Plane (Geometry)

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In Euclidean geometry, a plane is an abstract concept that models the common notion of a flat surface — without depressions, protrusions, holes and a boundary — that for any two of its points entirely contains the straight line joining them.

Assuming a common (intuitive, physical) idea of the geometry of space, "plane" can be defined in terms of distances, orthogonality, lines, coordinates etc. In a more abstract approach (vector spaces) planes are defined as two-dimensional affine subspaces. In an axiomatic approach, basic concepts of elementary geometry, such as "point", "line" and "plane", are undefined primitives.

(CC) Image: Dina Tsirelson
The surface of this metallic body consists of rectangles situated in various planes.

Non-axiomatic approach[edit]

Definitions[edit]

A remark[edit]

To define a plane is more complicated than it may seem.

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. However, this is not a good idea; such definitions are useless in mathematics, since they cannot be used when proving theorems. 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. Fortunately, it is possible to define a line 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.

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). 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[edit]

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,

${\displaystyle |AC|=|BC|}$

— 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)[edit]

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.

Every point F on the line DE belongs to the plane through given points A, B and C, provided that D belongs to AB, E belongs to BC, and D, E do not coincide with B.

Definition via lines[edit]

Let three points A, B and C be given, not lying on a line. 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. 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 the lines AB and BC (in distinct points).

Definition via Cartesian coordinates[edit]

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

${\displaystyle ax+by+cz=d}$.

Here real numbers a, b, c and d are parameters such that at least one of a, b, c does not vanish.

Some properties of planes[edit]

Most basic properties[edit]

For any three points not situated in the same straight line there exists one and only one plane that contains these three points.

If two points of a straight line lie in a plane, then every point of the line lies in that plane.

If two planes have a common point then they have at least a second point in common.

Every plane contains at least three points not lying in the same straight line, and the space contains at least four points not lying in a plane.

Further properties[edit]

Two planes either do not intersect (are parallel), or intersect in a line, or coincide.

A line either does not intersect a plane (is parallel to it), or intersects it in a single point, or is contained in the plane.

Two lines perpendicular to the same plane are parallel to each other (or coincide).

Two planes perpendicular to the same line are parallel to each other (or coincide).

Axiomatic approach[edit]

What is wrong with the definitions given above?[edit]

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 mathematical theory.

In the axiomatic approach points, lines and planes are undefined primitives.

The modern approach (below) defines planes in a completely different way.

How does it work[edit]

Axiomatic approach is similar to chess in the following aspect.

A 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. 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. No chess without rooks, no rooks outside chess! One must introduce the game, its pieces and their properties in a single combined definition.

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.^[1] The "most basic properties of planes" listed above are roughly the plane-related assumptions (Hilbert's axioms), while "further properties" are the first plane-related consequences (theorems).

Modern approach[edit]

The modern approach defines the three-dimensional Euclidean space more algebraically, via linear spaces and quadratic forms, namely, as a real affine space whose difference space is a three-dimensional inner product space. For further details see Affine space#Euclidean space and space (mathematics).

In this approach a plane in an n-dimensional affine space (n ≥ 2) is defined as a (proper or improper) two-dimensional affine subspace.

A less formal version of this approach uses points, vectors and scalar product (called also dot product or inner product) of vectors without mentioning linear and affine spaces. Optionally, Cartesian coordinates of points and vectors are used. See algebraic equations below. There, in particular, equivalence between the definition via right angles (orthogonality) and the definition via Cartesian coordinates is explained.

Plane geometry[edit]

Plane geometry (also called "planar geometry") is a part of solid geometry that restricts itself to a single plane ("the plane") treated as a geometric universe. 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 studies the properties of plane figures (and configurations). Plane figures in elementary geometry are sets of points, lines, line segments and sometimes curves that fall on the same plane. For example, triangles, polygons and circles. In plane geometry every figure is plane, in contrast to solid geometry.

Algebraic equations[edit]

In analytic geometry several closely related algebraic equations are known for a plane in three-dimensional Euclidean space. A few algebraic representations will be discussed.

Point-normal representation[edit]

PD Image
The equation for a plane. P is an arbitrary point in plane; ${\displaystyle \scriptstyle {\vec {d}}}$ and ${\displaystyle \scriptstyle {\vec {n}}_{0}}$ are collinear.

One such equation is illustrated in the figure. Point P is an arbitrary point in the plane and O (the origin) is drawn outside the plane. The point A in the plane is chosen such that vector

${\displaystyle {\vec {d}}={\overrightarrow {OA}}}$

is orthogonal to the plane. The collinear vector

${\displaystyle {\vec {n}}_{0}={\frac {1}{d}}{\vec {d}}\quad {\hbox{with}}\quad d=\left|{\vec {d}}\,\right|}$

is a unit (length 1) vector normal (perpendicular) to the plane which is known as the normal of the plane in point A. Note that d is the distance of O to the plane. The following relation holds for an arbitrary point P in the plane (according to the definition via right angles):

${\displaystyle \left({\vec {r}}-{\vec {d}}\;\right)\cdot {\vec {n}}_{0}=0\quad {\hbox{with}}\quad {\vec {r}}={\overrightarrow {OP}}\quad {\hbox{and}}\quad {\vec {r}}-{\vec {d}}={\overrightarrow {AP}}.}$

This equation for the plane can be rewritten in terms of coordinates with respect to a Cartesian frame with origin in O. Dropping arrows for component vectors (real triplets) that are written bold, we find

${\displaystyle \left(\mathbf {r} -\mathbf {d} \right)\cdot \mathbf {n} _{0}=0\Longleftrightarrow \mathbf {r} \cdot \mathbf {n} _{0}=\mathbf {d} \cdot \mathbf {n} _{0}\Longleftrightarrow xa_{0}+yb_{0}+zc_{0}=d}$

with

${\displaystyle \mathbf {d} =(a,\;b,\;c),\quad \mathbf {n} _{0}=(a_{0},\;b_{0},\;c_{0}),\quad \mathbf {r} =(x,\;y,\;z),}$

and

${\displaystyle \mathbf {d} \cdot \mathbf {n} _{0}=\mathbf {d} \cdot {\frac {1}{d}}\mathbf {d} ={\frac {1}{d}}\mathbf {d} ^{2}={\frac {d^{2}}{d}}=d={\sqrt {a^{2}+b^{2}+c^{2}}}.}$

The definition via Cartesian coordinates is thus derived from the definition via right angles.

Moreover, the Hesse normal form for the plane (called after the 19th century mathematician Ludwig Otto Hesse) is obtained,

${\displaystyle \mathbf {r} \cdot \mathbf {n} _{0}=d;}$

it is characterized by the use of a unit-length vector ${\displaystyle \mathbf {n} _{0}}$ rather than an arbitrary vector orthogonal to the plane.

Conversely, the definition via right angles can be derived from the definition via Cartesian coordinates as follows. Given a linear equation for a plane

${\displaystyle ax+by+cz=e,\,}$

we write

${\displaystyle \mathbf {r} =(x,\;y,\;z),\quad \mathbf {f} =(a,\;b,\;c),\quad {\hbox{and}}\quad \mathbf {d} =\left({\frac {e}{a^{2}+b^{2}+c^{2}}}\right)\mathbf {f} .}$

It follows that

${\displaystyle \mathbf {f} \cdot \mathbf {r} =e=\mathbf {f} \cdot \mathbf {d} .}$

Hence we find the same orthogonality relation,

${\displaystyle \mathbf {f} \cdot (\mathbf {r} -\mathbf {d} )=0\;\Longrightarrow \;(\mathbf {r} -\mathbf {d} )\cdot \mathbf {n} _{0}=0\quad {\hbox{with}}\quad \mathbf {n} _{0}={\frac {1}{\sqrt {a^{2}+b^{2}+c^{2}}}}\mathbf {f} }$

where f , d, and n₀ are collinear. The equation may also be written in the following mnemonically convenient form

${\displaystyle \mathbf {d} \cdot (\mathbf {r} -\mathbf {d} )=0,}$

which is the equation for a plane through a point A perpendicular to ${\displaystyle {\overrightarrow {OA}}={\vec {d}}}$.

Three-point representation[edit]

PD Image
Plane through points A, B, and C.

The figure shows a plane that by definition passes through three different points A, B, and C that are not on one line. The point P is an arbitrary point in the plane and the reference point O is again drawn outside the plane, but the case that the plane passes through O is not excluded. Referring to figure 2 we introduce the following definitions

${\displaystyle {\vec {a}}={\overrightarrow {OA}},\quad {\vec {b}}={\overrightarrow {OB}},\quad {\vec {c}}={\overrightarrow {OC}},\quad {\vec {r}}={\overrightarrow {OP}}.}$

Clearly the following two non-collinear vectors belong to the plane

${\displaystyle {\vec {u}}={\overrightarrow {AB}}={\vec {b}}-{\vec {a}},\quad {\vec {v}}={\overrightarrow {AC}}={\vec {c}}-{\vec {a}}.}$

Because a plane (an affine space), with a given fixed point as origin is a 2-dimensional linear space and two non-collinear vectors with "tails" in the origin are linearly independent, it follows that any vector in the plane can be written as a linear combination of these two non-collinear vectors. (This is also expressed as: Any vector in the plane can be decomposed into components along the two non-collinear vectors.) In particular, taking A as origin in the plane,

${\displaystyle {\overrightarrow {AP}}={\vec {r}}-{\vec {a}}=\lambda {\vec {u}}+\mu {\vec {v}},\qquad \lambda ,\mu \in \mathbb {R} .}$

The real numbers λ and μ specify the direction of ${\displaystyle {\overrightarrow {AP}}}$. Hence the following equation for the position vector ${\displaystyle {\vec {r}}}$ of the arbitrary point P in the plane:

${\displaystyle {\vec {r}}={\vec {a}}+\lambda {\vec {u}}+\mu {\vec {v}}}$

is known as the point-direction representation of the plane. This representation is equal to the three-point representation

${\displaystyle {\vec {r}}={\vec {a}}+\lambda ({\vec {b}}-{\vec {a}})+\mu ({\vec {c}}-{\vec {a}}),}$

where ${\displaystyle {\vec {a}}}$, ${\displaystyle {\vec {b}}}$, and ${\displaystyle {\vec {c}}}$ are the position vectors of the three points that define the plane.

Writing for the position vector of the arbitrary point P in the plane

${\displaystyle {\vec {r}}=(1-\lambda -\mu )\,{\vec {a}}+\lambda \,{\vec {b}}+\mu \,{\vec {c}}\;\equiv \;\xi _{1}\,{\vec {a}}+\xi _{2}\,{\vec {b}}+\xi _{3}\,{\vec {c}}\;,}$

we find that the real triplet (ξ₁, ξ₂, ξ₃) with ξ₁ + ξ₁ + ξ₁ = 1 forms a set of coordinates for P. The numbers {ξ₁, ξ₂, ξ₃ | ξ₁+ ξ₂+ ξ₃ = 1 } are known as the barycentric coordinates of P. It is trivial to go from barycentric coordinates to the "three-point representation",

${\displaystyle {\vec {r}}=\xi _{1}{\vec {a}}+\xi _{2}{\vec {b}}+\xi _{3}{\vec {c}}\quad {\hbox{with}}\quad \xi _{1}=1-\xi _{2}-\xi _{3}\;\Longleftrightarrow \;{\vec {r}}={\vec {a}}+\xi _{2}({\vec {b}}-{\vec {a}})+\xi _{3}({\vec {c}}-{\vec {a}}).}$

Beyond mathematics[edit]

In industry, a surface plate^[2] is a piece of cast iron or other appropriate material whose surface (or rather a part of it) is made as close as possible to a geometric plane (or rather a part of it, usually a square). An old method of their manufacturing is the three-plate method: three roughly flat surfaces become more and more flat when rubbing against each other: first and second; second and third; third and first; first and second again, and so on. It is possible to achieve a surface close to a plane up to 10^–5 of its size.

Reference[edit]