Locality and bounding-box quality of two-dimensional space-filling curves

Space-filling curves can be used to organise points in the plane into bounding-box hierarchies (such as R-trees). We develop measures of the bounding-box quality of space-filling curves that express how effective different space-filling curves are for this purpose. We give general lower bounds on the bounding-box quality measures and on locality according to Gotsman and Lindenbaum for a large class of space-filling curves. We describe a generic algorithm to approximate these and similar quality measures for any given curve. Using our algorithm we find good approximations of the locality and the bounding-box quality of several known and new space-filling curves. Surprisingly, some curves with relatively bad locality by Gotsman and Lindenbaum's measure, have good bounding-box quality, while the curve with the best-known locality has relatively bad bounding-box quality.     

Space filling curves and geodesic laminations

In this paper we study a class of connected fractals that admit a space filling curve. We prove that these curves are Hölder continuous and measure preserving. To these space filling curves we associate geodesic laminations satisfying among other properties that points joined by geodesics have the same image in the fractal under the space filling curve. The laminations help us to understand the geometry of the curves. We define an expanding dynamical system on the laminations.     

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed number of hyperflexes in the moduli space of curves of genus 3.

     

Sieving for rational points on hyperelliptic curves

We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations . We illustrate the practicality of the method with some examples of hyperelliptic curves of genus .

     

A complete set of addition laws for incomplete Edwards curves

Edwards curves were the first curves shown to have a complete addition law. However, the completeness of the addition law depends on the curve parameters and even a complete Edwards curve becomes incomplete over a quadratic field extension. This paper covers arbitrary Edwards curves and gives a set of two addition laws that for any pair of input points P₁, P₂ produce the sum P₁+P₂.     

参数曲线的全局凸性判别条件

基于平面曲线的二次微商,导出了二重点的判别条件,结合参数曲线的局部凸性条件,得到了参数闭曲线的充要条件。给出了参数曲线的拐点判别条件,从而得到了参数曲线局部凸的充要条件。     

基于广义逆矩阵的Ball曲线的降多阶逼近

给出了Ball曲线的一种降多阶逼近方法.将曲线的降多阶过程视为升阶的逆过程,利用广义逆矩阵的理论从而得到降阶曲线控制顶点的显式表示式.这种方法还考虑了原曲线与降阶曲线在两端点处分别达到(r,s)阶连续的情形(r≥0,s≥0).其次,给出了降阶误差界的估计.最后,给出数值例子.     

Special Points on the Product of Two Modular Curves

We prove, assuming the generalized Riemann hypothesis for imaginary quadratic fields, the following special case of a conjecture of Oort, concerning Zarsiski closures of sets of CM points in Shimura varieties. Let X be an irreducible algebraic curve in C², containing infinitely many points of which both coordinates are j-invariants of CM elliptic curves. Suppose that both projections from X to C are not constant. Then there is an integer m 1such that X is the image, under the usual map, of the modular curve Y₂0(m). The proof uses some number theory and some topological arguments.     

Some families of Mordell curves associated to cubic fields

We introduce some Mordell curves of two different natures both of which are associated to cubic fields. One set of them consists of those elliptic curves whose rational points over the rational number field are described by or closely related to cubic fields. The other is a one-parameter family of Mordell curves which gives all (cyclic) cubic twists and all quadratic twists of the Fermat curve X³+Y³+Z³=0.     

Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with even characteristic

This paper is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus 4 case and the finite fields are of even characteristics. The number of isomorphism classes is computed and the explicit formulae are given. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The result can be used in the classification problems and it is useful for further studies of hyperelliptic curve cryptosystems, e.g. it is of interest for research on implementing the arithmetics of curves of low genus for cryptographic purposes. It could also be of interest for point counting problems; both on moduli spaces of curves, and on finding the maximal number of points that a pointed hyperelliptic curve over a given finite field may have.     

3D Euler spirals for 3D curve completion

Shape completion is an intriguing problem in geometry processing with applications in CAD and graphics. This paper defines a new type of 3D curve, which can be utilized for curve completion. It can be considered as the extension to three dimensions of the 2D Euler spiral. We prove several properties of this curve - properties that have been shown to be important for the appeal of curves. We illustrate its utility in two applications. The first is “fixing” curves detected by algorithms for edge detection on surfaces. The second is shape illustration in archaeology, where the user would like to draw curves that are missing due to the incompleteness of the input model.     

Characteristic 2 approach to bivariate interpolation problems

We investigate bivariate Hermite interpolation problems in characteristic 2. Given a nonnegative integer t, we describe all the sub-linear systems generated by monomials, in which there is no curve passing through a general point with multiplicity at least 2 ^t . As an application, we show that a certain linear system of plane curves with ten base points is non-special.      

Approximate merging of B-spline curves and surfaces

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.     

Frontal Partner Curves on Unit Sphere S2

In this study, by using moving frame along frontal of Legendre curve, we define frontal partner curves on unit sphere S². We give the relationships between curvatures of Legendre curves and frontal partner curves are strengthen by an example.     

Curves on Generic Surfaces of High Degree Through a Complete Intersection in P 3

We consider general surfaces, S, of high degree containing a given complete intersection space curve, Y. We study integral curves in the subgroup of Pic(S) generated by Y and the plane section. We determine the cohomological invariants of these curves and classify the subcanonical ones. Then using these subcanonical curves we produce stable rank two vector bundles on P ³.     

Bisoptic curves of hyperbolas

Given a hyperbola, we study its bisoptic curves, i.e. the geometric locus of points through which passes a pair of tangents making a fixed angle θ or 180° ? θ. This question has been addressed in a previous paper for parabolas and for ellipses, showing hyperbolas and spiric curves, respectively. Here the requested geometric locus can be empty. If not, it is a punctured spiric curve, and two cases occur: the curve can have either one loop or two loops. Finally, we reconstruct explicitly the spiric curve as the intersection of a plane with a self-intersecting torus.     

Towers of 2-covers of hyperelliptic curves

In this article, we give a way of constructing an unramified Galois-cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian -group. The construction does not make use of the embedding of the curve in its Jacobian, and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplication-by- map of an embedding of the curve in its Jacobian.

We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. In particular the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2-covers of hyperelliptic curves.

As an application, we determine the rational points on the genus curve arising from the question of whether the sum of the first fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds.

     

Nowhere Vanishing Torsion Closed Curves Always Hide Twice

Convexity may imply points of vanishing torsion as the spatial 4-vertex theorem shows. We state here that for a simple closed curve to have nowhere vanishing torsion, it must violate convexity hiding at least twice inside its convex hull. Both the 4-vertex and Lsquo;hiding-twice results are generalized by obtaining a relation between the number of vanishing torsion points (vertices) of a closed space curve and the number of its components inside its convex hull. We also comment on elastic curves.     

Algorithms for multi-extremal mathematical programming problems employing the set of joint space-filling curves

Some powerful algorithms for multi-extremal non-convex-constrained optimization problems are based on reducing these multi-dimensional problems to those of one dimension by applying Peano-type space-filling curves mapping a unit interval on the real axis onto a multi-dimensional hypercube. Here is presented and substantiated a new scheme simultaneously employing several joint Peano-type scannings which conducts the property of nearness of points in many dimensions to a property of nearness of pre-images of these points in one dimension significantly better than in the case of a scheme with a single space-filling curve. Sufficient conditions of global convergence for the new scheme are investigated.This paper was presented at the II. IIASA Workshop on Global Optimization, Sopron (Hungary), December 9–14, 1990.