Point at infinity in the context of Riemann sphere

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (projective space) and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice versa.

Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in Euclidean geometry, the concept of an angle does not apply in projective geometry, because no measure of angles is invariant with respect to projective transformations, as is seen in perspective drawing from a changing perspective. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See Projective plane for the basics of projective geometry in two dimensions.

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 ⁠${\displaystyle k}$⁠.

In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity". For solving this problem, points at infinity have been introduced, in such a way that in a real projective plane, two distinct projective lines meet in exactly one point. The set of these points at infinity, the "horizon" of the visual perspective in the plane, is a real projective line. It is the set of directions emanating from an observer situated at any point, with opposite directions identified.

An example of a real projective line is the projectively extended real line, which is often called the projective line.

In geometry, a generalized circle, sometimes called a cline or circline, is a straight line or a circle, the curves of constant curvature in the Euclidean plane.

The natural setting for generalized circles is the extended plane, a plane along with one point at infinity through which every straight line is considered to pass. Given any three distinct points in the extended plane, there exists precisely one generalized circle passing through all three.

A gnomonic projection, also known as a central projection or rectilinear projection, is a perspective projection of a sphere, with center of projection at the sphere's center, onto any plane not passing through the center, most commonly a tangent plane. Under gnomonic projection every great circle on the sphere is projected to a straight line in the plane (a great circle is a geodesic on the sphere, the shortest path between any two points, analogous to a straight line on the plane). More generally, a gnomonic projection can be taken of any n-dimensional hypersphere onto a hyperplane.

The projection is the n-dimensional generalization of the trigonometric tangent which maps from the circle to a straight line, and as with the tangent, every pair of antipodal points on the sphere projects to a single point in the plane, while the points on the plane through the sphere's center and parallel to the image plane project to points at infinity; often the projection is considered as a one-to-one correspondence between points in the hemisphere and points in the plane, in which case any finite part of the image plane represents a portion of the hemisphere.

In geometry, the center of curvature of a curve is a point located at a distance from the curve equal to the radius of curvature lying on the curve normal vector. It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the centre of curvature. Cauchy defined the center of curvature C as the intersection point of two infinitely close normal lines to the curve. The locus of centers of curvature for each point on the curve comprise the evolute of the curve. This term is generally used in physics regarding the study of lenses and mirrors (see radius of curvature (optics)).

It can also be defined as the spherical distance between the point at which all the rays falling on a lens or mirror either seems to converge to (in the case of convex lenses and concave mirrors) or diverge from (in the case of concave lenses or convex mirrors) and the lens or mirror itself.It lies on the principal axis of a mirror or lens. In case of a convex mirror it lies behind the polished, or reflecting, surface and it lies in front of the reflecting surface in case of a concave mirror.

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 ⁠${\displaystyle k}$⁠.