NDLI: Exact integer hybrid subdivision and forward differencing of cubics

Content Provider	ACM Digital Library
Author	Klassen, R. Victor
Copyright Year	1994
Abstract	Forward differencing is widely used to generate rapidly large numbers of points at equally space parameter values along a curve. A failing of forward differencing is the tendency to generate many extraneous points for curves with highly nonuniform parameterizations. A key result is presented and proven, namely, that a few levels of subdivision, prior to initialization for forward differencing, can improve substantially the quality of the step size estimate, resulting in very few extra points. The initial subdivisions can be done without loss of the exact integer precision available in forward differencing. For small numbers of points—a common occurrence in fonts—exact subdivision is even faster than exact forward differencing. When exact subdivision is used in conjunction with a previously presented exact forward-differencing algorithm, arbitrary cubic curves may be rendered with 32-bit arithmetic and guaranteed single-pixel accuracy, in a grid with an address space as large as 0…7281, with no two generated points greater than one pixel apart. This is more steps than previously possible. Previous discussions of rendering using subdivision have concentrated not on distance but on straightness estimates, whereby subdivision can be stopped once a subcurve can be drawn safely using its polygonal approximation. In this article, bounds are also derived on the size of the control polygon after multiple levels of subdivision: these are used to determine bounds on the number of steps required for differencing. It is shown that any curve whose rasterization fits in a space of ω pixels requires no more than 9ω steps.
Starting Page	240
Ending Page	255
Page Count	16
File Format	PDF
ISSN	07300301
e-ISSN	15577368
DOI	10.1145/195784.197476
Volume Number	13
Issue Number	3
Journal	ACM Transactions on Graphics (TOG)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	1994-07-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Be´zier polynomials Parameterization Spline curves
Content Type	Text
Resource Type	Article
Subject	Computer Graphics and Computer-Aided Design

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