it, is a combination of the topology of the mesh and the position
|
|
data associated with its vertices.
|
|
|
|
.. image:: images/data_top_shape.png
|
|
:align: center
|
|
:width: 90%
|
|
:target: images/data_top_shape.png
|
|
|
|
When dealing with meshes there are advantages to separating the topology from the data,
|
|
and this is even more important when dealing with subdivision surfaces. The "shape"
|
|
referred to above is not just the shape of the mesh (the cage in this case) but could
|
|
be the shape of a refined cage or the limit surface. By observing the roles that both
|
|
the data and topology play in operations such as subdivision and evaluation, significant
|
|
advantages can be gained by managing data, topology and the associated computations
|
|
accordingly.
|
|
|
|
While the main purpose of subdivision surfaces is to use position data associated with
|
|
the vertices to define a smooth, continuous limit surface, there are many cases where
|
|
non-positional data is associated with a mesh. That data may often be interpolated
|
|
smoothly like position, but often it is preferred to interpolate it linearly or even
|
|
make it discontinuous along edges of the mesh. Texture coordinates and color are common
|
|
examples here.
|
|
|
|
Other than position, which is assigned to and associated with vertices, there are no
|
|
constraints on how arbitrary data can or should be associated or interpolated. Texture
|
|
coordinates, for example, can be assigned to create a completely smooth limit surface
|
|
like the position, linearly interpolated across faces, or even made discontinuous between
|
|
them. There are, however, consequences to consider -- both in terms of data management
|
|
and performance -- which are described below as the terminology and techniques used to
|
|
achieve each are defined.
|
|
|
|
----
|
|
|
|
Separating Data from Topology
|
|
*****************************
|
|
|
|
While the topology of meshes used to store subdivision surfaces is arbitrarily complex
|
|
and variable, the topology of the parametric patches that make up its limit surface are
|
|
simple and fixed. Bicubic B-Spline and Bezier patches are both defined by a simple 4x4
|
|
grid of control points and a set of basis functions for each point that collectively
|
|
form the resulting surface.
|
|
|
|
For such a patch, the position at a given parametric location is the result of the
|
|
combination of position data associated with its control points and the weights of the
|
|
corresponding basis functions (*weights* being the values of basis functions evaluated
|
|
at a parametric location). The topology and the basis functions remain the same, so we
|
|
can make use of the weights independent of the data. If the positions of the control
|
|
points change, we can simply recombine the new position data with the weights that we
|
|
just used and apply the same combination.
|
|
|
|
+----------------------------------------+----------------------------------------+----------------------------------------+
|
|
| .. image:: images/data_patch_top.png | .. image:: images/data_patch_1.jpg | .. image:: images/data_patch_2.jpg |
|
|
| :align: center | :align: center | :align: center |
|
|
| :width: 70% | :width: 95% | :width: 95% |
|
|
| :target: images/data_patch_top.png | :target: images/data_patch_1.jpg | :target: images/data_patch_2.jpg |
|
|
+----------------------------------------+----------------------------------------+----------------------------------------+
|
|
| The fixed topology of a parametric patch and two shapes resulting from two sets of positions. |
|
|
+----------------------------------------+----------------------------------------+----------------------------------------+
|
|
|
|
Similarly, for a piecewise surface, the position at a given parametric location is the
|
|
result of the single patch containing that parametric location evaluated at the given
|
|
position. The control points involved are the subset of control points associated with
|
|
that particular patch. If the topology of the surface is fixed, so too is the topology
|
|
of the collection of patches that comprise that surface. If the positions of those
|
|
control points change, we can recombine the new position data with the same weights for
|
|
the subset of points associated with the patch.
|
|
|
|
+----------------------------------------+----------------------------------------+----------------------------------------+
|
|
| .. image:: images/data_mesh_top.png | .. image:: images/data_mesh_1.jpg | .. image:: images/data_mesh_2.jpg |
|
|
| :align: center | :align: center | :align: center |
|
|
| :width: 70% | :width: 95% | :width: 95% |
|
|
| :target: images/data_mesh_top.png | :target: images/data_mesh_1.jpg | :target: images/data_mesh_2.jpg |
|
|
+----------------------------------------+----------------------------------------+----------------------------------------+
|
|
| More complex but fixed topology of a surface and two shapes resulting from two sets of positions. |
|
|
+----------------------------------------+----------------------------------------+----------------------------------------+
|
|
|
|
This holds for a piecewise surface of arbitrary topology. Regardless of how complex
|
|
the topology, as long as it remains fixed (i.e. relationships between vertices, edges
|
|
and faces does not change (or anything other settings affecting subdivision rules)),
|
|
the same techniques apply.
|
|
|
|
This is just one example of the value of separating computations involving topology from
|
|
those involving the data. Both subdivision and evaluation can be factored into steps
|
|
involving topology (computing the weights) and combining the data separately.
|
|
|
|
+---------------------------------------+---------------------------------------+---------------------------------------+
|
|
| .. image:: images/data_pose_1.jpg | .. image:: images/data_pose_2.jpg | .. image:: images/data_pose_3.jpg |
|
|
| :align: center | :align: center | :align: center |
|
|
| :width: 95% | :width: 95% | :width: 95% |
|
|
| :target: images/data_pose_1.jpg | :target: images/data_pose_2.jpg | :target: images/data_pose_3.jpg |
|
|
+---------------------------------------+---------------------------------------+---------------------------------------+
|
|
| Three shapes resulting from three sets of positions for the a mesh of complex but fixed topology. |
|
|
| (currently stand-in images until we have an animated character approved for publication) |
|
|
+---------------------------------------+---------------------------------------+---------------------------------------+
|
|
|
|
When the topology is fixed, enormous savings are possible by pre-computing information
|
|
associated with the topology and organizing the data associated with the control points in
|
|
a way that can be efficiently combined with it. This is key to understanding some of
|
|
the techniques used to process subdivision surfaces.
|
|
|
|
For a mesh of arbitrary topology, the control points of the underlying surface are the
|
|
vertices, and position data associated with them is most familiar. But there is nothing
|
|
that requires that the control points of a patch have to represent position -- the same
|
|
techniques apply regardless of the type of data involved.
|
|
|
|
----
|
|
|
|
Vertex and Varying Data
|
|
***********************
|
|
|
|
The most typical and fundamental operation is to evaluate a position on the surface, i.e.
|
|
evaluate the underlying patches of the limit surface using the (x,y,z) positions at the
|
|
vertices of the mesh. Given a parametric (u,v) location on one such patch, the data-independent
|
|
evaluation method first computes the weights and then combines the (x,y,z) vertex positions
|
|
resulting in an (x,y,z) position at that location. But the weights and their combination
|
|
can be applied to any data at the vertices, e.g. color, texture coordinates or anything
|
|
else.
|
|
|
|
Data associated with the vertices that is interpolated this way, including position, is said
|
|
to be "vertex" data or to have "vertex" interpolation. Specifying other data as "vertex"
|
|
data will result in it being smoothly interpolated in exactly the same way (using exactly the
|
|
same weights) as the position. So to capture a simple 2D projection of the surface for
|
|
texture coordinates, 2D values matching the (x,y) of the positions would be used.
|
|
|
|
If linear interpolation of data associated with vertices is desired instead, the data is said
|
|
to be "varying" data or to have "varying" interpolation. Here the non-linear evaluation of
|
|
the patches defining the smooth limit surface is ignored and weights for simple linear
|
|
interpolation are used. This is a common choice for texture coordinates as evaluation of
|
|
texture without the need of bicubic patches is computationally cheaper. The linear
|
|
interpolation will not capture the smoothness required of a true projection between the
|
|
vertices, but both vertex and varying interpolation have their uses.
|
|
|
|
+------------------------------------------+------------------------------------------+
|
|
| .. image:: images/data_vertex_uv.jpg | .. image:: images/data_varying_uv.jpg |
|
|
| :align: center | :align: center |
|
|
| :width: 95% | :width: 95% |
|
|
| :target: images/data_vertex_uv.jpg | :target: images/data_varying_uv.jpg |
|
|
| | |
|
|
| Projected texture smoothly interpolated | Projected texture linearly interpolated |
|
|
| from vertex data | from varying data |
|
|
+------------------------------------------+------------------------------------------+
|
|
|
|
Since both vertex and varying data is associated with vertices (a unique value assigned
|
|
to each), the resulting surface will be continuous -- piecewise smooth in the case of
|
|
vertex data and piecewise linear in the case of varying.
|
|
|
|
----
|
|
|
|
Face-Varying Data and Topology
|
|
******************************
|
|
|
|
In order to support discontinuities in data on the surface, unlike vertex and varying data,
|
|
there must be multiple values associated with vertices, edges and/or faces, in order for
|
|
a discontinuity to exist.
|
|
|
|
Discontinuities are made possible by assigning values to the corners of faces, similar
|
|
to the way in which vertices are assigned to the corners of faces when defining the
|
|
topology of the mesh. Recalling the assignment of vertices to faces:
|
|
|
|
.. image:: images/data_top_vertex.png
|
|
:align: center
|
|
:width: 90%
|
|
:target: images/data_top_vertex.png
|
|
|
|
Vertex indices are assigned to all corners of each face as part of mesh construction and
|
|
are often referred to as the face-vertices of an individual face or the mesh. All
|
|
face-vertices that share the same vertex index will be connected by that vertex and share
|
|
the same vertex data associated with it.
|
|
|
|
By assigning a different set of indices to the face-vertices -- indices not referring to
|
|
the vertices but some set of data to be associated with the corners of each face -- corners
|
|
that share the same vertex no longer need to share the same data value and the data can be
|
|
made discontinuous between faces:
|
|
|
|
.. image:: images/data_top_fvary.png
|
|
:align: center
|
|
:width: 90%
|
|
:target: images/data_top_fvary.png
|
|
|
|
This method of associating data values with the face-vertices of the mesh is said to be
|
|
assigning "face-varying" data for "face-varying" interpolation. An interpolated value
|
|
will vary continuously within a face (i.e. the patch of the limit surface associated
|
|
with the face) but not necessarily across the edges or vertices shared with adjacent
|
|
faces.
|
|
|
|
+---------------------------------------------------------------+
|
|
| .. image:: images/data_fvar_xyz.jpg |
|
|
| :align: center |
|
|
| :width: 60% |
|
|
| :target: images/data_fvar_xyz.jpg |
|
|
| |
|
|
| Disjoint face-varying UV regions applied to the limit surface |
|
|
+---------------------------------------------------------------+
|
|
|
|
The combination of associating data values not with the vertices (the control points)
|
|
but the face corners, and the resulting data-dependent discontinuities that result,
|
|
make this a considerably more complicated approach than vertex or varying. The added
|
|
complexity of the data alone is reason to only use it when necessary, i.e. when
|
|
discontinuities are desired and present.
|
|
|
|
Part of the complexity of dealing with face-varying data and interpolation is the way in
|
|
which the interpolation behavior can be defined. Where the data is continuous, the
|
|
interpolation can be specified to be as smooth as the underlying limit surface of vertex
|
|
data or simply linear as achieved with varying data.
|
|
Where the data is discontinuous -- across interior edges and around vertices -- the
|
|
discontinuities create boundaries for the data, and partition the underlying surface into
|
|
disjoint regions. The interpolation along these boundaries can also be specified as
|
|
smooth or linear in a number of ways (many of which have a historical basis).
|
|
|
|
A more complete description of the different linear interpolation options with face-varying
|
|
data and interpolation is given later. These options make it possible to treat the data as
|
|
either vertex or varying, but with the added presence of discontinuities.
|
|
|
|
An essential point to remember with face-varying interpolation is that each set of data
|
|
is free to have its own discontinuities -- this leads to each data set having both unique
|
|
topology and size.
|
|
|
|
The topology specified for a collection of face-varying data is referred to as a
|
|
*channel* and is unique to face-varying interpolation. Unlike vertex and varying
|
|
interpolation, which both associate a data value with a vertex, the number of values in
|
|
a face-varying channel is not fixed by the number of vertices or faces. The number of
|
|
indices assigned to the face-corners will be the same for all channels, but the number
|
|
of unique values referred to by these indices may not. We can take advantage of the
|
|
common mesh topology in areas where the data is continuous, but we lose some of those
|
|
advantages around the discontinuities. This results in the higher complexity and cost
|
|
of a face-varying channel compared to vertex or varying data. If the topology for a
|
|
channel is fixed, though, similar techniques can be applied to factor computation
|
|
related to the topology so that changes to the data can be processed efficiently.
|
|
|
|
----
|
|
|
|
Schemes and Options
|
|
===================
|
|
|
|
While previous sections have described subdivision surfaces in more general terms, this
|
|
section describes a number of common variations (often referred to as *extensions* to
|
|
the subdivision algorithms) and the ways that they are represented in OpenSubdiv.
|
|
|
|
The number and nature of the extensions here significantly complicate what are otherwise
|
|
fairly simple subdivision algorithms. Historically applications have supported either a
|
|
subset or have had varying implementations of the same feature. OpenSubdiv strives to
|
|
provide a consistent and efficient implementation of this feature set.
|
|
|
|
Given the varying presentations of some of these features elsewhere, the naming chosen
|
|
by OpenSubdiv is emphasized here.
|
|
|
|
Subdivision Schemes
|
|
*******************
|
|
|
|
OpenSubdiv provides two well known subdivision surface types -- Catmull-Clark (often referred
|
|
to more tersely as "Catmark") and Loop subdivision. Catmull-Clark is more widely used and
|
|
suited to quad-dominant meshes, while Loop is preferred for purely triangulated meshes.
|
|
|
|
The many examples from previous sections have illustrated the more popular Catmull-Clark
|
|
scheme. For an example of Loop:
|
|
|
|
+------------------------------------+------------------------------------+------------------------------------+------------------------------------+
|
|
| .. image:: images/loop_cage.jpg | .. image:: images/loop_sub1.jpg | .. image:: images/loop_sub2.jpg | .. image:: images/loop_surface.jpg |
|
|
| :align: center | :align: center | :align: center | :align: center |
|
|
| :width: 95% | :width: 95% | :width: 95% | :width: 95% |
|
|
| :target: images/loop_cage.jpg | :target: images/loop_sub1.jpg | :target: images/loop_sub2.jpg | :target: images/loop_surface.jpg|
|
|
+------------------------------------+------------------------------------+------------------------------------+------------------------------------+
|
|
|
|
*Note that while Loop subdivision has long been available, support for the limit surface
|
|
of Loop subdivision (i.e. arbitrary evaluation of the surface via patches) is not supported
|
|
prior to version 3.4.*
|
|
|
|
----
|
|
|
|
Boundary Interpolation Rules
|
|
****************************
|
|
|
|
Boundary interpolation rules control how subdivision and the limit surface behave for faces
|
|
adjacent to boundary edges and vertices.
|
|
|
|
The following choices are available via the enumeration *Sdc::Options::VtxBoundaryInterpolation*:
|
|
|
|
+----------------------------------+----------------------------------------------------------+
|
|
| Mode | Behavior |
|
|
+==================================+==========================================================+
|
|
| **VTX_BOUNDARY_NONE** | No boundary edge interpolation should occur; instead |
|
|
| | boundary faces are implicitly tagged as holes so that |
|
|
| | the boundary vertices continue to support the adjacent |
|
|
| | interior faces, but no surface corresponding to the |
|
|
| | boundary faces is generated |
|
|
+----------------------------------+----------------------------------------------------------+
|
|
| **VTX_BOUNDARY_EDGE_ONLY** | A sequence of boundary vertices defines a smooth curve |
|
|
| | to which the limit surface along boundary faces extends |
|
|
+----------------------------------+----------------------------------------------------------+
|
|
| **VTX_BOUNDARY_EDGE_AND_CORNER** | Similar to edge-only but the smooth curve resulting on |
|
|
| | the boundary is made to interpolate corner vertices |
|
|
| | (vertices with exactly one incident face) |
|
|
+----------------------------------+----------------------------------------------------------+
|
|
|
|
On a grid example:
|
|
|
|
.. image:: images/vertex_boundary.png
|
|
:align: center
|
|
:target: images/vertex_boundary.png
|
|
|
|
In practice, it is rare to use no boundary interpolation at all -- this feature has
|
|
its uses in allowing separate meshes to be seamlessly joined together by replicating
|
|
the vertices along boundaries, but these uses are limited. Given the global nature
|
|
of the setting, it is usually preferable to explicitly make the boundary faces holes
|
|
in the areas where surfaces from separate meshes are joined.
|
|
|
|
The remaining "edge only" and "edge and corner" choices are then solely distinguished
|
|
by whether or not the surface at corner vertices is smooth or sharp.
|
|
|
|
----
|
|
|
|
Face-varying Interpolation Rules
|
|
********************************
|
|
|
|
Face-varying interpolation rules control how face-varying data is interpolated both in the
|
|
interior of face-varying regions (smooth or linear) and at the boundaries where it is
|
|
discontinuous (constrained to be linear or "pinned" in a number of ways). Where the
|
|
topology is continuous and the interpolation chosen to be smooth, the behavior of
|
|
face-varying interpolation will match that of the vertex interpolation.
|
|
|
|
Choices for face-varying interpolation are most commonly available in the context of UVs
|
|
for texture coordinates and a number of names for such choices have evolved in different
|
|
applications over the years. The choices offered by OpenSubdiv cover a wide range of popular
|
|
applications. The feature is named face-varying *linear* interpolation -- rather than
|
|
*boundary* interpolation commonly used -- to emphasize that it can be applied
|
|
to the entire surface (not just boundaries) and that the effects are to make the surface
|
|
behave more linearly in various ways.
|
|
|
|
The following choices are available for the *Sdc::Options::FVarLinearInterpolation* enum --
|
|
the ordering here applying progressively more linear constraints:
|
|
|
|
+--------------------------------+-------------------------------------------------------------+
|
|
| Mode | Behavior |
|
|
+================================+=============================================================+
|
|
| **FVAR_LINEAR_NONE** | smooth everywhere the mesh is smooth |
|
|
+--------------------------------+-------------------------------------------------------------+
|
|
| **FVAR_LINEAR_CORNERS_ONLY** | linearly interpolate (sharpen or pin) 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 a simple 4x4
|
|
grid of quads segmented into three UV regions (their control point locations implied
|
|
by interpolation in the FVAR_LINEAR_ALL case):
|
|
|
|
.. image:: images/fvar_boundaries.png
|
|
:align: center
|
|
:width: 90%
|
|
:target: images/fvar_boundaries.png
|
|
|
|
(For those familiar, this shape and its assigned UV sets are available for inspection
|
|
in the "catmark_fvar_bound1" shape of OpenSubdiv's example and regression shapes.)
|
|
|
|
----
|
|
|
|
Semi-Sharp Creases
|
|
******************
|
|
|
|
Just as some types of parametric surfaces support additional shaping controls to
|
|
affect creasing along the boundaries between surface elements, OpenSubdiv provides
|
|
additional sharpness values or "weights" associated with edges and vertices to
|
|
achieve similar results over arbitrary topology.
|
|
|
|
Setting sharpness values to a maximum value (10 in this case -- a number chosen for
|
|
historical reasons) effectively modifies the subdivision rules so that the boundaries
|
|
between the piecewise smooth surfaces are infinitely sharp or discontinuous.
|
|
|
|
But since real world surfaces never really have infinitely sharp edges, especially
|
|
when viewed sufficiently close, it is often preferable to set the sharpness
|
|
lower than this value, making the crease "semi-sharp". A constant weight value
|
|
assigned to a sequence of edges connected edges therefore enables the creation of
|
|
features akin to fillets and blends without adding extra rows of vertices (though
|
|
that technique still has its merits):
|
|
|
|
.. image:: images/gtruck.png
|
|
:align: center
|
|
:height: 300
|
|
:target: images/gtruck.png
|
|
|
|
Sharpness values range from 0-10, with a value of 0 (or less) having no effect on the
|
|
surface and a value of 10 (or more) making the feature completely 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.
|
|
|
|
----
|
|
|
|
Other Options
|
|
*************
|
|
|
|
While the preceding options represent features available in a wide-variety of tools
|
|
and modeling formats, a few others exist whose recognition and adoption is more limited.
|
|
In some cases, they offer improvements to undesirable behavior of the subdivision
|
|
algorithms, but their effects are less than ideal.
|
|
|
|
Given both their limited effectiveness and lack of recognition, these options should be
|
|
used with caution.
|
|
|
|
----
|
|
|
|
Chaikin Rule
|
|
~~~~~~~~~~~~
|
|
|
|
The "Chaikin Rule" is a variation of the semi-sharp creasing method that attempts to
|
|
improve the appearance of creases along a sequence of connected edges when the sharpness
|
|
values differ. This choice modifies the subdivision of sharpness values using Chaikin's
|
|
curve subdivision algorithm to consider all sharpness values of edges around a common
|
|
vertex when determining the sharpness of child edges.
|
|
|
|
The creasing method can be set using the values defined in the enumeration
|
|
*Sdc::Options::CreasingMethod*:
|
|
|
|
+---------------------+---------------------------------------------+
|
|
| 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:
|
|
|
|
.. image:: images/chaikin.png
|
|
:align: center
|
|
:target: images/chaikin.png
|
|
|
|
----
|
|
|
|
"Triangle Subdivision" Rule
|
|
~~~~~~~~~~~~~~~~~~~~~~~~~~~
|
|
|
|
The triangle subdivision rule is a rule added to the Catmull-Clark scheme that
|
|
modifies the behavior at triangular faces to improve the undesirable surface
|
|
artefacts that often result in such areas.
|
|
|
|
+---------------------+---------------------------------------------+
|
|
| Mode | Behavior |
|
|
+=====================+=============================================+
|
|
| **TRI_SUB_CATMARK** | Default Catmark scheme weights |
|
|
+---------------------+---------------------------------------------+
|
|
| **TRI_SUB_SMOOTH** | "Smooth triangle" weights |
|
|
+---------------------+---------------------------------------------+
|
|
|
|
Cylinder example :
|
|
|
|
.. image:: images/smoothtriangles.png
|
|
:align: center
|
|
:height: 300
|
|
:target: images/smoothtriangles.png
|
|
|
|
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
|
|
|