Hessesche Normal Form Ebene Beispiel Essay
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]} 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. The vector represents the unitnormal vector of E or g, that points from the origin of the coordinate system to the plane (or line, in 2D). The distance is the distance from the origin to the plane (or line).
This equation is satisfied by all points P described by the location vector , which lie precisely in the plane E (or in 2D, on the line g).
Derivation/Calculation from the normal form[edit]
Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D.
In the normal form,
a plane is given by a normal vector as well as an arbitrary position vector of a point . The direction of is chosen to satisfy the following inequality
By dividing the normal vector by its magnitude, we obtain the unit (or normalized) normal vector
and the above equation can be rewritten as
Substituting
we obtain the Hesse normal form
In this diagram, d is the distance from the origin. Because holds for every point in the plane, it is also true at point Q (the point where the vector from the origin meets the plane E), with , per the definition of the Scalar product
The magnitude of is the shortest distance from the origin to the plane.
References[edit]
Alex Reichwein lehrt im Herbstsemester 2017 am Department of Political Sciencean der University of Copenhagen im Master IR.
NEWS - EVENTS 2017/18:
GGS-Sektion "Normen und (deren) Wandel in der Weltpolitik" - Lecture Series Programme 2017/18
Publikationsprojekte 2016/2017 (Auswahl):
- Hans J. Morgenthauund die Twenty Years' Crisis. Wiesbaden: Springer/VS.
- Die R2P: Norm oder Instrument für einen neuen Militärischen Humanitarismus? Berlin: LIT Verlag.
- Trends in European IR Theory. 8 Volumes, London/New York: Palgrave Macmillan.
Konferenzbeiträge & Workshops 2018
ISA's 59th Annual Convention San Francisco/CAL., 4-7 April 2018
- Gideon Rose 20 Years later: Neoclassical Realism Reconsidered: Theoretical Innovations (organized Panel and Roundtable, paper giver on the classical realist core in neoclassical realism)
- Foreign Policy in Times of Global Power Shifts: Insights from Neoclassical Realism (organized Panel, Chair and Discussant)
- Trends in European IR Theory: Advances, Doldrums, Decay (paper giver on the Genesis of the Realist Tradition in IR)
- Power of Rules and Rule of Power: Responsibility is Dead - Long Live Protection? (Roundtable Participant) Full Programme
EISA's 12th Pan-European Conference on IR, Prague, 12-15 September 2018 (in cooperation with the CEEISA)
Section Convenor Back to the Future? Realist Visions of Order, International Politics and Foreign Policy in Times of Crises, Conflict and the return of War in Europe (Section List)
Konferenzbeiträge & Workshops 2017:
- Rethinking the R2P: Towards a New Authoritarian Interventionism?5. Offene IB-Sektionstagung der DVPW,Bremen, 4-6 Oktober 2017.
- Reappraising Realism as a European Tradition in IR, EISA's 11 Pan-European Conference on IR, Barcelona, 13-16 September 2017.
- A dangerous responsibility: Protecting minorities under the R2P umbrella, EISA's IR Conference, Barcelona.
- Responsible Power: Germany’s emerging leadership role in Europe, NOPSA Workshops, Odense, University of Southern Denmark, 7-10 August 2017.
- Section Convenor and Chair Realist Visions of World Order and Foreign Policy in Times of Global Power Shifts, 5th Global Conference of the WISC/ TAIR,Taipei/Taiwan, 1-3 April 2017.
- A dangerous responsibility: Protecting national minorities, religious fellows or irredentism movements under the R2P umbrella, ISA's 58th Annual Convention, Baltimore/Maryland, 22-25 February 2017.
Auswahl neuerer Publikationen:
- Realism and European Foreign Policy, in: Jørgensen, K.E. et.al. (eds.), The SAGE Handbook of European Foreign Policy. London u.a.: SAGE, 99-120.
- Morgenthau, Vietnam und die Sorge um Amerika, in: Rohde, Chr./Troy, J. (Hg.), Macht, Recht, Demokratie: Zum Staatsverständnis von Hans J. Morgenthau.Baden Baden: Nomos, Reihe Staatsverständnisse), 95-139.
Alle Publikationen 2013-2017 am Seitenende; Gesamtverzeichnis unter "Publikationen"