A family of tangent continuous cubic algebraic splines

Reviewer: Vadim Shapiro

The algebraic splines studied in this paper are a subclass of piecewise cubic planar curves where each curve segment is a subset of some cubic curve f x,y =0 , constructed to smoothly interpolate vertices of a control triangle. Because cubic curves that are defined implicitly by f x,y =0 do not always admit a rational parametric form, such piecewise cubic curves are rarely used in commercial systems, but they possess many other attractive computational properties (including low-degree and local control), making them increasingly popular objects of study. The curve segments in this paper are further restricted to be subsets of convex ovals, and the main contributions of the paper are in establishing the conditions under which a given cubic curve is such an oval and in developing techniques for spline construction and limited shape control. Additional analysis of curvature conditions is promised in a subsequent paper. The paper is well written but contains many technical details that assume general familiarity with algebraic geometry (real projective spaces and Bezout's theorem) and topics in computer-aided geometric design (such as Be´zier construction). One disappointing aspect of the paper is its relatively narrow focus; for example, it contains virtually no discussion of how such splines could be used in practice, how they compare with other splines, or whether any of the described methods may generalize to three-dimensional algebraic cubic patches. On the other hand, the new results are described in sufficient detail to allow the development and implementation of a graphics package to construct and manipulate such algebraic cubic splines. Thus, the paper should interest researchers in geometric modeling and developers of CAD and graphics software.