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In geometry, a point is an abstract idealization of an exact position, without size, in physical space, [1] or its generalization to other kinds of mathematical spaces.As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, and higher-dimensional objects consist; conversely ...
Antipodal point, the point diametrically opposite to another point on a sphere, such that a line drawn between them passes through the centre of the sphere and forms a true diameter; Conjugate point, any point that can almost be joined to another by a 1-parameter family of geodesics (e.g., the antipodes of a sphere, which are linkable by any ...
The same relation, + + =, may be regarded as either the equation of a line or the equation of a point. In general, there is no difference either algebraically or logically between homogeneous coordinates of points and lines. So plane geometry with points as the fundamental elements and plane geometry with lines as the fundamental elements are ...
The following are the assumptions of the point-line-plane postulate: [1] Unique line assumption. There is exactly one line passing through two distinct points. Number line assumption. Every line is a set of points which can be put into a one-to-one correspondence with the real numbers. Any point can correspond with 0 (zero) and any other point ...
In this system, an arbitrary point O (the origin) is chosen on a given line. The coordinate of a point P is defined as the signed distance from O to P, where the signed distance is the distance taken as positive or negative depending on which side of the line P lies. Each point is given a unique coordinate and each real number is the coordinate ...
In geometry, an affine plane is a system of points and lines that satisfy the following axioms: [1] Any two distinct points lie on a unique line. Given any line and any point not on that line there is a unique line which contains the point and does not meet the given line. (Playfair's axiom) There exist four points such that no three are ...
The red dot represents the point at which the two lines intersect. In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point (sometimes called a ...
As any line in this extension of σ corresponds to a plane through O, and since any pair of such planes intersects in a line through O, one can conclude that any pair of lines in the extension intersect: the point of intersection lies where the plane intersection meets σ or the line at infinity. Thus the axiom of projective geometry, requiring ...
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