DOCSLIB.ORG

As a One Big

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
As a One Big Glimpses of geometry and graphics Przemys law Koprowski Contents Introduction 5 Part 1. Geometry of space 7 Chapter 1. Homogeneous coordinates and affine transforms 9 1. Projective space 9 2. Homogeneous coordinates 11 3. Vectors and normals 17 4. Isometries 19 5. Transformation decomposition 23 Implementation notes 25 Exercises 25 Bibliographic notes 25 Chapter 2. Projective space and quaternions 27 1. Projective maps 27 2. Projections and perspective 32 3. Duality and the Pl¨ucker-Grassmann coordinates 41 4. Quaternions and transform interpolation 54 Exercises 62 Bibliographic notes 63 Part 2. Geometry of curves 65 Chapter 3. Cubic curves 67 1. Introduction 67 2. B´eziersplines 71 3. Interpolation with B´ezier curves 75 4. Catmull-Rom splines 79 5. B-splines 85 6. Other cubic representations 89 7. Change of basis 90 8. Curve splitting 93 Implementation notes 95 Exercises 95 Bibliographic notes 96 Chapter 4. Rational parametric curves 103 1. Why rational splines? 103 2. B´eziercurves 106 3 4 CONTENTS 3. Uniform B-splines 112 4. NURBS 118 5. Tangent and curvature 124 6. On-curve alignment 124 7. Manifolds and geometric continuity 124 8. Arc-length parametrization 124 Exercises 124 Chapter 5. Spherical curves 125 1. Transformation interpolation revisited 125 Chapter 6. Algebraic curves 127 1. Affine algebraic curves 127 2. Resultants 128 3. Projective algebraic curves 133 4. Singular points on algebraic curves 138 5. Parametrization of algebraic curves 141 6. Implicitization of parametric curves 147 7. Intersection points 148 8. Hessian and the inflection points 150 9. Cubic curves 154 Implementation notes 163 Exercises 163 Bibliographic notes 163 Part 3. Geometry of solids and surfaces 165 Chapter 7. Polygons and polyhedra 167 1. Triangles 167 2. Convex polytopes 170 3. Convex polygons 171 Chapter 8. Parametric surfaces 173 Bibliography 175 Introduction \They believe the world is run by geometry. All lines and angles and numbers. That sort of thing [ ... ] can lead to some very unsound ideas." | T. Pratchett Part 1 Geometry of space CHAPTER 1 Homogeneous coordinates and affine transforms \Space is essentially one" | I. Kant \We cannot assert with Kant that the propositions of Euclidean geometry possess any universal truth" | A. D'Abro This chapter introduces the notion of homogeneous coordinates, which is one of the fundamental structures in the domain of geometric modeling. To this end we shall define a projective space, which is an extension of an affine space known from the basic linear algebra. We won't, however, present an in-depth exposition of projective geometry in this chapter and, after introducing a homogeneous coordi- nate system, we will restrict ourselves to the affine case, returning to the projective space in Chapter 2. In computer graphics, one constantly deals with basic geometric transforma- tions of R3 such as a rotation, scaling and translation. The former two transfor- mations are linear maps, but the last one is not linear. It is well known, that every linear transform can be represented with a use of matrix multiplication. Fix T T T 3 the canonical basis "1 = (1; 0; 0) , "2 = (0; 1; 0) and "3 = (0; 0; 1) of R . Let 3 τ End R be a linear endomorphism and aij be the i-th coordinate of τ("i), then we2 have 0 1 0 1 0 1 x a11 a12 a13 x τ @yA = @a21 a22 a23A @yA · . z a31 a32 a33 z Our aim is to develop an equally convenient and uniform1 representation for all affine endomorphisms. In fact we will eventually be able to represent an even wider class of functions known as projective maps. 1. Projective space In this section we introduce the notion of a projective space. Let n N be a fixed dimension (in practice we usually need n = 2 or n = 3). Define a relation2 ∼ on the set Rn+1 (0;:::; 0)T by the condition n _ u v u = cv: ∼ () c 0 2Rnf g 1If all the transforms are represented in the same, standardized, fashion, they can be easily implemented in a hardware (e.g. GPU). Build: January 28, 2021 10 Homogeneous coordinates and affine transforms In other words, two non-zero vectors are related if they are linearly dependent (i.e. they belong to the same line through the origin). It is clear that is an equivalence relation. ∼ Definition 1.1. The set PnR of equivalence classes of (n + 1)-dimensional vectors with respect to the relation is called the n-dimensional projective space. T ∼ The class of a vector (x0; : : : ; xn) is denoted by x0 : ::: : xn . J K Of course, in the above definition one can substitute any arbitrary field (e.g. the rationals, the complex numbers,. ) in place of the reals and define respectively the rational projective space PnQ, the complex projective space PnC and so on. We won't need them, however, till Chapter 6. Consider now the subset Ui of the projective space PnR consisting of all the (projective) points having non-zero i-th coordinate: n Ui := x0 : ::: : xn P R xi = 0 : J K 2 j 6 Defining PnR we have excluded the null vector of Rn+1, consequently every pro- n Sn jective point has at least one non-zero coordinate. It follows that P R = i=0 Ui. n Observe that there is a bijection Ui R defined by the formula $ x x x x T x0 : ::: : xi : ::: : xn ( 0=xi;:::; i−1=xi; i+1=xi;:::; n=xi) : J K 7! It allows us to identify Ui (for every 1 i n) with the affine n-dimensional space n n ≤ ≤ n R . The map Ui R is called a dehomogenization of Ui and its inverse R Ui ! ! is the homogenization of Rn. It is most convenient to use them for either the first or the last coordinate and the choice between these two cases is just a matter of taste. We will stick to the latter convention, where the maps have the forms: T homogenization : (x1; : : : ; xn) x1 : ::: : xn : 1 ; 7! Jx x K T dehomogenization : x0 : ::: : xn 1 : xn ( 0=x1;:::; n−1=xn) : J − K 7! Let us characterize the completion of the affine space Rn in the projective space PnR. n Proposition 1.2. There is a canonical bijection between P R Un and the n 1 n (n 1)-dimensional projective space P − R. − Proof. By the very definition Un = x0 : ::: : xn xn = 0 , hence its n J K j 6 completion P R Un consists of all the projective points of the form x0 : ::: : n J xn 1 : 0 . Of course some xi = 0 due to the way we have constructed the projective space.− DroppingK the last coordinate,6 which is constantly zero anyway, we arrive n 1 at P − R. n 0 The set P R Un is called the set of points at infinity. Observe that P R consists n of a single point by definition. Then, the projective line P1R decomposes into a sum of the affine line R1 and the unique point at infinity. Next, the projective plane P2R consists of the affine plane R2 and the line at infinity. Further, the projective 3-space P3R is just the affine 3-space R3 together with the plane at infinity, and so on. There are simple geometric models for these low-dimensional projective spaces. n The coordinates of a projective point x0 : ::: : xn P R are unique only up to a multiplication by a non-zero factor.J Thus, normalizingK 2 them, we may assume that x2 + + x2 = 1. Consequently, every projective point is represented by a 0 ··· n point of a n-dimensional unit sphere Sn Rn+1. Unfortunately, there are two such points. These antipodal points P and ⊂P of a sphere map to the same point in − Build: January 28, 2021 Homogeneous coordinates and affine transforms 11 P P P Q P Q − P − Q Q Q − − Q − P − Figure 1.1. Circle as a model of the projective line the projective space. Consequently, one constructs the projective space by gluing together antipodal points of the unit sphere. In particular, we may easily check that the real projective line P1R can be identified with a unit circle. Imagine a rubber loop representing a circle. Twist it into an \eight-shape" and fold it so that the two halves overlap, joining the points that were originally opposing each other (see Fig. 1.1). We get a circle again! 2. Homogeneous coordinates We will return to the projective space in the next chapter, but for the time being, let us concentrate on the affine space Rn, which we identify with the subset n T n Un P R. Every point P = (x1; : : : ; xn) R can be assigned its projective co- ⊂ 2 ordinates x1 : ::: : xn : 1 . We call them homogeneous coordinates of P and write J T K [x1; : : : ; xn; 1] . In what follows, we always use square brackets when working with homogeneous coordinates and parentheses when dealing with linear/affine coordi- T nates. The distinction between x0 : ::: : xn and [x0; : : : ; xn] is purely artificial for points, but will become importantJ once weK admit vectors into our discussion. A word of caution: the homogeneous coordinates of a point (x; y)T R2 in the 2 plane are [xw; yw; w]T for any non-zero w R. Hence all: [x; y; 1]T, [2x; 2y; 2]T, [ 10:7x; 10:7y; 10:7]T,. refer to the same2 point. Similarly, the homogeneous co- − − − T 3 T T ordinates of a point (x; y; z) R are [x; y; z; 1] as well as [x=3; y=3; z=3; 1=3] ,.
Load more

Recommended publications
  • Projective Geometry: a Short Introduction
    Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g.
    [Show full text]
  • Algebraic Geometry Michael Stoll
    Introductory Geometry Course No. 100 351 Fall 2005 Second Part: Algebraic Geometry Michael Stoll Contents 1. What Is Algebraic Geometry? 2 2. Affine Spaces and Algebraic Sets 3 3. Projective Spaces and Algebraic Sets 6 4. Projective Closure and Affine Patches 9 5. Morphisms and Rational Maps 11 6. Curves — Local Properties 14 7. B´ezout’sTheorem 18 2 1. What Is Algebraic Geometry? Linear Algebra can be seen (in parts at least) as the study of systems of linear equations. In geometric terms, this can be interpreted as the study of linear (or affine) subspaces of Cn (say). Algebraic Geometry generalizes this in a natural way be looking at systems of polynomial equations. Their geometric realizations (their solution sets in Cn, say) are called algebraic varieties. Many questions one can study in various parts of mathematics lead in a natural way to (systems of) polynomial equations, to which the methods of Algebraic Geometry can be applied. Algebraic Geometry provides a translation between algebra (solutions of equations) and geometry (points on algebraic varieties). The methods are mostly algebraic, but the geometry provides the intuition. Compared to Differential Geometry, in Algebraic Geometry we consider a rather restricted class of “manifolds” — those given by polynomial equations (we can allow “singularities”, however). For example, y = cos x defines a perfectly nice differentiable curve in the plane, but not an algebraic curve. In return, we can get stronger results, for example a criterion for the existence of solutions (in the complex numbers), or statements on the number of solutions (for example when intersecting two curves), or classification results.
    [Show full text]
  • Arxiv:1910.11630V1 [Math.AG] 25 Oct 2019 3 Geometric Invariant Theory 10 3.1 Quotients and the Notion of Stability
    Geometric Invariant Theory, holomorphic vector bundles and the Harder–Narasimhan filtration Alfonso Zamora Departamento de Matem´aticaAplicada y Estad´ıstica Universidad CEU San Pablo Juli´anRomea 23, 28003 Madrid, Spain e-mail: [email protected] Ronald A. Z´u˜niga-Rojas Centro de Investigaciones Matem´aticasy Metamatem´aticas CIMM Escuela de Matem´atica,Universidad de Costa Rica UCR San Jos´e11501, Costa Rica e-mail: [email protected] Abstract. This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Narasimhan filtration and, from which, correspondences with the GIT picture and results derived from stratifications on the moduli space are discussed. Keywords: Geometric Invariant Theory, Harder-Narasimhan filtration, moduli spaces, vector bundles, Higgs bundles, GIT stability, symplectic stability, stratifications. MSC class: 14D07, 14D20, 14H10, 14H60, 53D30 Contents 1 Introduction 2 2 Preliminaries 4 2.1 Lie groups . .4 2.2 Lie algebras . .6 2.3 Algebraic varieties . .7 2.4 Vector bundles . .8 arXiv:1910.11630v1 [math.AG] 25 Oct 2019 3 Geometric Invariant Theory 10 3.1 Quotients and the notion of stability . 10 3.2 Hilbert-Mumford criterion . 14 3.3 Symplectic stability . 18 3.4 Examples . 21 3.5 Maximal unstability . 24 2 git, hvb & hnf 4 Moduli Space of vector bundles 28 4.1 GIT construction of the moduli space . 28 4.2 Harder-Narasimhan filtration .
    [Show full text]
  • Vector Bundles and Projective Modules
    VECTOR BUNDLES AND PROJECTIVE MODULES BY RICHARD G. SWAN(i) Serre [9, §50] has shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective mo- dules over its coordinate ring. For some time, it has been assumed that a similar correspondence exists between topological vector bundles over a compact Haus- dorff space X and finitely generated projective modules over the ring of con- tinuous real-valued functions on X. A number of examples of projective modules have been given using this correspondence. However, no rigorous treatment of the correspondence seems to have been given. I will give such a treatment here and then give some of the examples which may be constructed in this way. 1. Preliminaries. Let K denote either the real numbers, complex numbers or quaternions. A X-vector bundle £ over a topological space X consists of a space F(£) (the total space), a continuous map p : E(Ç) -+ X (the projection) which is onto, and, on each fiber Fx(¡z)= p-1(x), the structure of a finite di- mensional vector space over K. These objects are required to satisfy the follow- ing condition: for each xeX, there is a neighborhood U of x, an integer n, and a homeomorphism <p:p-1(U)-> U x K" such that on each fiber <b is a X-homo- morphism. The fibers u x Kn of U x K" are X-vector spaces in the obvious way. Note that I do not require n to be a constant.
    [Show full text]
  • Classical Algebraic Geometry
    CLASSICAL ALGEBRAIC GEOMETRY Daniel Plaumann Universität Konstanz Summer A brief inaccurate history of algebraic geometry - Projective geometry. Emergence of ’analytic’geometry with cartesian coordinates, as opposed to ’synthetic’(axiomatic) geometry in the style of Euclid. (Celebrities: Plücker, Hesse, Cayley) - Complex analytic geometry. Powerful new tools for the study of geo- metric problems over C.(Celebrities: Abel, Jacobi, Riemann) - Classical school. Perfected the use of existing tools without any ’dog- matic’approach. (Celebrities: Castelnuovo, Segre, Severi, M. Noether) - Algebraization. Development of modern algebraic foundations (’com- mutative ring theory’) for algebraic geometry. (Celebrities: Hilbert, E. Noether, Zariski) from Modern algebraic geometry. All-encompassing abstract frameworks (schemes, stacks), greatly widening the scope of algebraic geometry. (Celebrities: Weil, Serre, Grothendieck, Deligne, Mumford) from Computational algebraic geometry Symbolic computation and dis- crete methods, many new applications. (Celebrities: Buchberger) Literature Primary source [Ha] J. Harris, Algebraic Geometry: A first course. Springer GTM () Classical algebraic geometry [BCGB] M. C. Beltrametti, E. Carletti, D. Gallarati, G. Monti Bragadin. Lectures on Curves, Sur- faces and Projective Varieties. A classical view of algebraic geometry. EMS Textbooks (translated from Italian) () [Do] I. Dolgachev. Classical Algebraic Geometry. A modern view. Cambridge UP () Algorithmic algebraic geometry [CLO] D. Cox, J. Little, D.
    [Show full text]
  • Chapter 12 the Cross Ratio
    Chapter 12 The cross ratio Math 4520, Fall 2017 We have studied the collineations of a projective plane, the automorphisms of the underlying field, the linear functions of Affine geometry, etc. We have been led to these ideas by various problems at hand, but let us step back and take a look at one important point of view of the big picture. 12.1 Klein's Erlanger program In 1872, Felix Klein, one of the leading mathematicians and geometers of his day, in the city of Erlanger, took the following point of view as to what the role of geometry was in mathematics. This is from his \Recent trends in geometric research." Let there be given a manifold and in it a group of transforma- tions; it is our task to investigate those properties of a figure belonging to the manifold that are not changed by the transfor- mation of the group. So our purpose is clear. Choose the group of transformations that you are interested in, and then hunt for the \invariants" that are relevant. This search for invariants has proved very fruitful and useful since the time of Klein for many areas of mathematics, not just classical geometry. In some case the invariants have turned out to be simple polynomials or rational functions, such as the case with the cross ratio. In other cases the invariants were groups themselves, such as homology groups in the case of algebraic topology. 12.2 The projective line In Chapter 11 we saw that the collineations of a projective plane come in two \species," projectivities and field automorphisms.
    [Show full text]
  • Example Sheet 1
    Part III 2015-16: Differential geometry [email protected] Example Sheet 1 3 1 1. (i) Do the charts '1(x) = x and '2(x) = x (x 2 R) belong to the same C differentiable structure on R? (ii) Let Rj, j = 1; 2, be the manifold defined by using the chart 'j on the topo- logical space R. Are R1 and R2 diffeomorphic? 2. Let X be the metric space defined as follows: Let P1;:::;PN be distinct points in the Euclidean plane with its standard metric d, and define a distance function (not a ∗ 2 ∗ metric for N > 1) ρ on R by ρ (P; Q) = minfd(P; Q); mini;j(d(P; Pi) + d(Pj;Q))g: 2 Let X denote the quotient of R obtained by identifying the N points Pi to a single point P¯ on X. Show that ρ∗ induces a metric on X (the London Underground metric). Show that, for N > 1, the space X cannot be given the structure of a topological manifold. 3. (i) Prove that the product of smooth manifolds has the structure of a smooth manifold. (ii) Prove that n-dimensional real projective space RP n = Sn={±1g has the struc- ture of a smooth manifold of dimension n. (iii) Prove that complex projective space CP n := (Cn+1 nf0g)=C∗ has the structure of a smooth manifold of dimension 2n. 4. (i) Prove that the complex projective line CP 1 is diffeomorphic to the sphere S2. (ii) Show that the natural map (C2 n f0g) ! CP 1 induces a smooth map of manifolds S3 ! S2, the Poincar´emap.
    [Show full text]
  • Vector Bundles and Projective Varieties
    VECTOR BUNDLES AND PROJECTIVE VARIETIES by NICHOLAS MARINO Submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics, Applied Mathematics, and Statistics CASE WESTERN RESERVE UNIVERSITY January 2019 CASE WESTERN RESERVE UNIVERSITY Department of Mathematics, Applied Mathematics, and Statistics We hereby approve the thesis of Nicholas Marino Candidate for the degree of Master of Science Committee Chair Nick Gurski Committee Member David Singer Committee Member Joel Langer Date of Defense: 10 December, 2018 1 Contents Abstract 3 1 Introduction 4 2 Basic Constructions 5 2.1 Elementary Definitions . 5 2.2 Line Bundles . 8 2.3 Divisors . 12 2.4 Differentials . 13 2.5 Chern Classes . 14 3 Moduli Spaces 17 3.1 Some Classifications . 17 3.2 Stable and Semi-stable Sheaves . 19 3.3 Representability . 21 4 Vector Bundles on Pn 26 4.1 Cohomological Tools . 26 4.2 Splitting on Higher Projective Spaces . 27 4.3 Stability . 36 5 Low-Dimensional Results 37 5.1 2-bundles and Surfaces . 37 5.2 Serre's Construction and Hartshorne's Conjecture . 39 5.3 The Horrocks-Mumford Bundle . 42 6 Ulrich Bundles 44 7 Conclusion 48 8 References 50 2 Vector Bundles and Projective Varieties Abstract by NICHOLAS MARINO Vector bundles play a prominent role in the study of projective algebraic varieties. Vector bundles can describe facets of the intrinsic geometry of a variety, as well as its relationship to other varieties, especially projective spaces. Here we outline the general theory of vector bundles and describe their classification and structure. We also consider some special bundles and general results in low dimensions, especially rank 2 bundles and surfaces, as well as bundles on projective spaces.
    [Show full text]
  • 4 Lines and Cross Ratios
    4 Lines and cross ratios At this stage of this monograph we enter a significant didactic problem. There are three concepts which are intimately related and which unfold their full power only if they play together. These concepts are performing calculations with geometric objects, determinants and determinant algebra and geometric incidence theorems. The reader may excuse that in a beginners book that makes only little assumptions on the pre-knowledge one has to introduce these concepts one after the other. Thereby we will sacrifice some mathe- matical beauty for clearness of exposition. Still we highly recommend to read the following chapters (at least) twice. In order to get an impression of the interplay of the different concepts. This and the next section is devoted to the relations of RP2 to calcula- tions in the underlying field R. For this we will first find methods to relate points in a projective plane to the coordinates over R. Then we will show that elementary operations like addition and multiplication can be mimicked in a purely geometric fashion. Finally we will use these facts to derive in- teresting statements about the structure of projective planes and projective transformations. 4.1 Coordinates on a line Assume that two distinct points [p] and [q] in R are given. How can we describe the set of all points on the line throughP these two points. It is clear that we can implicitly describe them by first calculating the homogeneous coordinates of the line through p and q and then selecting all points that are incident to this line.
    [Show full text]
  • A QUICK INTRODUCTION to ELLIPTIC CURVES This Writeup
    A QUICK INTRODUCTION TO ELLIPTIC CURVES This writeup sketches aspects of the theory of elliptic curves, first over fields of characteristic zero and then over arbitrary fields. 1. Elliptic curves in characteristic zero Let k denote any field of characteristic 0. Definition 1. An element α of an extension field K of k is algebraic over k if it satisfies some monic polynomial with coefficients in k. Otherwise α is transcenden- tal over k. A field extension K/k is algebraic if every α K is algebraic over k. A field k is algebraically closed if there is no proper algebraic∈ extension K/k; that is, in any extension field K of k, any element α that is algebraic over k in fact lies in k. An algebraic closure k of k is a minimal algebraically closed extension field of k. ′ Every field has an algebraic closure. Any two algebraic closures k and k of k ∼ ′ are k-isomorphic, meaning there exists an isomorphism k k fixing k pointwise, so we often refer imprecisely to “the” algebraic closure of−→k. A Weierstrass equation over k is any cubic equation of the form (1) E : y2 =4x3 g x g , g ,g k. − 2 − 3 2 3 ∈ Define the discriminant of the equation to be ∆= g3 27g2 k, 2 − 3 ∈ and if ∆ = 0 define the invariant of the equation to be 6 j = 1728g3/∆ k. 2 ∈ Definition. Let k be an algebraic closure of the field k. When a Weierstrass equation E has nonzero discriminant ∆ it is called nonsingular and the set 2 = (x, y) k satisfying E(x, y) E { ∈ }∪{∞} is called an elliptic curve over k.
    [Show full text]
  • C- CI C WC 2 the Hypercanonical Embedding
    Journal of Pure and Applied Algebra 19 (1980) 77-83 2 North-Holland Publishing Company TRANSCANONICAL EMBEDDINGS OF HYPERELLIPTIC CURVES David EISENBUD * Departrnenr of Ma(hemarics. Brandeis Universily. H’alrham, MA 02254, USA Throughout this paper, a curve will be an irreducible nonsingular one- dimensional projective variety over an algebraically closed field F of characteristic not 2. In Chapter 1 of his beautiful survey [7], Mumford undertakes to exhibit “every curve”. The nonhyperelliptic curves are described by the equations of their canonical embeddings [8, 91, but the hyperelliptic curves are given only as ramified double coverings of Pi. In this note we describe the equations and geometry of the “transcanonical” embeddings of hyperelliptic curves, completing, in a sense, the above exhibition. I would like to thank Sebastian Xambb, who contributed a number of ideas to the formulations below. We fix for the remainder of the paper a hyperelliptic curve C of genus gr2, and let Ko be the divisor corresponding to the unique map 0 of degree 2 from C to the projective line P’ [5, IV 5.31. From the Riemann-Roth formula and the criterion [j, IV 3.11 one sees easily that the complete linear series l(g+ k)Kd is very ample iff kz 1. We call the divisor (g-t k)Ko, for kz 1, the kth transcanonical divisor (since the canonical divisor is (g- l)Ko), and we write CL, for the image of C under the corresponding embedding in projective space; thus Ck is a nondegenerate (that is, not contained in a hyperplane) curve of degree 2g + 2k in P’+ 2k.
    [Show full text]
  • 1 16.01. Lecture 1. Complex Algebraic Curves in C2 and Real Algebraic Curves
    1 16.01. Lecture 1. Complex algebraic curves in C2 and real algebraic curves Let P (x; y) 2 C[x; y] be a polynomial in 2 variables with complex coefficients. P has no repeated factors if it cannot be written as P (x; y) = Q(x; y)2R(x; y) for non-constant polynomial Q. Definition 1.1. Let P 2 C[x; y] be a non-constant polynomial with no repeated factors. The complex algebraic curve defined by P is 2 C = f(x; y) 2 C j P (x; y) = 0g: We want to assume that curves are given by polynomials with no repeated factors because of the following Theorem 1.2 (Hilbert's Nullstellensatz). If P (x; y) and Q(x; y) are polynomials in C[x; y] then 2 2 f(x; y) 2 C j P (x; y) = 0g = f(x; y) 2 C j Q(x; y) = 0g if and only if there exist m; n 2 N such that P divides Qm and Q divides P n. In other words, if and only if P and Q have the same irreducible factors possibly occurring with different multiplicities. It follows from Theorem 1.2 that two polynomials P , Q with no repeated factors define the same complex algebraic curve C if and only if there exists λ 2 C∗ such that P = λQ. In general, one can define complex algebraic curves as equivalence classes of non- constant polynomials P 2 C[x; y] where P ∼ Q if and only if P and Q have the same irreducible factors.
    [Show full text]