Point at infinity in the context of "Projective geometry"

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects.

The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular points, inflection points and points at infinity. More advanced questions involve the topology of the curve and the relationship between curves defined by different equations.

In analytic geometry, an asymptote (/ˈæsɪmptoʊt/ ) of a curve is a straight line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.

The word "asymptote" derives from the Greek ἀσύμπτωτος (asumptōtos), which means "not falling together", from ἀ priv. "not" + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions (x, y) for:

for some coefficients a and b in K. The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition 4a + 27b ≠ 0, that is, being square-free in x.) It is always understood that the curve is really sitting in the projective plane, with the point O being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves; see § Elliptic curves over a general field below.)

In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity. The statement and the proof of many theorems of geometry are simplified by the resulting elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point (there is no "parallel" case).

There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a field K, commonly denoted P(K), as the set of one-dimensional subspaces of a two-dimensional K-vector space. This definition is a special instance of the general definition of a projective space.

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ${\displaystyle \infty }$ for infinity. With the Riemann model, the point ${\displaystyle \infty }$ is near to very large numbers, just as the point ${\displaystyle 0}$ is near to very small numbers.

The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as ${\displaystyle 1/0=\infty }$ well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.

In mathematics, a cubic plane curve , often called simply a cubic is a plane algebraic curve defined by a homogeneous polynomial of degree 3 in three variables ${\displaystyle F(X,Y,Z)}$ or by the corresponding polynomial in two variables ${\displaystyle f(x,y)=F(X,Y,1).}$ Starting from ⁠${\displaystyle f}$⁠, one can recover ⁠${\displaystyle F}$⁠ as ⁠${\displaystyle F(X,Y,Z)=Z^{3}f(X/Z,(Y/Z)}$⁠.

Typically, the coefficients of the polynomial belong to ${\displaystyle \mathbb {R} ,}$ but they may belong to any field ⁠${\displaystyle k}$⁠, in which case, one talks of a cubic defined over ⁠${\displaystyle k}$⁠. The points of the cubic are the points of the projective space of dimension three over the field of the complex numbers (or over an algebraic closure of ⁠${\displaystyle k}$⁠), whose projective coordinates satisfy the equation of the cubic$${\displaystyle F(X,Y,Z)=0.}$$A point at infinity of the cubic is a point such that ⁠${\displaystyle Z=0}$⁠. A real point of the cubic is a point with real coordinates. A point defined over ⁠${\displaystyle k}$⁠ is a point with coordinates in ⁠${\displaystyle k}$⁠.