Ronald N. Goldman
Abstract
In Part I, Markov chains were shown to be associated with solutions to several standard problems in computer-aided geometric design. Constraints on these Markov chains were also derived. Examples are given here of Markov chains that either satisfy some of these constraints or solve one of these problems. Subdivision matrices are also studied in special detail.
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Index Terms
- Markov chains and computer aided geometric design: Part II—examples and subdivision matrices
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Reviews
Paolo E. Sabella
This paper is a continuation of Part I [1], in which matrices M jk that transform curve definitions were identified as Markov chains in certain cases. That is, they satisfy :1S k M jk = 1 :9F:Y and M jk ?9T0 :9F:Y In addition, constraints on the Markov chains were given geometric interpretations. For example, if M is also a Descartes matrix, the curve possesses the variation diminishing property. In this paper, Goldman presents examples of Markov chains that occur in the solutions to certain geometric problems in CAD/CAM. In addition to casting new light on old problems, such as the Oslo Algorithm and subdivision matrices (the discrete B-spline and the subdivision matrix are shown to be Markov chains), the main contribution of this paper is that it provides a common framework in which to approach and analyze hitherto unrelated problems. Among the eight examples, two are particularly intriguing: (1)a parameterized family of transformations derived from symmetric random walks, and (2)the transformations obtained by repeatedly composing a transformation (products and powers). Even though the author admits that further investigation into the effects of (1) on the shape of given curves is warranted, in both cases the reader is left with acute curiosity to see a picture. Goldman's style is easy to follow; nevertheless, readers are forewarned that a familiarity with the mathematics of Bezier curves and B-splines is assumed for the intended audience. Enough details are provided for an eager implementor to attempt, in many cases shortcuts are indicated. However, where there exist previous solutions, such as [2] for knot insertions, no comparisons are given of the computational complexity. These two papers (Parts I and II) represent a starting point for a new approach. Despite the lack of graphical examples, they will remain a valuable reference.
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