OpenSubdiv/documentation/subdivision_surfaces.rst

..
     Copyright 2013 Pixar

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Subdivision Surfaces
--------------------

.. contents::
   :local:
   :backlinks: none

----

Introduction
============

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

.. image:: images/torus.png
   :align: center
   :height: 200

However, while they do provide a reliable smooth limit surface definition, bicubic
patch surfaces are limited to 2-dimensional topologies, which only describes a
very small fraction of real-world shapes. This fundamental parametric limitation
requires authoring tools to implementat 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.

.. raw:: html

    <center>
      <p align="center">
        <IMG src="images/tetra.0.jpg" style="width: 20%;">
        <IMG src="images/tetra.1.jpg" style="width: 20%;">
        <IMG src="images/tetra.2.jpg" style="width: 20%;">
        <IMG src="images/tetra.3.jpg" style="width: 20%;">
      </p>
    </center>

----

Manifold Geometry
*****************

Continuous limit surfaces require that the topology be a two-dimensional
manifold. It is therefore possible to model non-manifold geometry that cannot
be represented with a smooth C2 continuous limit. 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.

.. image:: images/nonmanifold_fan.png
   :align: center
   :target: images/nonmanifold_fan.png

With this configuration, it is unclear which face should contribute to the
limit surface, as 3 of them share the same edge (which incidentally breaks
half-edge cycles in said data-structures). Fan configurations are not limited
to 3 incident faces: any configuration where an edge is shared by more than
2 faces incurs the same problem.

----

Non-Manifold Disconnected Vertex
++++++++++++++++++++++++++++++++

A vertex is disconnected from any edge and face.

.. image:: images/nonmanifold_vert.png
   :align: center
   :target: images/nonmanifold_vert.png

This case is fairly trivial: there is no possible way to exact a limit surface here,
so the vertex simply has to be flagged as non-contributing, or discarded gracefully.

----

Boundary Interpolation Rules
============================

Boundary interpolation rules control how boundary face edges and facevarying data
are interpolated.

Vertex Data
***********

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

+------------------------+----------------------------------------------------------+
| Mode                   | Behavior                                                 |
+========================+==========================================================+
| 0 - **None**           | No boundary interpolation behavior should occur          |
|                        | (debug mode - boundaries are undefined)                  |
+------------------------+----------------------------------------------------------+
| 1 - **EdgeAndCorner**  | All the boundary edge-chains are sharp creases and       |
|                        | boundary vertices with exactly two incident edges are    |
|                        | sharp corners                                            |
+------------------------+----------------------------------------------------------+
| 2 - **EdgeOnly**       | All the boundary edge-chains are sharp creases; boundary |
|                        | vertices are not affected                                |
|                        |                                                          |
+------------------------+----------------------------------------------------------+

On a quad example:

.. image:: images/vertex_boundary.png
   :align: center
   :target: images/vertex_boundary.png


Facevarying Data
****************

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

+--------+----------------------------------------------------------+
| Mode   | Behavior                                                 |
+========+==========================================================+
| 0      | Bilinear interpolation (no smoothing)                    |
+--------+----------------------------------------------------------+
| 1      | Smooth UV                                                |
|        |                                                          |
|        |                                                          |
+--------+----------------------------------------------------------+
| 2      | Same as (1) but does not infer the presence of corners   |
|        | where two facevarying edges meet at a single faceA       |
|        |                                                          |
+--------+----------------------------------------------------------+
| 3      | Smooths facevarying values only near vertices that are   |
|        | not at a discontinuous boundary; all vertices on a       |
|        | discontinuous boundary are subdivided with a sharp rule  |
|        | (interpolated through).                                  |
|        | This mode is designed to be compatible with ZBrush and   |
|        | Maya's "smooth internal only" interpolation.             |
+--------+----------------------------------------------------------+

Unwrapped cube example:

.. image:: images/fvar_boundaries.png
   :align: center
   :target: images/fvar_boundaries.png

Propagate Corners
+++++++++++++++++

Facevarying interpolation mode 2 (*EdgeAndCorner*) can further be modified by the 
application of the *Propagate Corner* flag.

----

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.


.. image:: images/gtruck.jpg
   :align: center
   :height: 300
   :target: images/gtruck.jpg

----

Chaikin Rule
============

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

.. image:: images/chaikin.png
   :align: center
   :target: images/chaikin.png

----

Hierarchical Edits
==================

To understand the hierarchical aspect of subdivision, we realize that subdivision
itself leads to a natural hierarchy: after the first level of subdivision, each
face in a subdivision mesh subdivides to four quads (in the Catmull-Clark scheme),
or four triangles (in the Loop scheme). This creates a parent and child relationship
between the original face and the resulting four subdivided faces, which in turn
leads to a hierarchy of subdivision as each child in turn subdivides. A hierarchical
edit is an edit made to any one of the faces, edges, or vertices that arise anywhere
during subdivision. Normally these subdivision components inherit values from their
parents based on a set of subdivision rules that depend on the subdivision scheme.

A hierarchical edit overrides these values. This allows for a compact specification
of localized detail on a subdivision surface, without having to express information
about the rest of the subdivision surface at the same level of detail.

.. image:: images/hedit_example1.png
   :align: center
   :height: 300
   :target: images/hedit_example1.png

----

Hierarchical Edits Paths
************************

In order to perform a hierarchical edit, we need to be able to name the subdivision
component we are interested in, no matter where it may occur in the subdivision
hierarchy. This leads us to a hierarchical path specification for faces, since
once we have a face we can navigate to an incident edge or vertex by association.
We note that in a subdivision mesh, a face always has incident vertices, which are
labelled (in relation to the face) with an integer index starting at zero and in
consecutive order according to the usual winding rules for subdivision surfaces.
Faces also have incident edges, and these are labelled according to the origin
vertex of the edge.

.. image:: images/face_winding.png
   :align: center
   :target: images/face_winding.png

.. role:: red
.. role:: green
.. role:: blue

In this diagram, the indices of the vertices of the base face are marked in :red:`red`;
so on the left we have an extraordinary Catmull-Clark face with five vertices
(labeled :red:`0-4`) and on the right we have a regular Catmull-Clark face with four
vertices (labelled :red:`0-3`). The indices of the child faces are :blue:`blue`; note that in
both the extraordinary and regular cases, the child faces are indexed the same
way, i.e. the subface labeled :blue:`n` has one incident vertex that is the result of the
subdivision of the parent vertex also labeled :red:`n` in the parent face. Specifically,
we note that the subface :blue:`1` in both the regular and extraordinary face is nearest
to the vertex labelled :red:`1` in the parent.

The indices of the vertices of the child faces are labeled :green:`green`, and
this is where the difference lies between the extraordinary and regular case;
in the extraordinary case, vertex to vertex subdivision always results in a vertex
labeled :green:`0`, while in the regular case, vertex to vertex subdivision
assigns the same index to the child vertex. Again, specifically, we note that the
parent vertex indexed :red:`1` in the extraordinary case has a child vertex :green:`0`,
while in the regular case the parent vertex indexed :red:`1` actually has a child
vertex that is indexed :green:`1`. Note that this indexing scheme was chosen to
maintain the property that the vertex labeled 0 always has the lowest u/v
parametric value on the face.

.. image:: images/hedit_path.gif
   :align: center
   :target: images/hedit_path.gif

By appending a vertex index to a face index, we can create a vertex path
specification. For example, (:blue:`655` :green:`2` :red:`3` 0) specifies the 1st.
vertex of the :red:`3` rd. child face of the :green:`2` nd. child face of the of
the :blue:`655` th. face of the subdivision mesh.

----

Vertex Edits
************

XXXX

----

Edge Edits
**********

XXXX

----

Face Edits
**********

XXXX

----

Limitations
***********

XXXX

----

Uniform Subdivision
===================

Applies a uniform refinement scheme to the coarse faces of a mesh.

.. image:: images/uniform.gif
   :align: center
   :width: 300
   :target: images/uniform.gif

----

Feature Adaptive Subdivision
============================

Isolates extraordinary features by applying progressive refinement.

.. image:: images/subdiv_faceindex.png
   :align: center
   :target: images/subdiv_faceindex.png

.. image:: images/adaptive.gif
   :align: center
   :width: 300
   :target: images/adaptive.gif