COMPUTER AIDED GEOMETRIC DESIGN

a lntergraph Corporation, Huntsville, AL, USA b The University of Birminghara, Birmingham, UK
Received January 1995; revised September g homogeneous coordinates, a rational curve can be represented in a nonrational form. Based on such a nonrational representation of a curve, a simple method to identify inflection points and cusps on 2-D and 3-D rational curves is proposed. © 1997 Elsevier Science B.V.
where X(t), Y(t), and W(t) are polynomials of degree n in t E [a, hi. Using homogeneous coordinates, we may map r(t) into a 3-D space to obtain a nonrational representation of a curve; i.e., R(t) = (X(t), Y(t), W(t)). In CAGD applications, it is often desirable to find the conditions such that R(t) may or may not have inflection points and cusps. By generalising the approach in (Patterson, 1988), we shall derive such conditions for planar rational curves of any degree. A comprehensive treatment of planar rational curves can be found in (Walker, 1950) and a very readable treatment in (Primrose, 1955). To be consistent, we briefly review some relevant properties of planar rational curves and then derive the inflection point and cusp conditions for R(t). In a homogeneous coordinate system, the point (X, I/, W) is a loop if two distinct parameters tl, t2 give the same ratio for (X, Y, W), i.e.,
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line to the curve, it must meet the curve at least three times at an inflection point with parameter t. Thus, we have
X' (t) : X(t) = Y'(t) : Y(t) = W'(t) : W(t),
(1)
where a dash denotes differentiation with respect to t. If one or two of X(t), Y(t), W(t) are zero, relation (1) needs to be modified. For example, if X(t) = 0 (1) becomes X'(t) = O; Y'(t) : Y(t) = W'(t) : W(t) and if X(t) = Y(t) = 0 (1) becomes
X(tl):X(t2)
= Y ( t , ) : Y(t2) = W ( t , ) : W(t2).
A cusp may be regarded as the limit of a loop. Then, by using Taylor's expansion in the vicinity of t it can be proved (Primrose, 1955) that a cusp occurs if
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solving a univariate polynomial. There are some advantages with respect to our approach. Firstly, we can easily identify the multiplicity of an inflection point, which has a practical importance in manufacturing since the normal of a curve changes directions only at an inflection point with odd multiplicity. Secondly, the occurrence of cusps and inflection points on a 3-D curve corresponds to zeros of one univariate polynomial rather than zeros of three polynomials. Therefore, efficiency and stability in computing such zeros can be assured. In the subsequent sections, we shall assume that our rational parametric curves are properly parametrised; i.e., these curves cannot be identically described by a rational polynomial parametrisation of lower degree. From the theory of algebraic geometry it is known that such proper parametrisation always exists for rational curves. Detailed discussion on proper parametrisation can be found in (Sederberg, 1984, 1986).
Keywords: Rational parametric curves; Inflection points; Cusps
I. Introduction
In C A G D applications, it is often desirable to find the conditions such that a curve may or may not have cusps and inflection points. Given a properly parametrised rational curve a cusp may be considered as a point at which the first derivative vector of a curve vanishes, and an inflection point as a vanishing of the curvature vector. From this point of view, the detection of cusps and inflection points is addressed in (Patterson, 1988; DeRose and Stone, 1989; Manocha and Canny, 1992). From the theory of algebraic curves, a point of multiplicity r is a point at which every line intersects the curve r times. In particular, a point of multiplicity one is a simple point, and a point of multiplicity two is a double point. A loop of a curve is a double point where the curve crosses itself and has two distinct tangents. A cusp is a double point at which two (cuspidal) tangents are coincident. An inflection point of a curve is a simple point at which the tangent intersects the curve three or more times. Accordingly, we say such an inflection point is of multiplicity three or more. From this geometric point of view, we shall derive the inflection point and cusp conditions in the form of determinants. Consequently, inflection points and cusps can be identified by * Corresponding author. 0167-8396/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. ,all S0167-8396(96)00041-6