DOCSLIB.ORG

Elliptic Curve Handbook

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Elliptic Curve Handbook Elliptic Curve Handbook Ian Connell February, 1999 http://www.math.mcgill.ca/connell/ Foreword The first version of this handbook was a set of notes of about 100 pages handed out to the class of an introductory course on elliptic curves given in the 1990 fall semester at McGill University in Montreal. Since then I have added to the notes, holding to the principle: If I look up a certain topic a year from now I want all the details right at hand, not in an “exercise”, so if I’ve forgotten something I won’t waste time. Thus there is much that an ordinary text would either condense, or relegate to an exercise. But at the same time I have maintained a solid mathematical style with the thought of sharing the handbook. Montreal, August, 1996. 1 Contents 1 Introduction to Elliptic Curves 101 1.1 The a,b,c’s and ∆,j ...................... 101 1.2 Quartic to Weierstrass ..................... 105 1.3 Projective coordinates. ..................... 111 1.4 Cubic to Weierstrass: Nagell’s algorithm ........... 115 1.4.1 Example 1: Selmer curves ............... 117 1.4.2 Example 2: Desboves curves .............. 121 1.4.3 Example 3: Intersection of quadric surfaces ..... 123 1.5 Singular points. ......................... 125 − 1.5.1 Example: No E/Z has ∆ = 1 or 1 .......... 130 1.6 Affine coordinate ring, function field, generic points ..... 132 1.7 The group law: nonsingular case ............... 133 1.7.1 Halving points ..................... 140 1.7.2 The division polynomials ............... 145 1.7.3 Remarks on the group of division points ....... 151 1.8 The group law: singular case ................. 152 1.8.1 Examples over finite fields ............... 156 Appendix: introduction to apecs .................. 160 2 Formal Groups 201 2.1 Discrete valuations ....................... 202 2.1.1 Examples ........................ 206 2.1.2 The filtration Em(K) ................. 208 2.1.3 Finite extensions .................... 210 2.1.4 Gauss’s lemma ..................... 212 2.2 Krull domains .......................... 212 2.2.1 Dedekind domains ................... 216 2.2.2 One variable function fields .............. 217 2.2.3 Elliptic function fields ................. 223 2.3 The group of reversible power series ............. 225 3 4 CONTENTS 2.4 Hensel’s lemma ......................... 228 2.4.1 An application to P -adically reversible series .... 232 2.5 Applications to elliptic curves ................. 234 2.5.1 Infinitesimal shifts ................... 234 2.5.2 Reduction mod π : a first look ............ 235 2.5.3 Local expansions .................... 240 2.6 Formal groups .......................... 244 2.6.1 The additive and multiplicative formal groups .... 248 2.6.2 The formal group of an elliptic curve ......... 251 2.7 The invariant differential of a formal group ......... 253 2.7.1 The elliptic curve case ................. 258 2.8 Formal groups in characteristic p............... 261 2.9 Formal groups in characteristic 0 ............... 263 2.9.1 The formal logarithm ................. 263 2.9.2 Formal groups over discrete valuation rings ..... 264 2.10 The Nagell-Lutz theorem for Krull domains ......... 268 2.10.1 Nagell-Lutz for Z .................... 273 2.10.2 Nagell-Lutz for quadratic fields ............ 277 3 The Mordell-Weil theorem 301 3.1 F2-Krull domains ........................ 303 3.2 The weak Mordell-Weil theorm ................ 305 3.3 Heights ............................. 307 3.3.1 Heights in number fields ................ 308 3.3.2 Heights in function fields ............... 311 3.4 Completion of the proof of Mordell-Weil ........... 314 3.4.1 Function fields in characteristic 0 ........... 318 3.5 The canonical height ...................... 321 3.5.1 Calculating the canonical height: a first look .... 329 3.5.2 The successive minima ................. 333 3.6 Algorithms for Mordell-Weil bases: a first look ....... 336 3.6.1 Simple 2-descent .................... 340 3.6.2 Simple 2-descent over UFD’s ............. 345 3.6.3 Examples over Q .................... 348 3.6.4 The Hilbert norm residue symbol ........... 351 3.6.5 Continuation of examples over Q ........... 355 3.6.6 Second descent ..................... 366 3.6.7 A transcendental example ............... 371 3.7 Billing’s upper bound for the rank .............. 372 3.7.1 Examples ........................ 377 3.7.2 An example of Selmer ................. 381 CONTENTS 5 4 Twists 401 4.1 Isomorphisms of elliptic curves ................ 401 4.2 Simplified Weierstrass equations ............... 405 4.3 Twists, quadratic and otherwise ................ 408 4.4 The isomorphism algorithm .................. 414 4.5 Automorphisms and fields of definition ............ 417 4.6 Legendre and Deuring forms .................. 420 4.7 Finite fields ........................... 423 4.7.1 The trace of Frobenius: preliminaries ......... 423 4.7.2 An application of Burnside’s formula ......... 426 4.7.3 The trace of Frobenius: continuation ......... 431 4.7.4 Examples ........................ 436 4.7.5 A preview of some future topics ............ 439 5 Minimal Weierstrass Equations 501 5.1 Some definitions ........................ 502 5.2 Kraus’s theorem ........................ 503 5.2.1 The case v(p) = 1 ................... 507 5.2.2 The globalization of Kraus’s theorem to number fields 510 5.3 Local minimal forms and δv .................. 513 5.3.1 The special cases v(3) = 1 and v(2) = 1 ....... 516 5.4 Unramified base change .................... 518 5.5 Global minimal equations and AE .............. 523 5.6 The Laska-Kraus algorithm .................. 525 5.6.1 The case Z ....................... 525 5.6.2 The general number field case ............. 530 5.7 How twisting affects minimal models ............. 534 5.7.1 The local case ...................... 534 5.7.2 The global case ..................... 537 5.7.3 Minimal twists ..................... 540 Chapter 1 Introduction to Elliptic Curves. 1.1 The a, b, c ’s and ∆, j, . We begin with a series of definitions of elliptic curve in order of increasing generality and sophistication. These definitions involve technical terms which will be defined at some point in what follows. The most concrete definition is that of a curve E given by a nonsingular Weierstrass equation: 2 3 2 y + a1xy + a3y = x + a2x + a4x + a6. (1) The coefficients ai are in a field K and E(K) denotes the set of all solutions (x, y) K K, together with the point O “at infinity” — to be explained in 1.3. We∈ will× see later why the a’s are numbered in this way; to remember the Weierstrass§ equation think of the terms as being in a graded ring with weight of x = 2 "" y = 3 "" ai = i so that each term in the equation has weight 6. (This also “explains” the absence of a5.) A slightly more general definition is: a plane nonsingular cubic with a ra- tional point (rational means the coordinates are in the designated field K and does not refer to the rational field Q, unless of course K = Q). An example of such a curve that is not a Weierstrass equation is the Fermat curve x3 + y3 = 1, with points (x, y) = (1, 0), (0, 1), 101 102 CHAPTER 1. INTRODUCTION TO ELLIPTIC CURVES. assuming the characteristic of K, denoted char K, is not 3. In Corollary 1.4.2 we will see how to transform such an equation into Weierstrass form. More general still: a nonsingular curve of genus 1 with a rational point. (As we will explain later, conic sections — circles, ellipses, parabolas, and hyperbolas — have genus 0 which implies that they are not elliptic curves.) An example that is not encompassed by the previous definitions is y2 = 3x4 2, with points (x, y) = ( 1, 1), − ± ± assuming char K = 2, 3. Proposition 1.2.1 below explains how to transform such quartic equations6 into Weierstrass form (without using √4 3 or √ 2 !). Alternative terminology which emphasizes the algebraic group− structure: abelian variety of dimension 1. More abstractly: E is a scheme over a base scheme S (e.g. spec K) which is proper, flat and finitely presented, equipped with a section ... : there is little point to state all the technicalities at this time. Suffice it to say that the work of Tate, Mazur and many others makes it plain that it is essential to know the language of schemes to understand the deeper arithmetic properties of elliptic curves. (More easily said than done!) Now let us begin to fill in some details. Consider a Weierstrass equation (1), which we denote as E. If char K = 2 we can complete the square by defining 6 η = y + (a1x + a3)/2: b b b η2 = x3 + 2 x2 + 4 x + 6 (2) 4 2 4 2 2 where b2 = a1 + 4a2, b4 = a1a3 + 2a4, b6 = a3 + 4a6. (3) If char K = 3 we can complete the cube by setting ξ = x + b /12: 6 2 c c η2 = ξ3 4 ξ 6 (4) − 48 − 864 2 3 where c4 = b 24b4, c6 = b + 36b2b4 216b6. (5) 2 − − 2 − One then defines b = a2a a a a + 4a a + a a2 a2, (6) 8 1 6 − 1 3 4 2 6 2 3 − 4 2 3 2 and ∆ = b b8 8b 27b + 9b2b4b6. (7) − 2 − 4 − 6 The subscripts on the b’s and c’s are their weights. We refer to (1), (2) and (4) as the a-form, b-form and c-form respectively. The definitions (3) and (5)–(7) are made for all E, regardless of the characteristic of K, and the condition that the curve be nonsingular, and so define an elliptic curve, is that ∆ = 0, as we will explain in 1.5. Then one defines j = c3/∆. For example 6 § 4 when char K = 2, ∆ = 0 = a and a are not both zero. 6 ⇒ 1 3 1.1. THE A, B, C ’S AND ∆, J, . 103 Thus κ := 2y + a1x + a3 is nonzero† for every elliptic curve E in any characteristic. When char K = 2 we have κ = 2η and κ is determined up to sign by x.
Load more

Recommended publications
  • 13 Elliptic Function
    13 Elliptic Function Recall that a function g : C → Cˆ is an elliptic function if it is meromorphic and there exists a lattice L = {mω1 + nω2 | m, n ∈ Z} such that g(z + ω)= g(z) for all z ∈ C and all ω ∈ L where ω1,ω2 are complex numbers that are R-linearly independent. We have shown that an elliptic function cannot be holomorphic, the number of its poles are finite and the sum of their residues is zero. Lemma 13.1 A non-constant elliptic function f always has the same number of zeros mod- ulo its associated lattice L as it does poles, counting multiplicites of zeros and orders of poles. Proof: Consider the function f ′/f, which is also an elliptic function with associated lattice L. We will evaluate the sum of the residues of this function in two different ways as above. Then this sum is zero, and by the argument principle from complex analysis 31, it is precisely the number of zeros of f counting multiplicities minus the number of poles of f counting orders. Corollary 13.2 A non-constant elliptic function f always takes on every value in Cˆ the same number of times modulo L, counting multiplicities. Proof: Given a complex value b, consider the function f −b. This function is also an elliptic function, and one with the same poles as f. By Theorem above, it therefore has the same number of zeros as f. Thus, f must take on the value b as many times as it does 0.
    [Show full text]
  • Combination of Cubic and Quartic Plane Curve
    IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis.
    [Show full text]
  • ELLIPTIC FUNCTIONS (Approach of Abel and Jacobi)
    Math 213a (Fall 2021) Yum-Tong Siu 1 ELLIPTIC FUNCTIONS (Approach of Abel and Jacobi) Significance of Elliptic Functions. Elliptic functions and their associated theta functions are a new class of special functions which play an impor- tant role in explicit solutions of real world problems. Elliptic functions as meromorphic functions on compact Riemann surfaces of genus 1 and their associated theta functions as holomorphic sections of holomorphic line bun- dles on compact Riemann surfaces pave the way for the development of the theory of Riemann surfaces and higher-dimensional abelian varieties. Two Approaches to Elliptic Function Theory. One approach (which we call the approach of Abel and Jacobi) follows the historic development with motivation from real-world problems and techniques developed for solving the difficulties encountered. One starts with the inverse of an elliptic func- tion defined by an indefinite integral whose integrand is the reciprocal of the square root of a quartic polynomial. An obstacle is to show that the inverse function of the indefinite integral is a global meromorphic function on C with two R-linearly independent primitive periods. The resulting dou- bly periodic meromorphic functions are known as Jacobian elliptic functions, though Abel was actually the first mathematician who succeeded in inverting such an indefinite integral. Nowadays, with the use of the notion of a Rie- mann surface, the inversion can be handled by using the fundamental group of the Riemann surface constructed to make the square root of the quartic polynomial single-valued. The great advantage of this approach is that there is vast literature for the properties of the Jacobain elliptic functions and of their associated Jacobian theta functions.
    [Show full text]
  • Lecture 9 Riemann Surfaces, Elliptic Functions Laplace Equation
    Lecture 9 Riemann surfaces, elliptic functions Laplace equation Consider a Riemannian metric gµν in two dimensions. In two dimensions it is always possible to choose coordinates (x,y) to diagonalize it as ds2 = Ω(x,y)(dx2 + dy2). We can then combine them into a complex combination z = x+iy to write this as ds2 =Ωdzdz¯. It is actually a K¨ahler metric since the condition ∂[i,gj]k¯ = 0 is trivial if i, j = 1. Thus, an orientable Riemannian manifold in two dimensions is always K¨ahler. In the diagonalized form of the metric, the Laplace operator is of the form, −1 ∆=4Ω ∂z∂¯z¯. Thus, any solution to the Laplace equation ∆φ = 0 can be expressed as a sum of a holomorphic and an anti-holomorphid function. ∆φ = 0 φ = f(z)+ f¯(¯z). → In the following, we assume Ω = 1 so that the metric is ds2 = dzdz¯. It is not difficult to generalize our results for non-constant Ω. Now, we would like to prove the following formula, 1 ∂¯ = πδ(z), z − where δ(z) = δ(x)δ(y). Since 1/z is holomorphic except at z = 0, the left-hand side should vanish except at z = 0. On the other hand, by the Stokes theorem, the integral of the left-hand side on a disk of radius r gives, 1 i dz dxdy ∂¯ = = π. Zx2+y2≤r2 z 2 I|z|=r z − This proves the formula. Thus, the Green function G(z, w) obeying ∆zG(z, w) = 4πδ(z w), − should behave as G(z, w)= log z w 2 = log(z w) log(¯z w¯), − | − | − − − − near z = w.
    [Show full text]
  • Polynomial Curves and Surfaces
    Polynomial Curves and Surfaces Chandrajit Bajaj and Andrew Gillette September 8, 2010 Contents 1 What is an Algebraic Curve or Surface? 2 1.1 Algebraic Curves . .3 1.2 Algebraic Surfaces . .3 2 Singularities and Extreme Points 4 2.1 Singularities and Genus . .4 2.2 Parameterizing with a Pencil of Lines . .6 2.3 Parameterizing with a Pencil of Curves . .7 2.4 Algebraic Space Curves . .8 2.5 Faithful Parameterizations . .9 3 Triangulation and Display 10 4 Polynomial and Power Basis 10 5 Power Series and Puiseux Expansions 11 5.1 Weierstrass Factorization . 11 5.2 Hensel Lifting . 11 6 Derivatives, Tangents, Curvatures 12 6.1 Curvature Computations . 12 6.1.1 Curvature Formulas . 12 6.1.2 Derivation . 13 7 Converting Between Implicit and Parametric Forms 20 7.1 Parameterization of Curves . 21 7.1.1 Parameterizing with lines . 24 7.1.2 Parameterizing with Higher Degree Curves . 26 7.1.3 Parameterization of conic, cubic plane curves . 30 7.2 Parameterization of Algebraic Space Curves . 30 7.3 Automatic Parametrization of Degree 2 Curves and Surfaces . 33 7.3.1 Conics . 34 7.3.2 Rational Fields . 36 7.4 Automatic Parametrization of Degree 3 Curves and Surfaces . 37 7.4.1 Cubics . 38 7.4.2 Cubicoids . 40 7.5 Parameterizations of Real Cubic Surfaces . 42 7.5.1 Real and Rational Points on Cubic Surfaces . 44 7.5.2 Algebraic Reduction . 45 1 7.5.3 Parameterizations without Real Skew Lines . 49 7.5.4 Classification and Straight Lines from Parametric Equations . 52 7.5.5 Parameterization of general algebraic plane curves by A-splines .
    [Show full text]
  • FINITE FOURIER SERIES and OVALS in PG(2, 2H)
    J. Aust. Math. Soc. 81 (2006), 21-34 FINITE FOURIER SERIES AND OVALS IN PG(2, 2h) J. CHRIS FISHERY and BERNHARD SCHMIDT (Received 2 April 2004; revised 7 May 2005) Communicated by L. Batten Abstract We propose the use of finite Fourier series as an alternative means of representing ovals in projective planes of even order. As an example to illustrate the method's potential, we show that the set [w1 + wy' + w ~y' : 0 < j < 2;'| c GF(22/1) forms an oval if w is a primitive (2h + l)sl root of unity in GF(22'') and GF(22'') is viewed as an affine plane over GF(2;'). For the verification, we only need some elementary 'trigonometric identities' and a basic irreducibility lemma that is of independent interest. Finally, we show that our example is the Payne oval when h is odd, and the Adelaide oval when h is even. 2000 Mathematics subject classification: primary 51E20, 05B25. 1. Introduction In any finite projective plane of order q, an oval is a set of q + 1 points, no three of which are collinear. In the classical plane PG(2, q) over GF(g), a nondegenerate conic is the prototypical oval. If the order of a plane is even, then the tangents to an oval all pass through a point that is called the nucleus of the oval. We call an oval together with its nucleus a hyperoval. During the 1950s Beniamino Segre proved that in PG(2, q), (1) when q is odd, then there exist no ovals other than the conies, and (2) when q — 2\ coordinates may be chosen so that the points of a hyperoval are the elements of the set {(*, f{x), 1) | JC G GF(<?)} U {(0, 1,0), (1,0, 0)}, where / is a permutation polynomial of degree at most q — 2 for which /(0) = 0, /(I) = 1, and with the additional property that for all 5 in GF(^), the function /,.
    [Show full text]
  • Construction of Rational Surfaces of Degree 12 in Projective Fourspace
    Construction of Rational Surfaces of Degree 12 in Projective Fourspace Hirotachi Abo Abstract The aim of this paper is to present two different constructions of smooth rational surfaces in projective fourspace with degree 12 and sectional genus 13. In particular, we establish the existences of five different families of smooth rational surfaces in projective fourspace with the prescribed invariants. 1 Introduction Hartshone conjectured that only finitely many components of the Hilbert scheme of surfaces in P4 contain smooth rational surfaces. In 1989, this conjecture was positively solved by Ellingsrud and Peskine [7]. The exact bound for the degree is, however, still open, and hence the question concerning the exact bound motivates us to find smooth rational surfaces in P4. The goal of this paper is to construct five different families of smooth rational surfaces in P4 with degree 12 and sectional genus 13. The rational surfaces in P4 were previously known up to degree 11. In this paper, we present two different constructions of smooth rational surfaces in P4 with the given invariants. Both the constructions are based on the technique of “Beilinson monad”. Let V be a finite-dimensional vector space over a field K and let W be its dual space. The basic idea behind a Beilinson monad is to represent a given coherent sheaf on P(W ) as a homology of a finite complex whose objects are direct sums of bundles of differentials. The differentials in the monad are given by homogeneous matrices over an exterior algebra E = V V . To construct a Beilinson monad for a given coherent sheaf, we typically take the following steps: Determine the type of the Beilinson monad, that is, determine each object, and then find differentials in the monad.
    [Show full text]
  • NOTES on ELLIPTIC CURVES Contents 1
    NOTES ON ELLIPTIC CURVES DINO FESTI Contents 1. Introduction: Solving equations 1 1.1. Equation of degree one in one variable 2 1.2. Equations of higher degree in one variable 2 1.3. Equations of degree one in more variables 3 1.4. Equations of degree two in two variables: plane conics 4 1.5. Exercises 5 2. Cubic curves and Weierstrass form 6 2.1. Weierstrass form 6 2.2. First definition of elliptic curves 8 2.3. The j-invariant 10 2.4. Exercises 11 3. Rational points of an elliptic curve 11 3.1. The group law 11 3.2. Points of finite order 13 3.3. The Mordell{Weil theorem 14 3.4. Isogenies 14 3.5. Exercises 15 4. Elliptic curves over C 16 4.1. Ellipses and elliptic curves 16 4.2. Lattices and elliptic functions 17 4.3. The Weierstrass } function 18 4.4. Exercises 22 5. Isogenies and j-invariant: revisited 22 5.1. Isogenies 22 5.2. The group SL2(Z) 24 5.3. The j-function 25 5.4. Exercises 27 References 27 1. Introduction: Solving equations Solving equations or, more precisely, finding the zeros of a given equation has been one of the first reasons to study mathematics, since the ancient times. The branch of mathematics devoted to solving equations is called Algebra. We are going Date: August 13, 2017. 1 2 DINO FESTI to see how elliptic curves represent a very natural and important step in the study of solutions of equations. Since 19th century it has been proved that Geometry is a very powerful tool in order to study Algebra.
    [Show full text]
  • Lectures on the Combinatorial Structure of the Moduli Spaces of Riemann Surfaces
    LECTURES ON THE COMBINATORIAL STRUCTURE OF THE MODULI SPACES OF RIEMANN SURFACES MOTOHICO MULASE Contents 1. Riemann Surfaces and Elliptic Functions 1 1.1. Basic Definitions 1 1.2. Elementary Examples 3 1.3. Weierstrass Elliptic Functions 10 1.4. Elliptic Functions and Elliptic Curves 13 1.5. Degeneration of the Weierstrass Elliptic Function 16 1.6. The Elliptic Modular Function 19 1.7. Compactification of the Moduli of Elliptic Curves 26 References 31 1. Riemann Surfaces and Elliptic Functions 1.1. Basic Definitions. Let us begin with defining Riemann surfaces and their moduli spaces. Definition 1.1 (Riemann surfaces). A Riemann surface is a paracompact Haus- S dorff topological space C with an open covering C = λ Uλ such that for each open set Uλ there is an open domain Vλ of the complex plane C and a homeomorphism (1.1) φλ : Vλ −→ Uλ −1 that satisfies that if Uλ ∩ Uµ 6= ∅, then the gluing map φµ ◦ φλ φ−1 (1.2) −1 φλ µ −1 Vλ ⊃ φλ (Uλ ∩ Uµ) −−−−→ Uλ ∩ Uµ −−−−→ φµ (Uλ ∩ Uµ) ⊂ Vµ is a biholomorphic function. Remark. (1) A topological space X is paracompact if for every open covering S S X = λ Uλ, there is a locally finite open cover X = i Vi such that Vi ⊂ Uλ for some λ. Locally finite means that for every x ∈ X, there are only finitely many Vi’s that contain x. X is said to be Hausdorff if for every pair of distinct points x, y of X, there are open neighborhoods Wx 3 x and Wy 3 y such that Wx ∩ Wy = ∅.
    [Show full text]
  • A Brief Note on the Approach to the Conic Sections of a Right Circular Cone from Dynamic Geometry
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by EPrints Complutense A brief note on the approach to the conic sections of a right circular cone from dynamic geometry Eugenio Roanes–Lozano Abstract. Nowadays there are different powerful 3D dynamic geometry systems (DGS) such as GeoGebra 5, Calques 3D and Cabri Geometry 3D. An obvious application of this software that has been addressed by several authors is obtaining the conic sections of a right circular cone: the dynamic capabilities of 3D DGS allows to slowly vary the angle of the plane w.r.t. the axis of the cone, thus obtaining the different types of conics. In all the approaches we have found, a cone is firstly constructed and it is cut through variable planes. We propose to perform the construction the other way round: the plane is fixed (in fact it is a very convenient plane: z = 0) and the cone is the moving object. This way the conic is expressed as a function of x and y (instead of as a function of x, y and z). Moreover, if the 3D DGS has algebraic capabilities, it is possible to obtain the implicit equation of the conic. Mathematics Subject Classification (2010). Primary 68U99; Secondary 97G40; Tertiary 68U05. Keywords. Conics, Dynamic Geometry, 3D Geometry, Computer Algebra. 1. Introduction Nowadays there are different powerful 3D dynamic geometry systems (DGS) such as GeoGebra 5 [2], Calques 3D [7, 10] and Cabri Geometry 3D [11]. Focusing on the most widespread one (GeoGebra 5) and conics, there is a “conics” icon in the ToolBar.
    [Show full text]
  • Chapter 9: Conics Section 9.1 Ellipses
    Chapter 9: Conics Section 9.1 Ellipses ..................................................................................................... 579 Section 9.2 Hyperbolas ............................................................................................... 597 Section 9.3 Parabolas and Non-Linear Systems ......................................................... 617 Section 9.4 Conics in Polar Coordinates..................................................................... 630 In this chapter, we will explore a set of shapes defined by a common characteristic: they can all be formed by slicing a cone with a plane. These families of curves have a broad range of applications in physics and astronomy, from describing the shape of your car headlight reflectors to describing the orbits of planets and comets. Section 9.1 Ellipses The National Statuary Hall1 in Washington, D.C. is an oval-shaped room called a whispering chamber because the shape makes it possible for sound to reflect from the walls in a special way. Two people standing in specific places are able to hear each other whispering even though they are far apart. To determine where they should stand, we will need to better understand ellipses. An ellipse is a type of conic section, a shape resulting from intersecting a plane with a cone and looking at the curve where they intersect. They were discovered by the Greek mathematician Menaechmus over two millennia ago. The figure below2 shows two types of conic sections. When a plane is perpendicular to the axis of the cone, the shape of the intersection is a circle. A slightly titled plane creates an oval-shaped conic section called an ellipse. 1 Photo by Gary Palmer, Flickr, CC-BY, https://www.flickr.com/photos/gregpalmer/2157517950 2 Pbroks13 (https://commons.wikimedia.org/wiki/File:Conic_sections_with_plane.svg), “Conic sections with plane”, cropped to show only ellipse and circle by L Michaels, CC BY 3.0 This chapter is part of Precalculus: An Investigation of Functions © Lippman & Rasmussen 2020.
    [Show full text]
  • Dissertation Hyperovals, Laguerre Planes And
    DISSERTATION HYPEROVALS, LAGUERRE PLANES AND HEMISYSTEMS { AN APPROACH VIA SYMMETRY Submitted by Luke Bayens Department of Mathematics In partial fulfillment of the requirements For the Degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Spring 2013 Doctoral Committee: Advisor: Tim Penttila Jeff Achter Willem Bohm Chris Peterson ABSTRACT HYPEROVALS, LAGUERRE PLANES AND HEMISYSTEMS { AN APPROACH VIA SYMMETRY In 1872, Felix Klein proposed the idea that geometry was best thought of as the study of invariants of a group of transformations. This had a profound effect on the study of geometry, eventually elevating symmetry to a central role. This thesis embodies the spirit of Klein's Erlangen program in the modern context of finite geometries { we employ knowledge about finite classical groups to solve long-standing problems in the area. We first look at hyperovals in finite Desarguesian projective planes. In the last 25 years a number of infinite families have been constructed. The area has seen a lot of activity, motivated by links with flocks, generalized quadrangles, and Laguerre planes, amongst others. An important element in the study of hyperovals and their related objects has been the determination of their groups { indeed often the only way of distinguishing them has been via such a calculation. We compute the automorphism group of the family of ovals constructed by Cherowitzo in 1998, and also obtain general results about groups acting on hyperovals, including a classification of hyperovals with large automorphism groups. We then turn our attention to finite Laguerre planes. We characterize the Miquelian Laguerre planes as those admitting a group containing a non-trivial elation and acting tran- sitively on flags, with an additional hypothesis { a quasiprimitive action on circles for planes of odd order, and insolubility of the group for planes of even order.
    [Show full text]