Subdivision versus Tessellation

[Carlosan]

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The preceding sections illustrate subdivision surfaces as piecewise parametric surfaces of arbitrary topology. As piecewise parametric surfaces, they consist of a cage and the underlying surface defined by that cage.

Two techniques used to display subdivision surfaces are subdivision and tessellation. Both have their legitimate uses, but there is an important distinction between them:

• subdivision operates on a cage and produces a refined cage
• tessellation operates on a surface and produces a discretization of that surface

The existence and relative simplicity of the subdivision algorithm makes it easy to apply repeatedly to approximate the shape of the surface, but with the result being a refined cage, that approximation is not always very accurate. When compared to a cage refined to a different level, or a tessellation that uses points evaluated directly on the limit surface, the discrepancies can be confusing.

Subdivision

Subdivision is the process that gives "subdivision surfaces" their name, but it is not unique to them. Being piecewise parametric surfaces, let's first look at subdivision in the context of the simpler parametric patches that comprise them.

Subdivision is a special case of refinement, which is key to the success of some of the most widely used types of parametric patches and their aggregate surfaces. A surface can be "refined" when an algorithm exists such that more control points can be introduced while keeping the shape of the surface exactly the same. For interactive and design purposes, this allows a designer to introduce more resolution for finer control without introducing undesired side effects in the shape. For more analytical purposes, it allows the surface to be broken into pieces, often adaptively, while being faithful to the original shape.

One reason why both B-spline and Bezier patches are so widely used is that both of them can be refined. Uniform subdivision -- the process of splitting each of the patches in one or both of its directions -- is a special case of refinement that both of these patch types support:

B-Spline surface and its cage

Cage subdivided 1x

Cage subdivided 2x

In the cases illustrated above for B-Splines, the uniformly refined cages produce the same limit surface as the original (granted in more pieces). So it is fair to say that both uniform B-splines and Bezier surfaces are subdivision surfaces.

The limit surface remains the same with the many more control points (roughly 4x with each iteration of subdivision), and those points are closer to (but not on) the surface. It may be tempting to use these new control points to represent the surface, but using the same number of points evaluated at corresponding uniformly spaced parametric locations on the 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:

Uniform (3x3) 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.

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.