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Rendering Curves and Surfaces with Hybrid Subdivision
and Forward Differencing

We present a hybrid rendering algorithm for rendering parametric
curves and surfaces. The algorithm uses a series of Direct Rendering
Criteria (DRC) for determining whether the curve/surface can be directly
rendered by forward differencing with a constant step size.
The DRCs test the geometric
flatness of the curve/surface, its parametric uniformity, and the
ability to use only integer arithmetic in the
forward differencing algorithm. If any of
the DRCs is not fulfilled, the curve/surface is subdivided. The
location of the subdivision in parameter space is chosen to
increase the chances that the new segments will satisfy the DRCs.
For the integer arithmetic DRC we introduce a general method for
determining an alignment of the
forward differences. We show that for cubic (quartic)
curves this alignment enables up to 2^13 (2^11) forward
steps for curves lying in a 128K X 128K space.
The method is applicable to curves of any order.

* ACM Transactions on Graphics, * 10(4):323-341, October 1991.