Algebraic Curve Segments Research Articles

Greyhound racing ideal trajectory path generation for straight to bend based on jerk rate minimization

This paper presents methods for modelling and designing an ideal path trajectory between straight and bend track path segments for racing greyhounds. To do this, we numerically generate clothoid and algebraic curve segments for racing quadrupeds using a sequential vector transformation method as well as using a helper equation for approaching ideal clothoid segments that would respect greyhound kinematic parameters and boundary conditions of the track. Further, we look into the limitations of using a clothoid curve for racing dog track path design and propose a smooth composite curve for track transition design which roughly maintains G3 curvature continuity for smooth jerk to overcome limitations of a clothoid transition. Finally, we show results from race data modelling and past injury data, which provide a strong indication of clothoid curve segments improving the dynamics and safety of racing greyhounds while reducing injuries.

Identifying and approximating monotonous segments of algebraic curves using support function representation

Algorithms describing the topology of real algebraic curves search primarily the singular points and they are usually based on algebraic techniques applied directly to the curve equation. We adopt a different approach, which is primarily based on the identification and approximation of smooth monotonous curve segments, which can in certain cases cross the singularities of the curve. We use not only the primary algebraic equation of the planar curve but also (and more importantly) its implicit support function representation. This representation is also used for an approximation of the segments. This way we obtain an approximate graph of the entire curve which has several nice properties. It approximates the curve within a given Hausdorff distance. The actual error can be measured efficiently and behaves as O(N−3) where N is the number of segments. The approximate graph is rational and has rational offsets. In the simplest case it consists of arc segments which are efficiently represented via the support function. The question of topological equivalence of the approximate and precise graphs of the curve is also addressed and solved using bounding triangles and axis projections. The theoretical description of the whole procedure is accompanied by several examples which show the efficiency of our method.

PLANAR SHAPE MANIPULATION USING APPROXIMATE GEOMETRIC PRIMITIVES

We present robust algorithms for set operations and affine transformations on curved shapes in the plane using approximate geometric primitives. We use a refinement algorithm to ensure consistency. Its computational complexity is O (n log n + k) for an input of size n with k = O (n2) consistency violations. The output is as accurate as the geometric primitives. We validate our algorithms using sequences of six set operations and affine transforms on shapes bounded by algebraic curve segments of degree 1 to 6. We test generic and degenerate inputs.

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {rn(z)}n∈ℕ of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {{rn(z)}n∈ℕ in the complement CP1⧹Df, where Df is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {rn(z)}n∈ℕ. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.

Regular algebraic curve segments (III)—applications in interactive design and data fitting

In this paper (part three of the trilogy) we use low degree G 1 and G 2 continuous regular algebraic spline curves defined within parallelograms, to interpolate an ordered set of data points in the plane. We explicitly characterize curve families whose members have the required interpolating properties and possess a minimal number of inflection points. The regular algebraic spline curves considered here have many attractive features: They are easy to construct. There exist convenient geometric control handles to locally modify the shape of the curve. The error of the approximation is controllable. Since the spline curve is always inside the parallelogram, the error of the fit is bounded by the size of the parallelogram. The spline curve can be rapidly displayed, even though the algebraic curve segments are implicitly defined.

Regular algebraic curve segments (I)—Definitions and characteristics

In this paper (part one of a trilogy), we introduce the concept of a discriminating family of regular algebraic curves (real, nonsingular and connected). Several discriminating families are obtained which yield different characterizations of the Bernstein–Bézier (BB) bivariate polynomials over the plane triangle and the quadrilateral domain such that their zero contours are smooth and connected. These regular curve segments in BB basis can be smoothly joined together to form algebraic curve splines or A-splines. Algorithms for the efficient graphics display of these new A-spline families are also provided.

Regular algebraic curve segments (II)—Interpolation and approximation

In this paper (part two of the trilogy) we introduce three classes of reduced form D-regular algebraic curve splines and use them for interpolation and approximation of various algebraic curves. Explicit formulas for interpolation and approximation are also given in some low degree cases.

A-splines: local interpolation and approximation using Gk-continuous piecewise real algebraic curves

We provide sufficient conditions for the Bernstein–Bézier (BB) form of an implicitly defined bivariate polynomial over a triangle, such that the zero contour of the polynomial defines a smooth and single sheeted real algebraic curve segment. We call a piecewise G k -continuous chain of such real algebraic curve segments in BB-form as an A-spline (short for algebraic spline). We prove that the degree n A-splines can achieve in general G 2 n−3 continuity by local fitting and still have degrees of freedom to achieve local data approximation. As examples, we show how to construct locally convex cubic A-splines to interpolate and/or approximate the vertices of an arbitrary planar polygon with up to G 4 continuity, to fit discrete points and derivatives data, and approximate high degree parametric and implicitly defined curves. Additionally, we provide computable error bounds.

Voronoi diagram and medial axis algorithm for planar domains with curved boundaries — II: Detailed algorithm description

Details of algorithms to construct the Voronoi diagrams and medial axes of planars domain bounded by free-form (polynomial or rational) curve segments are presented, based on theoretical foundations given in the first installment Ramamurthy and Farouki, J. Comput. Appl. Math. (1999) 102 119–141 of this two-part paper. In particular, we focus on key topological and computational issues that arise in these constructions. The topological issues include: (i) the data structures needed to represent various geometrical entities — bisectors, Voronoi regions, etc., and (ii) the Boolean operations (i.e., union, intersection, and difference) on planar sets required by the algorithm. Specifically, representations for the Voronoi polygons of boundary segments, and for individual Voronoi diagram or medial axis edges, are proposed. Since these edges may be segments of (a) nonrational algebraic curves (curve/curve bisectors); (b) rational curves (point/curve bisectors); or (c) straight lines (point/point bisectors), data structures tailored to each of these geometrical entities are introduced. The computational issues addressed include the curve intersection algorithms required in the Boolean operations, and iterative schemes used to precisely locate bifurcation or “ n-prong” points ( n ⩾ 3) of the Voronoi diagram and medial axis. A selection of computed Voronoi diagram and medial axis examples is included to illustrate the capabilities of the algorithm.

Recursive algebraic curve fitting and rendering

We describe a recursive algorithm that uses quadratic algebraic curve segments to vectorize digital images. The closeness of fitting and the smoothness of connection between curve segments are ensured by a recursive algebraic curve fitting and a subsequent fine-tuning procedure. The idea is to provide an alternative way to vectorize outside parametric schemes, while maintaining the precision of parametric vectorization. We can also have all the new features of algebraic representation; for instance, the implicit forms and unique insights into curve shapes and control point weights. We also present a triangular quadtree rendering scheme for displaying algebraic curves. These algorithms combine features from both parametric and algebraic schemes to meet different requirements for curve fitting.

Convex hulls of objects bounded by algebraic curves

We present an0(n ·do(1)) algorithm to compute the convex hull of a curved object bounded by0(n) algebraic curve segments of maximum degreed.

A localized method for intersecting plane algebraic curve segments

We present a local method for the computation of the intersections of plane algebraic curve segments. The conventional method of intersection is global, because it must first find all of the intersections between two curves before it can restrict the segments in question; hence, it cannot take advantage of situations dealing with the intersection of short-curve segments on complex curves. Our local method, on the other hand, will directly find only those intersections that lie on the segments, as it is based upon an extension of methods for tracing along a curve.

A point-inclusion algorithm for a domain with boundary composed of algebraic curve segments

In this paper an algorithm is given which decides, for a domain with boundary composed of planar algebraic curve segments, whether an arbitrary point is inside or external to the domain. The algebraic curve segments of the algorithm are assumed to be contained in rectangles or other polygons (which may not contain other pieces of the curve). We discuss which curve segments have this representation. When the curves are conic sections, we give some simple rules for constructing the representation.

Generation of configuration space obstacles: The case of moving algebraic curves

We present algebraic algorithms to generate the boundary of planar configuration space obstacles arising from the translatory motion of objects amongst obstacles. Both the boundaries of the objects and obstacles are given by segments of algebraic curves.

Tracking parameterized algebraic curves on raster displays

An algorithm is presented for the tracking of parameterized algebraic curve segments on raster displays. Such segments are generally obtained by projecting the edges of an algebraic surface boundary representation describing a solid object. We sub-divide the curve segments such that each resultant segment is properly bounded by a guiding triangle. The triangle is used to approximate the segment sequentially and unambiguously on the integer coordinates of the display. The algorithm is both efficient and robust as opposed to existing parametric and algebraic tracking techniques.

Planar piecewise algebraic curves

An approach is described for piecing together segments of planar algebraic curves with derivative continuity. The application of piecewise algebraic curves to area modelling (the two-dimensional analogue of solid modelling) is discussed. A technique is presented for expressing a planar rational parametric curve as an algebraic curve segment. An upper bound is derived for the farthest distance between two algebraic curves (one of which may also be a parametric curve) within a specified region.