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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 in computer graphics, motion planning, and collision detection. ==Algebraic form== In vector notation, a plane can be expressed as the set of points for which : where is a normal vector to the plane and is a point on the plane. (The notation denotes the dot product of the two vector and .) The vector equation for a line is : where is a vector in the direction of the line, is a point on the line, and is a scalar in the real number domain. Substitute the equation for the line into the equation for the plane gives : Expanding gives : And solve for : If then the line and plane are parallel. There will be two cases: if 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 there is a single point of intersection. The value of can be calculated and the point of intersection is given by :. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Line–plane intersection」の詳細全文を読む スポンサード リンク
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