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Plane geometry3/13/2023 ![]() The space contains (at least) four points not lying in the same plane. If two planes have a common point then they have (at least) a second point in common.Įvery plane contains (at least) three points not lying in the same straight line. If (at least) two points of a given straight line lie in a given plane, then all points of this line lie in that plane. Some properties of planes Most basic propertiesįor any three points not situated in the same straight line there exists one and only one plane that contains these three points. Here real numbers a, b, c and d are parameters such that at least one of a, b, c does not vanish. In terms of Cartesian coordinates x, y, z ascribed to every point of the space, a plane is the set of points whose coordinates satisfy the linear equation a x + b y + c z = d. In other words, this plane is the set of all points F such that either F coincides with B or there exists a line through F that intersects the lines AB and BC (in distinct points). The union of all these lines, together with the point B, is a plane. Consider the lines DE for all points D (different from B) on the line AB and all points E (also different from B) on the line BC. Let three points A, B and C be given, not lying on a line. This is the plane orthogonal to the line AB through the middle point of the line segment AB.Įvery point F on the line DE belongs to the plane through given points A, B and C, provided that D belongs to the line AB, E belongs to the line BC, and D, E do not coincide with B. The set of all points C that are equally far from A and B - that is, Let two different points A and B be given. ![]() Likewise, a line segment is not a line.īelow, all points, lines and planes are situated in the space (assumed to be a three-dimensional Euclidean space), and by lines we mean straight lines. Note that a part of a plane is not a plane. Any other definition is equally acceptable provided that it is equivalent to these. However, a circle determines its center and radius uniquely for a plane, the situation is different.įour equivalent definitions of "plane" are given below. Similarly, a plane is a set of points chosen according to their relation to some given objects (points, lines etc). A circle is a set of points chosen according to their relation to some given parameters (center and radius). The definitions of "plane" given below may be compared with the definition of a circle as consisting of those points in a plane that are a given distance (the radius) away from a given point (the center). They are criticized afterwards, see axiomatic approach. Still, the definitions given below are tentative. Fortunately, it is possible to define a plane via more elementary notions, and this way is preferred in mathematics. (See also Theory (mathematics)#Defined or undefined.) Planes are treated by elementary geometry, but the notions of surface 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 plane as a surface with zero curvature, where a surface is defined as a geometric object having length and breadth but no depth. To define a plane is more complicated than it may seem. Non-axiomatic approach Definitions A remark 2.1 What is wrong with the definitions given above?.1.1.5 Definition via Cartesian coordinates.1.1.3 Definition via right angles (orthogonality).
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