Euclidean planes in three-dimensional space

A {{vanchor|plane segment}} or {{vanchor|planar region}} (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.{{efn|More precisely, an algebraic entity analogous to an equivalence class of oriented plane segments.}}

A face is a plane segment bounding a solid object.{{cite book | title = Merriam-Webster's Collegiate Dictionary | edition = Eleventh | publisher = Merriam-Webster | location = Springfield, MA | year = 2004}}

A slab is a region bounded by two parallel planes.

A parallelepiped is a region bounded by three pairs of parallel planes.

{{Main|Euclidean geometry}}

Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry.{{harvnb|Eves|1963|loc=p. 19}} 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.{{Citation

|last= Joyce

|first= D. E.|author-link=David E. Joyce (mathematician)

|title= Euclid's Elements, Book I, Definition 7

|publisher= Clark University

|year= 1996

|url= http://aleph0.clarku.edu/~djoyce/java/elements/bookI/defI7.html

|access-date= 8 August 2009}} 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.

File:Planes parallel.svg

A plane is a ruled surface.

{{excerpt|Euclidean plane|templates=-General geometry}}

This section is solely concerned with planes embedded in three dimensions: specifically, in Cartesian product.

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.

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.

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 {{math|r₀}} be the position vector of some point {{math|1=P₀ = (x₀, y₀, z₀)}}, and let {{math|1=n = (a, b, c)}} be a nonzero vector. The plane determined by the point {{math|P₀}} and the vector {{mvar|n}} consists of those points {{math|P}}, with position vector {{mvar|r}}, such that the vector drawn from {{math|P₀}} to {{math|P}} is perpendicular to {{mvar|n}}. 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 {{mvar|r}} such that

$$\backslash boldsymbol\{n\}\; \backslash cdot\; (\backslash boldsymbol\{r\}-\backslash boldsymbol\{r\}\_0)=0.$$

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.{{harvnb|Anton|1994|loc=p. 155}} This is just a linear equation

$$ax\; +\; by\; +\; cz\; +\; d\; =\; 0,$$

where

$$d\; =\; -(ax\_0\; +\; by\_0\; +\; cz\_0),$$

which is the expanded form of $-\; \backslash boldsymbol\{n\}\; \backslash cdot\; \backslash boldsymbol\{r\}\_0.$

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 {{math|a}}, {{math|b}}, {{math|c}}, and {{math|d}} are constants and {{math|a}}, {{math|b}}, and {{math|c}} are not all zero, then the graph of the equation

$$ax\; +\; by\; +\; cz\; +\; d\; =\; 0,$$

is a plane having the vector {{math|1=n = (a, b, c)}} as a normal.{{harvnb|Anton|1994|loc=p. 156}} This familiar equation for a plane is called the general form of the equation of the plane or just the plane equation.{{Citation

|title= Plane

|url= http://mathworld.wolfram.com/Plane.html

|year= 2009

|author= Weisstein, Eric W.

|journal= MathWorld--A Wolfram Web Resource

|access-date= 2009-08-08

}}

Thus for example a regression equation of the form {{math|1=y = d + ax + cz}} (with {{math|1=b = −1}}) establishes a best-fit plane in three-dimensional space when there are two explanatory variables.

Alternatively, a plane may be described parametrically as the set of all points of the form

$$\backslash boldsymbol\{r\}\; =\; \backslash boldsymbol\{r\}\_0\; +\; s\; \backslash boldsymbol\{v\}\; +\; t\; \backslash boldsymbol\{w\},$$

File:PlaneR.jpg

where {{mvar|s}} and {{mvar|t}} range over all real numbers, {{mvar|v}} and {{mvar|w}} are given linearly independent vectors defining the plane, and {{math|r₀}} is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors {{mvar|v}} and {{mvar|w}} can be visualized as vectors starting at {{math|r₀}} and pointing in different directions along the plane. The vectors {{mvar|v}} and {{mvar|w}} can be perpendicular, but cannot be parallel.

Let {{math|1=p₁ = (x₁, y₁, z₁)}}, {{math|1=p₂ = (x₂, y₂, z₂)}}, and {{math|1=p₃ = (x₃, y₃, z₃)}} be non-collinear points.

The plane passing through {{math|p₁}}, {{math|p₂}}, and {{math|p₃}} can be described as the set of all points (x,y,z) that satisfy the following determinant equations:

$$\backslash begin\{vmatrix\}$$

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{vmatrix} = \begin{vmatrix}

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{vmatrix} = 0.

To describe the plane by an equation of the form $ax\; +\; by\; +\; cz\; +\; d\; =\; 0$, solve the following system of equations:

$$ax\_1\; +\; by\_1\; +\; cz\_1\; +\; d\; =\; 0$$

$$ax\_2\; +\; by\_2\; +\; cz\_2\; +\; d\; =\; 0$$

$$ax\_3\; +\; by\_3\; +\; cz\_3\; +\; d\; =\; 0.$$

This system can be solved using Cramer's rule and basic matrix manipulations. Let

$$D\; =\; \backslash begin\{vmatrix\}$$

x_1 & y_1 & z_1 \\

x_2 & y_2 & z_2 \\

x_3 & y_3 & z_3

\end{vmatrix}.

If {{math|D}} is non-zero (so for planes not through the origin) the values for {{math|a}}, {{math|b}} and {{math|c}} can be calculated as follows:

$$a\; =\; \backslash frac\{-d\}\{D\}\; \backslash begin\{vmatrix\}$$

1 & y_1 & z_1 \\

1 & y_2 & z_2 \\

1 & y_3 & z_3

\end{vmatrix}

$$b\; =\; \backslash frac\{-d\}\{D\}\; \backslash begin\{vmatrix\}$$

x_1 & 1 & z_1 \\

x_2 & 1 & z_2 \\

x_3 & 1 & z_3

\end{vmatrix}

$$c\; =\; \backslash frac\{-d\}\{D\}\; \backslash begin\{vmatrix\}$$

x_1 & y_1 & 1 \\

x_2 & y_2 & 1 \\

x_3 & y_3 & 1

\end{vmatrix}.

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.

This plane can also be described by the {{Section link|Euclidean planes in three-dimensional space|Point–normal form and general form of the equation of a plane|nopage=y}} prescription above. A suitable normal vector is given by the cross product

$$\backslash boldsymbol\; n\; =\; (\; \backslash boldsymbol\; p\_2\; -\; \backslash boldsymbol\; p\_1\; )\; \backslash times\; (\; \backslash boldsymbol\; p\_3\; -\; \backslash boldsymbol\; p\_1\; ),$$

and the point {{math|r₀}} can be taken to be any of the given points {{math|p₁}}, {{math|p₂}} or {{math|p₃}}{{citation

|last= Dawkins

|first= Paul

|title= Calculus III

|chapter= Equations of Planes

|chapter-url= http://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfPlanes.aspx

}} (or any other point in the plane).

{{excerpt|Distance from a point to a plane}}

{{excerpt|Line–plane intersection}}

{{excerpt|Plane–plane intersection}}

{{excerpt|Intersection (geometry)#A sphere and a plane }}

{{see also|Surface#In the physical sciences}}

File:Plane wave wavefronts 3D.svgs 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.{{cite web | last=Evans | first=Jon | title=Smoothest surface ever is a mirror for atoms | website=New Scientist | date=2008-08-22 | url=https://www.newscientist.com/article/dn14587-smoothest-surface-ever-is-a-mirror-for-atoms/ | access-date=2023-03-05}}

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.

File:Webysther 20211009102044 - Parque Geológico do Varvito.jpg, Brazil]]

{{Main|Miller index}}

File:Miller Indices Cubes2.svg

The attitude of a lattice plane is the orientation of the line normal to the plane,

{{cite book |author=William Anthony Granville |title=Elements of the Differential and Integral Calculus |page=[https://archive.org/details/elementsdiffere02goog/page/n293 275] |url=https://archive.org/details/elementsdiffere02goog |chapter=§178 Normal line to a surface |publisher=Ginn & Company |year=1904

}} 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),

{{cite book |title=A Treatise on the Mathematical Theory of Elasticity| volume=1 |author=Augustus Edward Hough Love |url=https://books.google.com/books?id=-GkSAAAAIAAJ&pg=PA79 |page=79 ff |year=1892 |publisher=Cambridge University Press

}}

{{cite book |author1=Marcus Frederick Charles Ladd |author2=Rex Alfred Palmer |chapter-url=https://books.google.com/books?id=vNJrAe36BBMC&pg=PA62 |chapter=§2.3 Families of planes and interplanar spacings |page=62 ff |isbn=0-306-47454-9 |edition=4th |publisher=Springer |year=2003 |title=Structure Determination by X-Ray Crystallography

}} so the family of planes has an attitude common to all its constituent planes.

{{Main|Strike and dip}}

File:StrikeLine&Dip.JPG

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.

{{cite book |title=Structural Analysis and Synthesis: A Laboratory Course in Structural Geology |author1=Stephen Mark Rowland |author2=Ernest M. Duebendorfer |author3=Ilsa M. Schiefelbein |chapter-url=https://books.google.com/books?id=IWnmBEtmg2MC&q=%22attitude+of+a+line%22&pg=PR3 |chapter=Attitudes of lines and planes |isbn=978-1-4051-1652-7 |edition=3rd |publisher=Wiley-Blackwell |year=2007 |page=1 ff

}}

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.

{{main cat|Planes (geometry)}}

{{Div col|colwidth=20em}}

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

{{Div col end}}

{{notelist}}

{{reflist}}

• {{citation|last=Anton|first=Howard|title=Elementary Linear Algebra|edition=7th|publisher=John Wiley & Sons|year=1994|isbn=0-471-58742-7}}
• {{citation|last=Eves|first=Howard|title=A Survey of Geometry|publisher=Allyn and Bacon, Inc.|place=Boston|year=1963|volume= I}}
• {{cite book |title=Solid Geometry|url=https://archive.org/details/solidgeometry00hobbgoog|first=Charles Austen|last=Hobbs|publisher=Cambridge, G.H. Kent|year=1921|pages=[https://dn790007.ca.archive.org/0/items/solidgeometry00hobbgoog/solidgeometry00hobbgoog.pdf#page=151 396-400]|lccn=21016427}}

{{Commons category}}

• {{SpringerEOM|title=Plane|id=p/p072790}}
• {{MathWorld|title=Plane|urlname=Plane}}
• [http://www.wdl.org/en/item/2850/ "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.

{{Authority control}}

{{DEFAULTSORT:Plane (Geometry)}}

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