The implementation of a library of basic functions for the construction and analysis of planar quintic Pythagorean-hodograph (PH) curves is presented using the complex representation. The special algebraic structure of PH curves permits exact algorithms for the computation of key properties, such as arc length, elastic bending energy, and offset (parallel) curves. Single planar PH quintic segments are constructed as interpolants to first-order Hermite data (end points and derivatives), and this construction is then extended to open or closed C 2 PH quintic spline curves interpolating a sequence of points in the plane. The nonlinear nature of PH curves incurs a multiplicity of formal solutions to such interpolation problems, and a key aspect of the algorithms is to efficiently single out the unique “good” interpolant among them.

Abstract The use of rotation‐minimizing directed frames (RMDFs) for defining smoothly varying camera orientations along given spatial paths, in real or virtual environments, is proposed. A directed frame on a space curve${\bf r}(\xi)$ is a varying orthonormal basis$({\bf o},{\bf p},{\bf q})$ for ℝ3such that${\bf o}(\xi)={\bf r}(\xi)/|{\bf r}(\xi)|$ coincides with the unit polar vector from the origin to each curve point, and such a frame is rotation‐minimizing if its angular velocity vector${\bf \omega}$ maintains a vanishing component alongo. To facilitate computation of rotation‐minimizing directed frames, it is shown that the basic theory is equivalent to the established theory for rotation‐minimizing adapted frames—for which one frame vector coincides with the tangent${\bf t}(\xi)={\bf r}'(\xi)/|{\bf r}'(\xi)|$ at each curve point—if one replaces the given space curve by its anti‐hodograph (i.e., indefinite integral). A family of polynomial curves on which RMDFs can be computed exactly by a rational function integration, the Pythagorean (P) curves, is also introduced, together with algorithms for their construction. Copyright © 2009 John Wiley & Sons, Ltd.

The design, implementation, and testing of a C++ software library for univriate polynomials in Bernstein form is described. By invoking the class environment and operator overloading, each polynomial in an expression is interpreted as an object compatible with the arithmetic operations and other common functions (subdivision, degree, elevation, differentiation and integration, compoistion, greatest common divisor, real-root solving, etc.) for polynomials in Bernstein form. The library allows compact and intuitive implementation of lengthy manipulation of Bernstein-form polynomials, which often arise in computer graphics and computer-aided design and manufacturing applications. A series of empirical tests indicates that the library functions are typically very accurate and reliable, even for polynomials of surprisingly high degree.

An n-th Minkowski root /spl otimes//sup 1/n/ A of a given complex set A is defined by the property {z/sub 1/z/sub 2/...z/sub n/|z/sub i//spl isin//spl otimes//sup 1/n/ A}/spl equiv/A, i.e., the set of all products of n independently chosen values from /spl otimes//sup 1/n/ is identical to A. Hence, the n-th Minkowski power of /spl otimes//sup 1/n/A yields the original set A. Minkowski root extractions are fundamental operations in the Minkowski geometric algebra of complex sets: depending on the nature of A, subtle issues concerning the existence, uniqueness and minimality or maximality of /spl otimes//sup 1/n/A may arise. For a domain A with a smooth boundary that is strictly logarithmically convex, we show that each connected component of the "ordinary" root A/sup 1/n/={z|z/sup n//spl isin/A} is a Minkowski n-th root. /spl otimes//sup 1/n/A has a more intricate structure, however when /spl part/A has logarithmic inflections. For example, if A is a circular disk, /spl otimes//sup 1/n/A is (a single loop of) the "n-th order" ovals of Cassini or lemniscate of Bernoulli when 0/spl notin/A or 0/spl isin//spl part/A, respectively. But when 0 is in the interior of A, a composite curve (portions of the Cassini ovals and a higher-order curve) is required to describe /spl otimes//sup 1/n/A.

An NC system that machines a curved shape at fixed depth of cut experiences time-varying cutting forces due to the ‘curvature effect’—the material removal rate is higher than nominal in concave regions, and lower in convex regions. A curvature-dependent feedrate function that automatically compensates for this effect is formulated, and it is shown that, for Pythagorean-hodograph (PH) curves, the periodic real time computation of reference points in accordance with this function can be analytically reduced to a sequence of root-finding problems for simple monotone functions. Empirical results from an implementation of this variable-feedrate interpolators on an open-architecture CNC milling machine are presented and compared with results from fixed-feedrate interpolators. The curvature-compensated feedrate scheme has important potential applications in ensuring part accuracy and in optimizing part programs consistent with a prescribed accuracy.

In general, two quadric surfaces intersect in a nonsingular quartic space curve. Under special circumstances, however, this intersection may “degenerate” into a quartic with a double point, or a composite of lines, conics, and twisted cubics whose degrees, counted over the complex projective domain, sum to four. Such degenerate forms are important since they occur with surprising frequency in practice and, unlike the generic case, they admit rational parameterizations. Invoking concepts from classical algebraic geometry, we formulate the condition for a degenerate intersection in terms of the vanishing of a polynomial expression in the quadric coefficients. When this is satisfied, we apply a multivariate polynomial factorization algorithm to the projecting cone of the intersection curve. Factors of this cone which correspond to intersection components “at infinity” may be removed a priori. A careful examination of the remaining cone factors then facilitates the identification and parameterization of the various real, affine intersection elements that may arise: isolated points, lines, conics, cubics, and singular quartics. The procedure is essentially automatic (avoiding the tedium of case-by-case analyses), encompasses the full range of quadric forms, and is amenable to implementation in exact (symbolic) arithmetic.