Kig can handle any classical object of the dynamic geometry, but also: # The center of curvature and osculating circle of a curve; # The dilation, generic affinity, inversion, projective application, homography and harmonic homology; # The hyperbola with given asymptotes; # The Bézier curves (2nd and 3rd degree); # The polar line of a point and pole of a line with respect to a conic section; # The asymptotes of a hyperbola; # The cubic curve through 9 points; # The cubic curve with a double point through 6 points; # The cubic curve with a cusp through 4 points.
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Without idempotency, total symmetric quasigroups correspond to the geometric notion of extended Steiner triple, also called Generalized Elliptic Cubic Curve (GECC).
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The plane algebraic curve (a cubic curve) of equation crosses itself at the origin (0,0). The origin is a double point of this curve. It is singular because a single tangent may not be correctly defined there.
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A cubic curve may have a singular point, in which case it has a parametrization in terms of a projective line. Otherwise a non-singular cubic curve is known to have nine points of inflection, over an algebraically closed field such as the complex numbers. This can be shown by taking the homogeneous version of the Hessian matrix, which defines again a cubic, and intersecting it with C; the intersections are then counted by Bézout's theorem. However, only three of these points may be real, so that the others cannot be seen in the real projective plane by drawing the curve.
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This does depend on having a K-rational point, which serves as the point at infinity in Weierstrass form. There are many cubic curves that have no such point, for example when K is the rational number field. The singular points of an irreducible plane cubic curve are quite limited: one double point, or one cusp. A reducible plane cubic curve is either a conic and a line or three lines, and accordingly have two double points or a tacnode (if a conic and a line), or up to three double points or a single triple point (concurrent lines) if three lines.
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The map :\sigma: P^2 \times P^1 \to P^5 is known as the Segre threefold. It is an example of a rational normal scroll. The intersection of the Segre threefold and a three-plane P^3 is a twisted cubic curve.
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Suppose that the equation :y^2 = x^3 + ax^2 + bx + c defines a non-singular cubic curve with integer coefficients a, b, c, and let D be the discriminant of the cubic polynomial on the right side: :D = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2.
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As x goes to positive infinity, the slope of the line from the origin to the point (x, x2) also goes to positive infinity. As x goes to negative infinity, the slope of the same line goes to negative infinity. Compare this to the variety V(y − x3). This is a cubic curve.
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Successive division may be used. Such a fitting procedure tries to fit the curve with a single cubic curve; if the fit is acceptable, then the procedure stops. Otherwise, it selects some advantageous point along the curve and breaks the curve into two parts. It then fits the parts while keeping the joint tangent.
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The only 'easy' cases are those for d = 1, for an elliptic curve with linear span the projective plane or projective 3-space. In the plane, every elliptic curve is given by a cubic curve. In P3, an elliptic curve can be obtained as the intersection of two quadrics. In general abelian varieties are not complete intersections.
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The Weierstrass functions are doubly periodic; that is, they are periodic with respect to a lattice Λ; in essence, the Weierstrass functions are naturally defined on a torus T = C/Λ. This torus may be embedded in the complex projective plane by means of the map :z \mapsto [1 :\wp(z) : \wp'(z)/2] This map is a group isomorphism of the torus (considered with its natural group structure) with the chord-and-tangent group law on the cubic curve which is the image of this map. It is also an isomorphism of Riemann surfaces from the torus to the cubic curve, so topologically, an elliptic curve is a torus. If the lattice Λ is related by multiplication by a non-zero complex number c to a lattice cΛ, then the corresponding curves are isomorphic.
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In a letter to Euler, Cramer pointed out that the cubic curves x3 − x = 0 and y3 − y = 0 intersect in precisely 9 points (each equation represents a set of three parallel lines x = −1, x = 0, x = +1; and y = −1, y = 0, y = +1 respectively). Hence 9 points are not sufficient to uniquely determine a cubic curve in degenerate cases such as these.
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The actual projective embeddings are complicated (see equations defining abelian varieties) when n > 1, and are really coextensive with the theory of theta-functions of several complex variables (with fixed modulus). There is nothing as simple as the cubic curve description for n = 1. Computer algebra can handle cases for small n reasonably well. By Chow's theorem, no complex torus other than the abelian varieties can 'fit' into projective space.
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It is harder to determine whether a curve of genus 1 has a rational point. The Hasse principle fails in this case: for example, by Ernst Selmer, the cubic curve 3x3 \+ 4y3 \+ 5z3 = 0 in P2 has a point over all completions of Q, but no rational point.Silverman (2009), Remark X.4.11. The failure of the Hasse principle for curves of genus 1 is measured by the Tate–Shafarevich group.
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For another example, first consider the affine cubic curve :y^2 = x^3 - x. in the 2-dimensional affine space (over a field of characteristic not two). It has the associated cubic homogeneous polynomial equation: :y^2z = x^3 - xz^2, which defines a curve in P2 called an elliptic curve. The curve has genus one (genus formula); in particular, it is not isomorphic to the projective line P1, which has genus zero.
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The Darboux cubic may be defined from the de Longchamps point, as the locus of points X such that X, the isogonal conjugate of X, and the de Longchamps point are collinear. It is the only cubic curve invariant of a triangle that is both isogonally self-conjugate and centrally symmetric; its center of symmetry is the circumcenter of the triangle.. The de Longchamps point itself lies on this curve, as does its reflection the orthocenter.
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The Clebsch transfer of the discriminant of a binary cubic is a contravariant F of ternary cubics of degree 4 and class 6, giving the dual cubic of a cubic curve. The Cayleyan P of a ternary cubic is a contravariant of degree 3 and class 3. The quippian Q of a ternary cubic is a contravariant of degree 5 and class 3. The Hermite contravariant Π is another contravariant of ternary cubics of degree 12 and class 9.
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In this scheme, any t out of n shares may be used to recover the secret. The system relies on the idea that you can fit a unique polynomial of degree to any set of t points that lie on the polynomial. It takes two points to define a straight line, three points to fully define a quadratic, four points to define a cubic curve, and so on. That is, it takes t points to define a polynomial of degree .
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A morphism of varieties is finite if the inverse image of every point is finite and the morphism is proper. A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve X in the affine plane A2 defined by x2 = y3 is not normal, because there is a finite birational morphism A1 → X (namely, t maps to (t3, t2)) which is not an isomorphism. By contrast, the affine line A1 is normal: it cannot be simplified any further by finite birational morphisms.
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Unaware of the solution of the geometry by Leonhard Euler, Rankine cited the cubic curve (a polynomial curve of degree 3), which is an approximation of the Euler spiral for small angular changes in the same way that a parabola is an approximation to a circular curve. Marie Alfred Cornu (and later some civil engineers) also solved the calculus of the Euler spiral independently. Euler spirals are now widely used in rail and highway engineering for providing a transition or an easement between a tangent and a horizontal circular curve.
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The tangent- chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group E(Q)/2E(Q) which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for E(Q) to be finitely- generated; and it shows that the rank is finite. This turns out to be the essential difficulty.
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The Nagell–Lutz theorem generalizes to arbitrary number fields and more general cubic equations.See, for example, Theorem VIII.7.1 of Joseph H. Silverman (1986), "The arithmetic of elliptic curves", Springer, . For curves over the rationals, the generalization says that, for a nonsingular cubic curve whose Weierstrass form :y^2 +a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 has integer coefficients, any rational point P=(x,y) of finite order must have integer coordinates, or else have order 2 and coordinates of the form x=m/4, y=n/8, for m and n integers.
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When working in the projective plane, we can define a group structure on any smooth cubic curve. In Weierstrass normal form, such a curve will have an additional point at infinity, O, at the homogeneous coordinates [0:1:0] which serves as the identity of the group. Since the curve is symmetrical about the x-axis, given any point P, we can take −P to be the point opposite it. We take −O to be just O. If P and Q are two points on the curve, then we can uniquely describe a third point, P + Q, in the following way.
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The Sylvester–Gallai theorem was posed as a problem by . suggests that Sylvester may have been motivated by a related phenomenon in algebraic geometry, in which the inflection points of a cubic curve in the complex projective plane form a configuration of nine points and twelve lines (the Hesse configuration) in which each line determined by two of the points contains a third point. The Sylvester–Gallai theorem implies that it is impossible for all nine of these points to have real coordinates. claimed to have a short proof of the Sylvester–Gallai theorem, but it was already noted to be incomplete at the time of publication.
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The paradox was first published by Colin Maclaurin. Cramer and Leonhard Euler corresponded on the paradox in letters of 1744 and 1745 and Euler explained the problem to Cramer. It has become known as Cramer's paradox after featuring in his 1750 book Introduction à l'analyse des lignes courbes algébriques, although Cramer quoted Maclaurin as the source of the statement. At about the same time, Euler published examples showing a cubic curve which was not uniquely defined by 9 pointsEuler, L. "Sur une contradiction apparente dans la doctrine des lignes courbes." Mémoires de l'Académie des Sciences de Berlin 4, 219-233, 1750 and discussed the problem in his book Introductio in analysin infinitorum.
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Four task positions yield six relative displacement poles, and Burmester selected four to form the opposite pole quadrilateral, which he then used to graphically generate the circling point curve (Kreispunktcurven). Burmester also showed that the circling point curve was a circular cubic curve in the moving body. Five positions: To reach five task positions, Burmester intersects the circling point curve generated by the opposite pole quadrilateral for a set of four of the five task positions, with the circling point curve generated by the opposite pole quadrilateral for different set of four task positions. Five poses imply ten relative displacement poles, which yields four different opposite pole quadrilaterals each having its own circling point curve.
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A normal complex variety X has the property, when viewed as a stratified space using the classical topology, that every link is connected. Equivalently, every complex point x has arbitrarily small neighborhoods U such that U minus the singular set of X is connected. For example, it follows that the nodal cubic curve X in the figure, defined by x2 = y2(y + 1), is not normal. This also follows from the definition of normality, since there is a finite birational morphism from A1 to X which is not an isomorphism; it sends two points of A1 to the same point in X. Curve y2 = x2(x + 1) More generally, a scheme X is normal if each of its local rings :OX,x is an integrally closed domain.
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In a plane cubic model three points sum to zero in the group if and only if they are collinear. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions. The intersection of two quadric surfaces is, in general, a nonsingular curve of genus one and degree four, and thus an elliptic curve, if it has a rational point. In special cases, the intersection either may be a rational singular quartic or is decomposed in curves of smaller degrees which are not always distinct (either a cubic curve and a line, or two conics, or a conic and two lines, or four lines).
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Image for 9-points theorem, special case, when both and are unions of 3 lines In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane . The original form states: :Assume that two cubics and in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes through any eight of the points also passes through the ninth point. A more intrinsic form of the Cayley–Bacharach theorem reads as follows: :Every cubic curve on an algebraically closed field that passes through a given set of eight points also passes through a certain (fixed) ninth point , counting multiplicities.
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Also, both curves are rational, as they are parameterized by x, and the Riemann-Roch theorem implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular. Thus many of the properties of algebraic varieties, including birational equivalence and all the topological properties, depend on the behavior "at infinity" and so it is natural to study the varieties in projective space. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For these reasons, projective space plays a fundamental role in algebraic geometry.
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These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given field K. Each point P imposes a single linear condition on F, if we ask that C pass through P. Therefore, we can find some cubic curve through any nine given points, which may be degenerate, and may not be unique, but will be unique and non-degenerate if the points are in general position; compare to two points determining a line and how five points determine a conic. If two cubics pass through a given set of nine points, then in fact a pencil of cubics does, and the points satisfy additional properties; see Cayley–Bacharach theorem. Singular cubic . A parametrization is given by .
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A twisted cubic curve, the subject of Atiyah's first paper Atiyah's early papers on algebraic geometry (and some general papers) are reprinted in the first volume of his collected works. As an undergraduate Atiyah was interested in classical projective geometry, and wrote his first paper: a short note on twisted cubics. He started research under W. V. D. Hodge and won the Smith's prize for 1954 for a sheaf-theoretic approach to ruled surfaces, which encouraged Atiyah to continue in mathematics, rather than switch to his other interests—architecture and archaeology. His PhD thesis with Hodge was on a sheaf-theoretic approach to Solomon Lefschetz's theory of integrals of the second kind on algebraic varieties, and resulted in an invitation to visit the Institute for Advanced Study in Princeton for a year.
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As x goes to positive infinity, the slope of the line from the origin to the point (x, x3) goes to positive infinity just as before. But unlike before, as x goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. So the behavior "at infinity" of V(y − x3) is different from the behavior "at infinity" of V(y − x2). The consideration of the projective completion of the two curves, which is their prolongation "at infinity" in the projective plane, allows us to quantify this difference: the point at infinity of the parabola is a regular point, whose tangent is the line at infinity, while the point at infinity of the cubic curve is a cusp.
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An elliptic curve may be defined as any curve of genus one with a rational point: a common model is a nonsingular cubic curve, which suffices to model any genus one curve. In this model the distinguished point is commonly taken to be an inflection point at infinity; this amounts to requiring that the curve can be written in Tate- Weierstrass form, which in its projective version is :y^2z + a_1 xyz + a_3 yz^2 = x^3 + a_2 x^2z + a_4 xz^2 + a_6 z^3. If the characteristic of the field is different from 2 and 3, then a linear change of coordinates allows putting a_1=a_2=a_3=0, which gives the classical Weierstrass form :y^2 = x^3 + p x + q. Elliptic curves carry the structure of an abelian group with the distinguished point as the identity of the group law.
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On early railroads, because of the low speeds and wide-radius curves employed, the surveyors were able to ignore any form of easement, but during the 19th century, as speeds increased, the need for a track curve with gradually increasing curvature became apparent. Rankine's 1862 "Civil Engineering" cites several such curves, including an 1828 or 1829 proposal based on the "curve of sines" by William Gravatt, and the curve of adjustment by William Froude around 1842 approximating the elastic curve. The actual equation given in Rankine is that of a cubic curve, which is a polynomial curve of degree 3, at the time also known as a cubic parabola. In the UK, only from 1845, when legislation and land costs began to constrain the laying out of rail routes and tighter curves were necessary, were the principles beginning to be applied in practice.
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In the opposite direction, a variety X over a number field k is said to have potentially dense rational points if there is a finite extension field E of k such that the E-rational points of X are Zariski dense in X. Frédéric Campana conjectured that a variety is potentially dense if and only if it has no rational fibration over a positive-dimensional orbifold of general type.Campana (2004), Conjecture 9.20. A known case is that every cubic surface in P3 over a number field k has potentially dense rational points, because (more strongly) it becomes rational over some finite extension of k (unless it is the cone over a plane cubic curve). Campana's conjecture would also imply that a K3 surface X (such as a smooth quartic surface in P3) over a number field has potentially dense rational points.
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The Sylvester–Gallai theorem also does not directly apply to geometries in which points have coordinates that are pairs of complex numbers or quaternions, but these geometries have more complicated analogues of the theorem. For instance, in the complex projective plane there exists a configuration of nine points, Hesse's configuration (the inflection points of a cubic curve), in which every line is non-ordinary, violating the Sylvester–Gallai theorem. Such a configuration is known as a Sylvester–Gallai configuration, and it cannot be realized by points and lines of the Euclidean plane. Another way of stating the Sylvester–Gallai theorem is that whenever the points of a Sylvester–Gallai configuration are embedded into a Euclidean space, preserving colinearities, the points must all lie on a single line, and the example of the Hesse configuration shows that this is false for the complex projective plane.
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In most cases, the tangent will intersect a second point R and we can take its opposite. However, if P happens to be an inflection point (a point where the concavity of the curve changes), we take R to be P itself and P + P is simply the point opposite itself. For a cubic curve not in Weierstrass normal form, we can still define a group structure by designating one of its nine inflection points as the identity O. In the projective plane, each line will intersect a cubic at three points when accounting for multiplicity. For a point P, −P is defined as the unique third point on the line passing through O and P. Then, for any P and Q, P + Q is defined as −R where R is the unique third point on the line containing P and Q. Let K be a field over which the curve is defined (i.e.
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The reason of the interest for Diophantine equations, in the elliptic curve case, is that K may not be algebraically closed. There can exist curves C that have no point defined over K, and which become isomorphic over a larger field to E, which by definition has a point over K to serve as identity element for its addition law. That is, for this case we should distinguish C that have genus 1, from elliptic curves E that have a K-point (or, in other words, provide a Diophantine equation that has a solution in K). The curves C turn out to be torsors over E, and form a set carrying a rich structure in the case that K is a number field (the theory of the Selmer group). In fact a typical plane cubic curve C over Q has no particular reason to have a rational point; the standard Weierstrass model always does, namely the point at infinity, but you need a point over K to put C into that form over K. This theory has been developed with great attention to local analysis, leading to the definition of the Tate-Shafarevich group.
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