Line-plane intersection
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In analytic geometry, the intersection of a line and a plane can be the empty set, a point, or a line. Distinguishing these cases, and determining equations for the point and line in the latter cases have use, for example, in computer graphics, motion planning, and collision detection.
[edit] Equations in 3D Euclidean space
[edit] Parametric form
A line is described by all points that are a given direction from a point. Thus a line can be represented as
where and are two distinct points along the line.
Similarly a plane can be represented as
where , k = 0,1,2 are three points in the plane.
The point at which the line intersects the plane is therefore described by setting the line equal to the plane in the parametric equation:
This can be simplified to
which can be expressed in matrix form as:
The point of intersection is then equal to
[edit] Usage
If the solution satisfies the condition , then the intersection point is on the line between and .
If the solution satisfies
then the intersection point is in the plane inside the triangle spanned by the three points , and .
This problem is typically solved by expressing it in matrix form, and inverting it: