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For a plane given by the general form plane equation + + + =, the vector = (,,) is a normal. For a plane whose equation is given in parametric form (,) = + +, where is a point on the plane and , are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both and , which can be found as the cross product =.
Normal vector in red, line in green, point O shown in blue. The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in or a plane in Euclidean space or a hyperplane in higher dimensions. [1][2] It is primarily used for calculating distances (see point-plane distance and point-line distance).
A space curve; the vectors T, N and B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.
Normal plane (geometry) Saddle surface with normal planes in directions of principal curvatures. A normal plane is any plane containing the normal vector of a surface at a particular point. The normal plane also refers to the plane that is perpendicular to the tangent vector of a space curve; (this plane also contains the normal vector) see ...
Conversely, it is easily shown that if a, b, c, and d are constants and a, b, and c are not all zero, then the graph of the equation + + + =, is a plane having the vector n = (a, b, c) as a normal. [5] This familiar equation for a plane is called the general form of the equation of the plane or just the plane equation. [6]
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b ...
Line–plane intersection. 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 ...
The vector area of a surface can be interpreted as the (signed) projected area or "shadow" of the surface in the plane in which it is greatest; its direction is given by that plane's normal. For a curved or faceted (i.e. non-planar) surface, the vector area is smaller in magnitude than the actual surface area.