Line -- plane intersection - wikipedia Line -- plane intersection This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. ( December 2009 ) The three possible plane - line relationships in three dimensions. ( Shown in each case is only a portion of the plane, which extends infinitely far. ) In analytic geometry, the intersection of a line and a plane in three - dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point. Distinguishing these cases, and determining equations for the point and line in the latter cases, have use in computer graphics, motion planning, and collision detection. Contents 1 Algebraic form 2 Parametric form 3 Uses 4 See also 5 External links Algebraic form In vector notation, a plane can be expressed as the set of points p ( displaystyle mathbf ( p ) ) for which ( p − p 0 ) ⋅ n = 0 ( displaystyle ( mathbf ( p ) - mathbf ( p_ ( 0 ) ) ) cdot mathbf ( n ) = 0 ) where n ( displaystyle mathbf ( n ) ) is a normal vector to the plane and p 0 ( displaystyle mathbf ( p_ ( 0 ) ) ) is a point on the plane. ( The notation a ⋅ b ( displaystyle mathbf ( a ) cdot mathbf ( b ) ) denotes the dot product of the vectors a ( displaystyle mathbf ( a ) ) and b ( displaystyle mathbf ( b ) ). ) The vector equation for a line is p = d l + l 0 d ∈ R ( displaystyle mathbf ( p ) = d mathbf ( l ) + mathbf ( l_ ( 0 ) ) quad d in mathbb ( R ) ) where l ( displaystyle mathbf ( l ) ) is a vector in the direction of the line, l 0 ( displaystyle mathbf ( l_ ( 0 ) ) ) is a point on the line, and d ( displaystyle d ) is a scalar in the real number domain. Substituting the equation for the line into the equation for the plane gives ( d l + l 0 − p 0 ) ⋅ n = 0. ( displaystyle ( d mathbf ( l ) + mathbf ( l_ ( 0 ) ) - mathbf ( p_ ( 0 ) ) ) cdot mathbf ( n ) = 0. ) Expanding gives d l ⋅ n + ( l 0 − p 0 ) ⋅ n = 0. ( displaystyle d mathbf ( l ) cdot mathbf ( n ) + ( mathbf ( l_ ( 0 ) ) - mathbf ( p_ ( 0 ) ) ) cdot mathbf ( n ) = 0. ) And solving for d ( displaystyle d ) gives d = ( p 0 − l 0 ) ⋅ n l ⋅ n. ( displaystyle d = ( ( mathbf ( p_ ( 0 ) ) - mathbf ( l_ ( 0 ) ) ) cdot mathbf ( n ) over mathbf ( l ) cdot mathbf ( n ) ). ) If l ⋅ n = 0 ( displaystyle mathbf ( l ) cdot mathbf ( n ) = 0 ) then the line and plane are parallel. There will be two cases : if ( p 0 − l 0 ) ⋅ n = 0 ( displaystyle ( mathbf ( p_ ( 0 ) ) - mathbf ( l_ ( 0 ) ) ) cdot mathbf ( n ) = 0 ) then the line is contained in the plane, that is, the line intersects the plane at each point of the line. Otherwise, the line and plane have no intersection. If l ⋅ n ≠ 0 ( displaystyle mathbf ( l ) cdot mathbf ( n ) \neq 0 ) there is a single point of intersection. The value of d ( displaystyle d ) can be calculated and the point of intersection is given by d l + l 0 ( displaystyle d mathbf ( l ) + mathbf ( l_ ( 0 ) ) ). Parametric form The intersection of line and plane. A line is described by all points that are a given direction from a point. Thus a general point on a line can be represented as l a + ( l b − l a ) t, t ∈ R ( displaystyle mathbf ( l ) _ ( a ) + ( mathbf ( l ) _ ( b ) - mathbf ( l ) _ ( a ) ) t, quad t in mathbb ( R ) ) where l a = ( x a, y a, z a ) ( displaystyle mathbf ( l ) _ ( a ) = ( x_ ( a ), y_ ( a ), z_ ( a ) ) ) and l b = ( x b, y b, z b ) ( displaystyle mathbf ( l ) _ ( b ) = ( x_ ( b ), y_ ( b ), z_ ( b ) ) ) are two distinct points along the line. Similarly a general point on a plane can be represented as p 0 + ( p 1 − p 0 ) u + ( p 2 − p 0 ) v, u, v ∈ R ( displaystyle mathbf ( p ) _ ( 0 ) + ( mathbf ( p ) _ ( 1 ) - mathbf ( p ) _ ( 0 ) ) u+ ( mathbf ( p ) _ ( 2 ) - mathbf ( p ) _ ( 0 ) ) v, quad u, v in mathbb ( R ) ) where p k = ( x k, y k, z k ) ( displaystyle mathbf ( p ) _ ( k ) = ( x_ ( k ), y_ ( k ), z_ ( k ) ) ), k = 0, 1, 2 ( displaystyle k = 0, 1, 2 ) are three points in the plane which are not co-linear. The point at which the line intersects the plane is therefore described by setting the point on the line equal to the point on the plane, giving the parametric equation : l a + ( l b − l a ) t = p 0 + ( p 1 − p 0 ) u + ( p 2 − p 0 ) v ( displaystyle mathbf ( l ) _ ( a ) + ( mathbf ( l ) _ ( b ) - mathbf ( l ) _ ( a ) ) t = mathbf ( p ) _ ( 0 ) + ( mathbf ( p ) _ ( 1 ) - mathbf ( p ) _ ( 0 ) ) u+ ( mathbf ( p ) _ ( 2 ) - mathbf ( p ) _ ( 0 ) ) v ) This can be rewritten as l a − p 0 = ( l a − l b ) t + ( p 1 − p 0 ) u + ( p 2 − p 0 ) v, ( displaystyle mathbf ( l ) _ ( a ) - mathbf ( p ) _ ( 0 ) = ( mathbf ( l ) _ ( a ) - mathbf ( l ) _ ( b ) ) t+ ( mathbf ( p ) _ ( 1 ) - mathbf ( p ) _ ( 0 ) ) u+ ( mathbf ( p ) _ ( 2 ) - mathbf ( p ) _ ( 0 ) ) v, ) which can be expressed in matrix form as : ( x a − x 0 y a − y 0 z a − z 0 ) = ( x a − x b x 1 − x 0 x 2 − x 0 y a − y b y 1 − y 0 y 2 − y 0 z a − z b z 1 − z 0 z 2 − z 0 ) ( t u v ) ( displaystyle ( \begin ( bmatrix ) x_ ( a ) - x_ ( 0 ) y_ ( a ) - y_ ( 0 ) z_ ( a ) - z_ ( 0 ) end ( bmatrix ) ) = ( \begin ( bmatrix ) x_ ( a ) - x_ ( b ) &x_ ( 1 ) - x_ ( 0 ) &x_ ( 2 ) - x_ ( 0 ) y_ ( a ) - y_ ( b ) &y_ ( 1 ) - y_ ( 0 ) &y_ ( 2 ) - y_ ( 0 ) z_ ( a ) - z_ ( b ) &z_ ( 1 ) - z_ ( 0 ) &z_ ( 2 ) - z_ ( 0 ) end ( bmatrix ) ) ( \begin ( bmatrix ) t \u \v end ( bmatrix ) ) ) The point of intersection is then equal to l a + ( l b − l a ) t ( displaystyle mathbf ( l ) _ ( a ) + ( mathbf ( l ) _ ( b ) - mathbf ( l ) _ ( a ) ) t ) If the line is parallel to the plane then the vectors l b − l a ( displaystyle mathbf ( l ) _ ( b ) - mathbf ( l ) _ ( a ) ), p 1 − p 0 ( displaystyle mathbf ( p ) _ ( 1 ) - mathbf ( p ) _ ( 0 ) ), and p 2 − p 0 ( displaystyle mathbf ( p ) _ ( 2 ) - mathbf ( p ) _ ( 0 ) ) will be linearly dependent and the matrix will be singular. This situation will also occur when the line lies in the plane. If the solution satisfies the condition t ∈ ( 0, 1 ), ( displaystyle t in ( 0, 1 ), ), then the intersection point is on the line between l a ( displaystyle mathbf ( l ) _ ( a ) ) and l b ( displaystyle mathbf ( l ) _ ( b ) ). If the solution satisfies u, v ∈ ( 0, 1 ), ( u + v ) ≤ 1, ( displaystyle u, v in ( 0, 1 ), ; ; ; ( u + v ) leq 1, ) then the intersection point is in the plane inside the triangle spanned by the three points p 0 ( displaystyle mathbf ( p ) _ ( 0 ) ), p 1 ( displaystyle mathbf ( p ) _ ( 1 ) ) and p 2 ( displaystyle mathbf ( p ) _ ( 2 ) ). This problem is solved by inverting the matrix : ( t u v ) = ( x a − x b x 1 − x 0 x 2 − x 0 y a − y b y 1 − y 0 y 2 − y 0 z a − z b z 1 − z 0 z 2 − z 0 ) − 1 ( x a − x 0 y a − y 0 z a − z 0 ). ( displaystyle ( \begin ( bmatrix ) t \u \v end ( bmatrix ) ) = ( \begin ( bmatrix ) x_ ( a ) - x_ ( b ) &x_ ( 1 ) - x_ ( 0 ) &x_ ( 2 ) - x_ ( 0 ) y_ ( a ) - y_ ( b ) &y_ ( 1 ) - y_ ( 0 ) &y_ ( 2 ) - y_ ( 0 ) z_ ( a ) - z_ ( b ) &z_ ( 1 ) - z_ ( 0 ) &z_ ( 2 ) - z_ ( 0 ) end ( bmatrix ) ) ^ ( - 1 ) ( \begin ( bmatrix ) x_ ( a ) - x_ ( 0 ) y_ ( a ) - y_ ( 0 ) z_ ( a ) - z_ ( 0 ) end ( bmatrix ) ). ) Uses In the ray tracing method of computer graphics a surface can be represented as a set of pieces of planes. The intersection of a ray of light with each plane is used to produce an image of the surface. In vision - based 3D reconstruction, a subfield of computer vision, depth values are commonly measured by so - called triangulation method, which finds the intersection between light plane and ray reflected toward camera. The algorithm can be generalised to cover intersection with other planar figures, in particular, the intersection of a polyhedron with a line. See also Plücker coordinates # Plane - line meet calculating the intersection when the line is expressed by Plücker coordinates. External links Intersections of Lines, Segments and Planes ( 2D & 3D ) from GeomAlgorithms.com Retrieved from `` https://en.wikipedia.org/w/index.php?title=Line -- plane_intersection&oldid = 807938437 '' Categories : Euclidean geometry Computational physics Geometric algorithms Hidden categories : Articles lacking sources from December 2009 All articles lacking sources Talk Contents About Wikipedia Čeština ភាសា ខ្មែរ Українська 中文 Edit links This page was last edited on 30 October 2017, at 23 : 05. Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia ® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. About Wikipedia

Question: how to name the intersection of a line and a plane
Answer: empty set