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In a different model of elliptic geometry, lines are represented by Euclidean planes passing through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two points determining a unique line) that make them suitable representations for lines in this geometry.
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 ...
Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, [a] which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental ...
Configuration (geometry), an arrangement of lines and a set of points with all lines containing the same number of points and all points belonging to the same number of lines Arrangement (space partition) , a partition of the plane given by overlaid curves or of a higher dimensional space by overlaid surfaces, without requiring the curves or ...
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 ...
Each curve in this example is a locus defined as the conchoid of the point P and the line l.In this example, P is 8 cm from l. In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.
So plane geometry with points as the fundamental elements and plane geometry with lines as the fundamental elements are equivalent except for interpretation. This leads to the concept of duality in projective geometry, the principle that the roles of points and lines can be interchanged in a theorem in projective geometry and the result will ...
Conversely, the polar line (or polar) of a point Q in a circle C is the line L such that its closest point P to the center of the circle is the inversion of Q in C. If a point A lies on the polar line q of another point Q, then Q lies on the polar line a of A. More generally, the polars of all the points on the line q must pass through its pole Q.