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Hesse normal form  

2014-09-11 01:41:54|  分类: 专业 |  标签: |举报 |字号 订阅

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The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in \mathbb{R}^2 or a plane in Euclidean space \mathbb{R}^3 or a hyperplane in higher dimensions.[1] It is primarily used for calculating distances, and is written in vector notation as

\vec r \cdot \vec n_0 - d = 0.\,

This equation is satisfied by all points P described by the location vector \vec r, which lie precisely in the plane E (or in 2D, on the line g).

The vector \vec n_0 represents the unit normal vector of E or g, that points from the origin of the coordinate system to the plane (or line, in 2D). The distance d \ge 0 is the distance from the origin to the plane (or line). The dot \cdot indicates the scalar product or dot product

The normal segment for a given line is defined to be the line segment drawn from the origin perpendicular to the line. This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of a straight line on the plane is given by:

 y sin theta + x cos theta - p = 0,,

where θ is the angle of inclination of the normal segment (the oriented angle from the unit vector of the x axis to this segment), and p is the (positive) length of the normal segment. The normal form can be derived from the general form by dividing all of the coefficients by

frac{|c|}{-c}sqrt{a^2 + b^2}.

This form is also called the Hesse normal form,[12] after the German mathematician Ludwig Otto Hesse.

Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, θ and p, to be specified. Note that if p > 0, then θ is uniquely defined modulo 2π. On the other hand, if the line is through the origin (c = 0, p = 0), one drops the |c|/(?c) term to compute sinθ and cosθ, and θ is only defined modulo π.

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