User:Boris Tsirelson/Sandbox1
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, the definitions given below are tentative. They are criticized afterwards, see "axiomatic approach".
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.
Four equivalent definitions of plane are given below. Any other definition is equally acceptable provided that it is equivalent to these. Note that a part of a plane is not a plane. Likewise, a line segment is not a line.
Below, 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.
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 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 point B.
Definition via cartesian coordinates
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 . Here real numbers a, b, c and d are parameters such that at least one of a, b, c does not vanish.
Axiomatic approach
What is wrong with the definitions given above?
The definitions given above assume implicitly that the 3-dimensional Euclidean space is already defined, together with (at least one of) such notions as distances, angles, straight lines, Cartesian coordinates, while planes are not defined yet. However, this situation never appears in mathematics.
In the axiomatic approach points, lines and planes are undefined primitives.
The modern approach (see below) defines planes in a completely different way.