Euclidean planes in three-dimensional space

Plane equation in normal form

In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space ${ℝ}^3$. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin. While a pair of real numbers ${ℝ}^2$ suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their embedding in the ambient space ${ℝ}^3$.

Derived concepts

A Template:Vanchor or Template:Vanchor (or simply "plane", in lay use) is a planar surface region; it is analogous to a line segment. A bivector is an oriented plane segment, analogous to directed line segments.Template:Efn A face is a plane segment bounding a solid object. A slab is a region bounded by two parallel planes. A parallelepiped is a region bounded by three pairs of parallel planes.

Background

Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry. He selected a small core of undefined terms (called common notions) and postulates (or axioms) which he then used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the Elements, it may be thought of as part of the common notions. Euclid never used numbers to measure length, angle, or area. The Euclidean plane equipped with a chosen Cartesian coordinate system is called a Cartesian plane; a non-Cartesian Euclidean plane equipped with a polar coordinate system would be called a polar plane.

Three parallel planes.

A plane is a ruled surface.

Representation

This section is solely concerned with planes embedded in three dimensions: specifically, in [[Cartesian product|Template:Math]].

Determination by contained points and lines

In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following:

• Three non-collinear points (points not on a single line).
• A line and a point not on that line.
• Two distinct but intersecting lines.
• Two distinct but parallel lines.

Properties

The following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues:

• Two distinct planes are either parallel or they intersect in a line.
• A line is either parallel to a plane, intersects it at a single point, or is contained in the plane.
• Two distinct lines perpendicular to the same plane must be parallel to each other.
• Two distinct planes perpendicular to the same line must be parallel to each other.

Point–normal form and general form of the equation of a planeTemplate:Anchor

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".

Specifically, let Template:Math be the position vector of some point Template:Math, and let Template:Math be a nonzero vector. The plane determined by the point Template:Math and the vector Template:Mvar consists of those points Template:Math, with position vector Template:Mvar, such that the vector drawn from Template:Math to Template:Math is perpendicular to Template:Mvar. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points Template:Mvar such that $$𝒏\cdot (𝒓-{𝒓}_0)=0.$$ The dot here means a dot (scalar) product.
Expanded this becomes $$a(x-x_0)+b(y-y_0)+c(z-z_0)=0,$$ which is the point–normal form of the equation of a plane. This is just a linear equation $$ax+by+cz+d=0,$$ where $$d=-(a{x}_0+b{y}_0+c{z}_0),$$ which is the expanded form of $-𝒏\cdot {𝒓}_0.$

In mathematics it is a common convention to express the normal as a unit vector, but the above argument holds for a normal vector of any non-zero length.

Conversely, it is easily shown that if Template:Math, Template:Math, Template:Math, and Template:Math are constants and Template:Math, Template:Math, and Template:Math are not all zero, then the graph of the equation $$ax+by+cz+d=0,$$ is a plane having the vector Template:Math as a normal. This familiar equation for a plane is called the general form of the equation of the plane or just the plane equation.

Thus for example a regression equation of the form Template:Math (with Template:Math) establishes a best-fit plane in three-dimensional space when there are two explanatory variables.

Describing a plane with a point and two vectors lying on it

Alternatively, a plane may be described parametrically as the set of all points of the form $$𝒓={𝒓}_0+s𝒗+t𝒘,$$

Vector description of a plane

where Template:Mvar and Template:Mvar range over all real numbers, Template:Mvar and Template:Mvar are given linearly independent vectors defining the plane, and Template:Math is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors Template:Mvar and Template:Mvar can be visualized as vectors starting at Template:Math and pointing in different directions along the plane. The vectors Template:Mvar and Template:Mvar can be perpendicular, but cannot be parallel.

Describing a plane through three points

Let Template:Math, Template:Math, and Template:Math be non-collinear points.

Method 1

The plane passing through Template:Math, Template:Math, and Template:Math can be described as the set of all points (x,y,z) that satisfy the following determinant equations: $$\left|\begin{array}{ccc}x-x_1& y-y_1& z-z_1\\ x_2-x_1& y_2-y_1& z_2-z_1\\ x_3-x_1& y_3-y_1& z_3-z_1\end{array}\right|=\left|\begin{array}{ccc}x-x_1& y-y_1& z-z_1\\ x-x_2& y-y_2& z-z_2\\ x-x_3& y-y_3& z-z_3\end{array}\right|=0.$$

Method 2

To describe the plane by an equation of the form $ax+by+cz+d=0$, solve the following system of equations: $$a{x}_1+b{y}_1+c{z}_1+d=0$$ $$a{x}_2+b{y}_2+c{z}_2+d=0$$ $$a{x}_3+b{y}_3+c{z}_3+d=0.$$

This system can be solved using Cramer's rule and basic matrix manipulations. Let $$D=\left|\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right|.$$

If Template:Math is non-zero (so for planes not through the origin) the values for Template:Math, Template:Math and Template:Math can be calculated as follows: $$a=\frac{-d}{D}\left|\begin{array}{ccc}1& y_1& z_1\\ 1& y_2& z_2\\ 1& y_3& z_3\end{array}\right|$$ $$b=\frac{-d}{D}\left|\begin{array}{ccc}{x}_1& 1& z_1\\ x_2& 1& z_2\\ x_3& 1& z_3\end{array}\right|$$ $$c=\frac{-d}{D}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|.$$

These equations are parametric in d. Setting d equal to any non-zero number and substituting it into these equations will yield one solution set.

Method 3

This plane can also be described by the Template:Section link prescription above. A suitable normal vector is given by the cross product $$𝒏=({𝒑}_2-{𝒑}_1)\times ({𝒑}_3-{𝒑}_1),$$ and the point Template:Math can be taken to be any of the given points Template:Math, Template:Math or Template:Math (or any other point in the plane).

Occurrence in nature

The wavefronts of a plane wave traveling in 3-space

A plane serves as a mathematical model for many physical phenomena, such as specular reflection in a plane mirror or wavefronts in a traveling plane wave. The free surface of undisturbed liquids tends to be nearly flat (see flatness). The flattest surface ever manufactured is a quantum-stabilized atom mirror. In astronomy, various reference planes are used to define positions in orbit. Anatomical planes may be lateral ("sagittal"), frontal ("coronal") or transversal. In geology, beds (layers of sediments) often are planar. Planes are involved in different forms of imaging, such as the focal plane, picture plane, and image plane.

Beds of sedimentary rock at Parque Geológico do Varvito, Itu, São Paulo, Brazil

Miller indices

Planes with different Miller indices in cubic crystals

The attitude of a lattice plane is the orientation of the line normal to the plane, and is described by the plane's Miller indices. In three-space a family of planes (a series of parallel planes) can be denoted by its Miller indices (hkl), so the family of planes has an attitude common to all its constituent planes.

Strike and dip

Strike line and dip of a plane describing attitude relative to a horizontal plane and a vertical plane perpendicular to the strike line

Many features observed in geology are planes or lines, and their orientation is commonly referred to as their attitude. These attitudes are specified with two angles.

For a line, these angles are called the trend and the plunge. The trend is the compass direction of the line, and the plunge is the downward angle it makes with a horizontal plane.

For a plane, the two angles are called its strike (angle) and its dip (angle). A strike line is the intersection of a horizontal plane with the observed planar feature (and therefore a horizontal line), and the strike angle is the bearing of this line (that is, relative to geographic north or from magnetic north). The dip is the angle between a horizontal plane and the observed planar feature as observed in a third vertical plane perpendicular to the strike line.

See also

• Dihedral angle
• Flat (geometry)
• Half-plane
• Hyperplane
• Plane coordinates
• Plane of incidence
• Plane of rotation
• Plane orientation
• Polygon

References

• Template:Citation
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External links

• Template:SpringerEOM
• Template:MathWorld
• "Easing the Difficulty of Arithmetic and Planar Geometry" is an Arabic manuscript, from the 15th century, that serves as a tutorial about plane geometry and arithmetic.