Clifford Algebra, Spin Representation, and Rational Parameterization of Curves and Surfaces

The Pythagorean hodograph (PH) curves are characterized by certain Pythagorean n-tuple identities in the polynomial ring, involving the derivatives of the curve coordinate functions. Such curves have many advantageous properties in computer aided geometric design. Thus far, PH curves have been studied in 2- or 3-dimensional Euclidean and Minkowski spaces. The characterization of PH curves in each of these contexts gives rise to different combinations of polynomials that satisfy further complicated identities. We present a novel approach to the Pythagorean hodograph curves, based on Clifford algebra methods, that unifies all known incarnations of PH curves into a single coherent framework. Furthermore, we discuss certain differential or algebraic geometric perspectives that arise from this new approach.     

On Rational Surfaces Ruled by Conics

Abstract

We study projective rational surfaces ruled by conics, describing their singularities and special fibres. In particular, if Sis smooth, we give a “canonical” procedure to determine a minimal model among the geometrically ruled surfaces birational to S.     

Canal Surfaces with Quaternions

Quaternions are more usable than three Euler angles in the three dimensional Euclidean space. Thus, many laws in different fields can be given by the quaternions. In this study, we show that canal surfaces and tube surfaces can be obtained by the quaternion product and by the matrix representation. Also, we show that the equation of canal surface given by the different frames of its spine curve can be obtained by the same unit quaternion. In addition, these surfaces are obtained by the homothetic motion. Then, we give some results.     

Anticanonical Rational Surfaces

A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically closed field of arbitrary characteristic. Applications, to be treated in separate papers, include questions involving: points in good position, birational models of rational surfaces in projective space, and resolutions for 0-dimensional subschemes of defined by complete ideals.

     

Rational Surfaces with Many Nodes

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain n b ₂ –2 disjoint smooth rational curves with self-intersection –2, where b ₂ is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with p _g = 0 and K² = 8, 9 which admit an automorphism of order 2.     

有理曲线和曲面的分片多项式逼近

本文利用摄动的思想,以摄动有理曲线（曲面）的系数的无穷模作为优化目标,给出了用多项式曲线（曲面）逼近有理曲线（曲面）的一种新方法．同以前的各种方法相比,该方法不仅收敛而且具有更快的收敛速度,并且可以与细分技术相结合,得到有理曲线与曲面的整体光滑、分片多项式的逼近．     

Integral Models of Extremal Rational Elliptic Surfaces

Miranda and Persson classified all extremal rational elliptic surfaces in characteristic zero. We show that each surface in Miranda and Persson's classification has an integral model with good reduction everywhere (except for those of type X ₁₁(j), which is an exceptional case), and that every extremal rational elliptic surface over an algebraically closed field of characteristic p > 0 can be obtained by reducing one of these integral models mod p.     

Very Ampleness of d-standard Classes on Rational Surfaces

We consider a rational surface X_r, obtained by blowing up ² along a curvilinear zero-dimensional subscheme of length r of the regular locus of a reduced irreducible plane curve of degree d, with d 4; and we give sufficient conditions for d-standard classes to be very ample (resp. base point free or non special) on such a rational surface X_r.     

On The Postulation of The Multiples of The Nodes for Generic Nodal Curves on Generic Rational Surfaces

Fix integers x > 0, m₁ ≥ … ≥ ^m _x > 0 and P₁,…,P_x ∈ P² such that no 3 of them are collinear. Let C ? P² a “ general ” degree d plane curve with an ordinary point with multiplicity m _i at each P _i and y further singularities which are ordinary nodes. Fix any A ? Sing(C){P₁,…, P_x} and any integer m > 0. Here we study the postulation of the fat points m A ? ∪_Q∈AmQ.     

Rank 2 Ample Vector Bundles on Some Smooth Rational Surfaces

Several classification results for ample vector bundles of rank 2 on Hirzebruch surfaces, and on Del Pezzo surfaces, are obtained. In particular, we classify rank 2 ample vector bundles with _c2 less than seven on Hirzebruch surfaces, and with _c2 less than four on Del Pezzo surfaces.     

On Some Special Surfaces Connected with Convex Surfaces of the Lobachevskii Space

With a closed convex surface in a Lobachevskii space we associate four special surfaces: the inscribed and circumscribed spheres, a sphere rolling freely over the inner side of , and an equidistant surface over whose inner side rolls freely. We find an exact dependence between these four special surfaces.     

Classification of Singular Fibers on Rational Elliptic Surfaces in Characteristic Three

ABSTRACT

We list all possible configurations of singular fibers on rational elliptic surfaces in characteristic three and prove that no others exist. In total, we find 267 distinct configurations. Configurations of purely multiplicative fibers in characteristic three are compared to those in characteristics zero and two.     

Stable Norms of Surfaces: Local Structure of the Unit Ball at Rational Directions

Let (M, g) be a closed orientable surface, equipped with a smooth Finsler metric. The metric induces a norm on the real homology of M, called a stable norm. We show this norm is neither strictly convex, nor smooth. Submitted: December 1996, final version: June 1997