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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 =.
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]
The normal section of a surface at a particular point is the curve produced by the intersection of that surface with a normal plane. [1] [2] [3] The curvature of the normal section is called the normal curvature. If the surface is bow or cylinder shaped, the maximum and the minimum of these curvatures are the principal curvatures.
The line with equation ax + by + c = 0 has slope -a/b, so any line perpendicular to it will have slope b/a (the negative reciprocal). Let (m, n) be the point of intersection of the line ax + by + c = 0 and the line perpendicular to it which passes through the point (x 0, y 0). The line through these two points is perpendicular to the original ...
[1] [2] Such a drawing is called a plane graph, or a planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are ...
Distance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue. In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane, a plane in Euclidean space, or a hyperplane in higher dimensions.
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the ...
In either the coordinate or vector formulations, one may verify that the given point lies on the given plane by plugging the point into the equation of the plane. To see that it is the closest point to the origin on the plane, observe that p {\displaystyle \mathbf {p} } is a scalar multiple of the vector v {\displaystyle \mathbf {v} } defining ...