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Zbl 0831.65013
Pottmann, Helmut
Curve design with rational Pythagorean-hodograph curves.
(English)
[J] Adv. Comput. Math. 3, No.1-2, 147-170 (1995). ISSN 1019-7168; ISSN 1572-9044/e

The author treats the problem of effectively constructing parallel curves (offset curves'' in engineering language). The problem with parallel curves in computational geometry naturally is the irrationality introduced by the computation of the unit normal. Since the unit normal can be rationally parametrized as point on the unit circle as function of the support angle, the author takes as the object of his study dual curves, i.e., curves represented as envelopes of their tangents.\par In the dual plane, each Bézier point yields a Bézier line, and these together with their Farin lines give an easily manipulated formalism for the computation of convex arcs. [If two consecutive Bézier lines in the projective plane are ${\bold x} \cdot {\bold B}_{(i)} = 0$, then the Farin line is represented by ${\bold B}_{(i)} + {\bold B}_{(i + 1)}]$. If a dual curve is rational of degree $m$, then the corresponding point-curve is rational of degree $2n - 2$.\par The author then shows that effective approximations that yield usable offset curves can be based on dual curves of degrees 4 and 5. In his algorithm, one has first to find all inflection points and vertices of a curve (since they will correspond to singularities of the dual curve) and then has to take as Bézier points these singularities and for each parameter interval so defined at least one more interpolating point.\par The formulas for the interpolating curve depend on the nature of the Bézier points used as endpoints; complete formulas are given for the four possible kinds to obtain approximations that are curvature- continuous.
MSC 2000:
*65D17 Computer aided design (modeling of curves and surfaces)
65D18 Computer graphics and computational geometry
51N15 Projective analytic geometry

Keywords: offset curves; parallel curves; computational geometry; Bézier point; Bézier line; Farin lines; dual curve; algorithm

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