"of a tangent from the point to the circle. The power of a point is also known as the point's circle power or the power of a circle with respect to the point. The power of point P ""(see in Figure 1)"" can be defined equivalently as the product of distances from the point P to the two intersection points of any ray emanating from P. For example, in Figure 1, a ray emanating from P intersects the circle in two points, M and N, whereas a tangent ray intersects the circle in one point T; the horizontal ray from"
"meant to suggest that the line continues indefinitely in the positive and negative directions. Another convention uses only one arrowhead which indicates the direction in which numbers grow. The line continues indefinitely in the positive and negative directions according to the rules of geometry which define a line without endpoints as an ""infinite line"", a line with one endpoint as a ""ray"", and a line with two endpoints as a ""line segment"". If a particular number is farther to the right on the number line than is another number, then the first number is greater than the second (equivalently, the"
"as the complement of a ray or curve in the complex plane going from 0 (inclusive) to infinity in some direction. In this case, the curve is known as a branch cut. For example, the principal branch has a branch cut along the negative real axis. If the function ""L""(""z"") is extended to be defined at a point of the branch cut, it will necessarily be discontinuous there; at best it will be continuous ""on one side"", like Log ""z"" at a negative real number. Each branch ""L""(""z"") of log ""z"" on an open set ""U"" is an inverse of"
"are called points at infinity, and are considered to lie on a new line, the line at infinity. There is a point at infinity corresponding to each direction (numerically given by the slope of a line), informally defined as the limit of a point that moves in that direction away from the origin. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point on the Euclidean plane, for any non-zero real number ""Z"", the triple is called a ""set of homogeneous coordinates"" for the point. By this"
"including Euclid's definition as 'that which has no part' and through the use of algebra or nested sets. In many areas of geometry, such as analytic geometry, differential geometry, and topology, all objects are considered to be built up from points. However, there has been some study of geometry without reference to points. Euclid described a line as ""breadthless length"" which ""lies equally with respect to the points on itself"". In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line"
"we may consider ""A"" as decomposing this line into two parts. Each such part is called a ray (or half-line) and the point ""A"" is called its ""initial point"". The point A is considered to be a member of the ray. Intuitively, a ray consists of those points on a line passing through ""A"" and proceeding indefinitely, starting at ""A"", in one direction only along the line. However, in order to use this concept of a ray in proofs a more precise definition is required. Given distinct points ""A"" and ""B"", they determine a unique ray with initial point ""A""."
"Line segment In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent"
"Path (topology) In mathematics, a path in a topological space ""X"" is a continuous function ""f"" from the unit interval ""I"" = [0,1] to ""X"" The ""initial point"" of the path is ""f""(0) and the ""terminal point"" is ""f""(1). One often speaks of a ""path from ""x"" to ""y"""" where ""x"" and ""y"" are the initial and terminal points of the path. Note that a path is not just a subset of ""X"" which ""looks like"" a curve, it also includes a parameterization. For example, the maps ""f""(""x"") = ""x"" and ""g""(""x"") = ""x"" represent two different paths from 0"
"points at infinity. They constitute a line in ""K""P — namely, the line arising from the plane in ""K"" — called the line at infinity. The points at infinity are the ""extra"" points where parallel lines intersect in the construction of the extended real plane; the point (0, ""x"", ""x"") is where all lines of slope ""x"" / ""x"" intersect. Consider for example the two lines in the affine plane ""K"". These lines have slope 0 and do not intersect. They can be regarded as subsets of ""KP via the embedding above, but these subsets are not lines in ""KP."
"Point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the ""end"" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we ""forget"" which points were added. This holds for a geometry over any field, and more generally over any division ring. In the real case, a point at infinity completes a line into a topologically closed curve."
"drawn starting at the origin. A line segment is drawn rightwards by the magnitude of the first coefficient (the coefficient of the highest-power term) (so that with a negative coefficient the segment will end left of the origin). From the end of the first segment another segment is drawn upwards by the magnitude of the second coefficient, then left by the magnitude of the third, and down by the magnitude of the fourth, and so on. The sequence of directions (not turns) is always rightward, upward, leftward, downward, then repeating itself. Thus each turn is counterclockwise. The process continues for"
"Tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that ""just touches"" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where ""f"" is the derivative of ""f"". A similar definition applies to space curves and curves in ""n""-dimensional Euclidean space. As it"
"Projectively extended real line In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the number line by a point denoted . It is thus the set formula_1 (where formula_2 is the set of the real numbers) with the standard arithmetic operations extended where possible, sometimes denoted by formula_3 The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line. More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute"
"Line (geometry) The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. Lines are an idealization of such objects. Until the 17th century, lines were defined as the ""[…] first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width. […] The straight line is that which is equally extended between its"
"Projective line In mathematics, 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 resultant 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"
"route between a source and a destination. Continued fraction }</math> In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers"
"also not one-point compactifications. Point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the ""end"" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we ""forget"" which points were added. This holds for a geometry over any field, and more generally over any division ring. In the real case, a point at infinity completes a line into"
"Limit (mathematics) In mathematics, a limit is the value that a function (or sequence) ""approaches"" as the input (or index) ""approaches"" some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit is usually written as and is read as ""the limit of of as approaches equals "". The fact that a function"
"""V"" is an infinite-dimensional (locally convex Hausdorff) topological vector space along lines similar to the finite-dimensional case. The main concerns of Gustave Choquet were in potential theory. Choquet theory has become a general paradigm, particularly for treating convex cones as determined by their extreme rays, and so for many different notions of ""positivity"" in mathematics. The two ends of a line segment determine the points in between: in vector terms the segment from ""v"" to ""w"" consists of the λ""v"" + (1 − λ)""w"" with 0 ≤ λ ≤ 1. The classical result of Hermann Minkowski says that in Euclidean"
"Line group A line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice. However, line groups may have more than one dimension, and they may involve those dimensions in its isometries or symmetry transformations. One constructs a line group by taking a point group in the full dimensions of the space, and then adding translations or offsets along the line to each of the point group's elements, in the fashion of constructing a space group. These offsets include the repeats, and a"
"Line at infinity In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line. In projective geometry, any pair of lines always intersects at some point, but parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on"
"Balbis In geometry, a balbis is a geometric shape that can be colloquially defined as a single (primary) line that is terminated by a (secondary) line at one endpoint and by a (secondary) line at the other endpoint. The terminating secondary lines are at right angles to the primary line. Its parallel sides are of indefinite lengths and can be infinitely long. The word ""balbis"" comes from the ancient Greek word βαλβίς, meaning a rope between two posts used to indicate the start and finish of a race. The most common example of a balbis is the capital letter 'H',"
"as a subset ""L"" that can be parametrized as for some vectors formula_4. Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points. In geometry, it is sometimes defined that a point ""B"" is between two other points ""A"" and ""C"", if the distance ""AB"" added to the distance ""BC"" is equal to the distance ""AC"". Thus in formula_7 the line segment with endpoints and is the following collection of points: In an axiomatic treatment of geometry, the notion of betweenness is"
"direction from the origin to the second point is defined as the ""positive direction"". This line segment may be translated along the line to build longer segments whose lengths correspond to multiples of the unit distance. In this manner real numbers can be associated to points on the line (as the distance from the origin to the point) and these are the Cartesian coordinates of the points on what may now be called the real line. As an alternate way to establish the metric, instead of choosing two points on the line, choose one point to be the origin, a"
"segments is usually reduced by making ""equivalent"" any pair having the same length and orientation. This application of an equivalence relation dates from Giusto Bellavitis’s introduction of the concept of equipollence of directed line segments in 1835. Analogous to straight line segments above, one can define arcs as segments of a curve. Line segment In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open"
"evaluates to positive or negative. A line splits a plane into halves and the half-plane that has a negative f(x,y) can be called the negative half-plane, and the other half can be called the positive half-plane. This observation is very important in the remainder of the derivation. Clearly, the starting point is on the line only because the line is defined to start and end on integer coordinates (though it is entirely reasonable to want to draw a line with non-integer end points). Keeping in mind that the slope is less-than-or-equal-to one, the problem now presents itself as to whether"
"complex field, the set of points at infinity is a manifold. In artistic drawing and technical perspective, the projection on the picture plane of the point at infinity of a class of parallel lines is called their vanishing point. In hyperbolic geometry, points at infinity are typically named ideal points. Unlike Euclidean and elliptic geometries, each line has two points at infinity: given a line ""l"" and a point ""P"" not on ""l"", the right- and left-limiting parallels converge asymptotically to different points at infinity. All points at infinity together form the Cayley absolute or boundary of a hyperbolic plane."
"at infinity, is known as the extended complex plane. We speak of a single ""point at infinity"" when discussing complex analysis. There are two points at infinity (positive, and negative) on the real number line, but there is only one point at infinity (the north pole) in the extended complex plane. Imagine for a moment what will happen to the lines of latitude and longitude when they are projected from the sphere onto the flat plane. The lines of latitude are all parallel to the equator, so they will become perfect circles centered on the origin . And the lines"
"End (topology) In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the ""ideal boundary"" of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification. The notion of an end of a topological space was introduced by . Let ""X"" be a topological space, and suppose that is an ascending sequence of compact subsets of ""X"" whose interiors cover ""X"". Then ""X"" has one"
"line. This is described by the following equation: Given two points (""x"",""y"") and (""x"",""y""), the change in ""x"" from one to the other is (""run""), while the change in ""y"" is (""rise""). Substituting both quantities into the above equation generates the formula: The formula fails for a vertical line, parallel to the ""y"" axis (see Division by zero), where the slope can be taken as infinite, so the slope of a vertical line is considered undefined. Suppose a line runs through two points: ""P"" = (1, 2) and ""Q"" = (13, 8). By dividing the difference in ""y""-coordinates by the"
"More generally, there is a ""k""-dimensional tangent space at each point of a ""k""-dimensional manifold in the ""n""-dimensional Euclidean space. Tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that ""just touches"" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where ""f"""
"By techniques of analytic continuation, this function can be extended uniquely to an analytic function (still denoted ""ζ""(""z"")) on the whole complex plane except at the point ""z"" = 1, where it has a simple pole. Infinite product In mathematics, for a sequence of complex numbers ""a"", ""a"", ""a"", ... the infinite product is defined to be the limit of the partial products ""a""""a""...""a"" as ""n"" increases without bound. The product is said to ""converge"" when the limit exists and is not zero. Otherwise the product is said to ""diverge"". A limit of zero is treated specially in order to"
"point). Phase line (mathematics) In mathematics, a phase line is a diagram that shows the qualitative behaviour of an autonomous ordinary differential equation in a single variable, formula_1. The phase line is the 1-dimensional form of the general formula_2-dimensional phase space, and is very simple to analyze. A line, usually vertical, represents an interval of the domain of the derivative. The critical points (i.e., roots of the derivative, points formula_3 such that formula_4) are indicated, and the intervals between the critical points have their signs indicated with arrows: an interval over which the derivative is positive has an arrow pointing"