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.
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).
It is written in vector notation as
The dot indicates the scalar product or dot product.
Vector points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E. The vector represents the unit normal vector of plane or line E. The distance is the shortest distance from the origin O to the plane or line.
^Bôcher, Maxime (1915), Plane Analytic Geometry: With Introductory Chapters on the Differential Calculus, H. Holt, p. 44.
^John Vince: Geometry for Computer Graphics. Springer, 2005, ISBN 9781852338343, pp. 42, 58, 135, 273
The Hessenormalform named after Otto Hesse, is an equation used in analytic geometry, and describes a line in R2{\displaystyle \mathbb {R} ^{2}} or a...
Hessian matrix, the Hessenormalform, the Hesse configuration, the Hessian group, Hessian pairs, Hesse's theorem, Hesse pencil, and the Hesse transfer principle...
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical...
the Hessenormalform of a line to calculate the distance | P l | {\displaystyle |Pl|} .) An ellipse possesses the following property: The normal at a...
{w} } is the (not necessarily normalized) normal vector to the hyperplane. This is much like Hessenormalform, except that w {\displaystyle \mathbf {w}...
f_{1})^{2}+(y-f_{2})^{2}} (the left side of the equation uses the Hessenormalform of a line to calculate the distance | P l | {\displaystyle |Pl|} )...
{\displaystyle \ n_{x}au+n_{y}bv+n_{z}cw=d.} Let muu + mvv + mww = δ be the Hessenormalform of the new plane and m = [ m u m v m w ] {\displaystyle \;\mathbf...
for computational reasons, Duda and Hart proposed the use of the Hessenormalform r=xcosθ+ysinθ,{\displaystyle r=x\cos \theta +y\sin \theta ,} where...
asymptote is b {\displaystyle b} (the semi-minor axis). From the Hessenormalform b x ± a y a 2 + b 2 = 0 {\displaystyle {\tfrac {bx\pm ay}{\sqrt {a^{2}+b^{2}}}}=0}...
the triangle by three linear equations in two unknowns given by the hessenormalforms of the corresponding lines. The solution of this overdetermined system...
Paderborn, Germany Hepatocyte nuclear factor Hermite normal form Hessenormalform Hoosier National Forest, in Indiana, United States Hydrazinium nitroformate...
Emil Julius Gumbel: "Gumbel distribution" Otto Hesse: "Hessian curve", "Hessian matrix", "Hessenormalform" Leo Koenigsberger Sofia Kovalevskaya: "Cauchy–Kowalevski...
Grand Duchess of Hesse and by Rhine through her marriage to Prince Georg Donatus, pretender to the throne of the Grand Duchy of Hesse. She was also the...
Ernest Louis, Grand Duke of Hesse and by Rhine, Prince Friedrich, Alexandra Feodorovna, Empress of Russia, Princess Marie of Hesse and by Rhine Prince Alfred...
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued...
original on 9 October 2022. Hesse, p. 47 Hesse, p. 39 Hesse, p. 41 Hesse, p. 49 Hesse, p. 44 Saxton, Vol. 3, p. 164 Hesse, p. 51 Plemenkov, p. 236 Saxton...
third inflection point; the nine points and 12 lines formed in this way form a realization of the Hesse configuration. Given an isogeny f : E → E ′ {\displaystyle...
SL2(3) of order 24. It also acts on the Hesse pencil of elliptic curves, and forms the automorphism group of the Hesse configuration of the 9 inflection points...
Darmstadt (German: [ˈdaʁmʃtat] ) is a city in the state of Hesse in Germany, located in the southern part of the Rhine-Main-Area (Frankfurt Metropolitan...
"Frank ford on the Main"), is the most populous city in the German state of Hesse. Its 791,000 inhabitants as of 2022 make it the fifth-most populous city...
princess, as well as Princess of Hesse-Kassel and Princess of Hanover through her successive marriages to Prince Christoph of Hesse and Prince George William...
battle starts. In normal fights, the player can drag their pointer across the circles representing each Sinner's attack, either to form a chain of attacks...
Probability for Data Science. Michigan Publishing. p. 18. ISBN 978-1-60785-746-4. Hesse, C. (1990). "Rates of convergence for the empirical distribution function...
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