They can measure angles with accuracy about 10 –6 of the right angle. In geodesy, theodolites use light rays as straight lines. In this approach a line in an n-dimensional affine space ( n ≥ 1) is defined as a (proper or improper) one-dimensional affine subspace. The Euclidean plane, that is, the two-dimensional Euclidean space is defined similarly. For further details see Affine space#Euclidean space and space (mathematics). The modern approach defines the three-dimensional Euclidean space more algebraically, via linear spaces and quadratic forms, namely, as an affine space whose difference space is a three-dimensional inner product space. The "most basic properties of lines" listed above are roughly the line-related assumptions (Hilbert's axioms), while "further properties" are first line-related consequences (theorems). It is possible to exclude plane-related axioms thus obtaining axioms of Euclidean plane geometry. Likewise, Euclidean space, its points, lines, planes and their properties are introduced simultaneously in a set of 20 assumptions known as Hilbert's axioms of Euclidean geometry (solid). No chess without rooks, no rooks outside chess! One must introduce the game, its pieces and their properties in a single combined definition. And conversely, the whole chess game cannot be defined before each piece is defined the properties of the rook are an indispensable part of the rules of the game. The modern approach (below) defines lines in a completely different way.Īxiomatic approach is similar to chess in the following aspect.Ī chess piece, say a rook, cannot be defined before the whole chess game is defined, since such a phrase as "the rook moves horizontally or vertically, forward or back, through any number of unoccupied squares" makes no sense unless it is already known that "chess is played on a square board of eight rows and eight columns" etc. In the axiomatic approach points and lines are undefined primitives. However, this situation never appears in mathematical theory. The definitions given above assume implicitly that the Euclidean plane (or alternatively the 3-dimensional Euclidean space) is already defined, together with such notions as distances and/or Cartesian coordinates, while lines are not defined yet. Two lines perpendicular to the same line are parallel to each other (or coincide).Īxiomatic approach What is wrong with the definitions given above? Two lines either do not intersect (are parallel), or intersect in a single point, or coincide. There exist three points not lying on a line.įor every line and every point outside the line there exists one and only one line through the given point which does not intersect the given line. Most basic propertiesįor every two different points there exists one and only one line that contains these two points. Here real numbers a, b and c are parameters such that at least one of a, b does not vanish.īelow, all points and lines are situated in the plane (assumed to be a two-dimensional Euclidean space). A point B is said to lie between points A and C if The other two definitions apply in plane geometry only.įirst we define betweenness via distances. The first definition (via betweenness) works both in plane geometry and in solid geometry. However, this definition does not work in plane geometry. We could also define a line as the intersection of two planes (neither parallel nor coinciding). In particular, a line segment is not a line. Note that a part of a line is not a line. Any other definition is equally acceptable provided that it is equivalent to these. Three equivalent definitions of line are given below. They are criticized afterwards, see axiomatic approach. Still, the definitions given below are tentative. Fortunately, it is possible to define a line via more elementary notions, and this way is preferred in mathematics. Straight lines are treated by elementary geometry, but the notions of curves and curvature are not elementary, they need more advanced mathematics and more sophisticated definitions. However, this is not a good idea such definitions are useless in mathematics, since they cannot be used when proving theorems. It is tempting to define a line as a curve of zero curvature, where a curve is defined as a geometric object having length but no breadth or depth. To define a line is more complicated than it may seem. In other words, plane geometry is the theory of the two-dimensional Euclidean space, while solid geometry is the theory of the three-dimensional Euclidean space. Plane geometry (called also "planar geometry") is a part of solid geometry that restricts itself to a single plane ("the plane") treated as a geometric universe. Lines are treated both in plane geometry and in solid geometry. 2.1 What is wrong with the definitions given above?.1.1.6 Definition via Cartesian coordinates.1.1.5 Definition via right angles (orthogonality) in disguise.
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