User:Boris Tsirelson/Sandbox1: Difference between revisions
imported>Boris Tsirelson |
imported>Boris Tsirelson |
||
| Line 26: | Line 26: | ||
Several equivalent definitions of plane given below may be compared with the definition of [[Circle (mathematics)|circle]] as consisting of those points in a plane that are a given distance (the radius) away from a given point (the center). A circle is a set of points chosen according to their relation to some given parameters (center and radius). Similarly, a plane is a set of points chosen according to their relation to some given objects (points, lines etc). However, a circle determines its center and radius uniquely; for a plane, the situation is different. | Several equivalent definitions of plane given below may be compared with the definition of [[Circle (mathematics)|circle]] as consisting of those points in a plane that are a given distance (the radius) away from a given point (the center). A circle is a set of points chosen according to their relation to some given parameters (center and radius). Similarly, a plane is a set of points chosen according to their relation to some given objects (points, lines etc). However, a circle determines its center and radius uniquely; for a plane, the situation is different. | ||
Below, all points, lines and planes are situated in a three-dimensional Euclidean space, and by lines we mean [[Line (geometry)|straight lines]]. | |||
====Definition via distances==== | ====Definition via distances==== | ||
Let two different points ''A'' and ''B'' be given | Let two different points ''A'' and ''B'' be given. The set of all points ''C'' that are equally far from ''A'' and ''B'' (that is, <math>|AC|=|BC|</math>) is a plane. | ||
This is the plane orthogonal to the line ''AB'' through the middle point of the line segment ''AB''. | This is the plane orthogonal to the line ''AB'' through the middle point of the line segment ''AB''. | ||
====Definition via | ====Definition via right angles (orthogonality)==== | ||
Let two different points ''A'' and ''B'' be given | Let two different points ''A'' and ''B'' be given. The set of all points ''C'' such that the lines ''AB'' and ''AC'' are orthogonal (that is, the angle ''BAC'' is [[Right angle (geometry)|right]]) is a plane. | ||
This is the plane orthogonal to the line ''AB'' through the point ''A''. | This is the plane orthogonal to the line ''AB'' through the point ''A''. | ||
| Line 41: | Line 43: | ||
====Definition via [[Line (geometry)|straight lines]]==== | ====Definition via [[Line (geometry)|straight lines]]==== | ||
Let three points ''A'', ''B'' and ''C'' be given | Let three points ''A'', ''B'' and ''C'' be given, not lying on a line. Consider the lines ''DE'' for all points ''D'' on the line ''AB'' different from ''B'' and all points ''E'' on the line ''BC'' different from ''B''. The [[union]] of all these lines, together with the point ''B'', is a plane. | ||
This is the plane through ''A'', ''B'' and ''C''. | This is the plane through ''A'', ''B'' and ''C''. | ||
Revision as of 01:14, 29 March 2010
Plane
Non-axiomatic approach
Definitions
A remark
To define a plane is more complicated than it may seem.
It is tempting to define a plane as a surface with zero curvature (or something like that). However, this is not a good idea, since the notions of surface and curvature are much more complicated than the notion of plane. In fact, several different notions of surface are introduced by topology and differential geometry, and several different notions of curvature are introduced by differential geometry; these are far beyond elementary mathematics. Fortunately, it is possible to define a plane via more elementary notions, and this way is preferred in mathematics. Still, some problems remain, see "axiomatic approach" below.
Several equivalent definitions of plane given below may be compared with the definition of circle as consisting of those points in a plane that are a given distance (the radius) away from a given point (the center). A circle is a set of points chosen according to their relation to some given parameters (center and radius). Similarly, a plane is a set of points chosen according to their relation to some given objects (points, lines etc). However, a circle determines its center and radius uniquely; for a plane, the situation is different.
Below, all points, lines and planes are situated in a three-dimensional Euclidean space, and by lines we mean straight lines.
Definition via distances
Let two different points A and B be given. The set of all points C that are equally far from A and B (that is, ) is a plane.
This is the plane orthogonal to the line AB through the middle point of the line segment AB.
Definition via right angles (orthogonality)
Let two different points A and B be given. The set of all points C such that the lines AB and AC are orthogonal (that is, the angle BAC is right) is a plane.
This is the plane orthogonal to the line AB through the point A.
Definition via straight lines
Let three points A, B and C be given, not lying on a line. Consider the lines DE for all points D on the line AB different from B and all points E on the line BC different from B. The union of all these lines, together with the point B, is a plane.
This is the plane through A, B and C.
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 both the line AB and the line BC, and not at their intersection B.