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Eccentricitas


(Redirectum de Excentricitas)



Eccentricitas (-atis, f.)[1] sive excentricitas[2] seu eccentrotes[3] in mathematica est parametrum sectionis conicae quod a littera \({\displaystyle e}\) aut \({\displaystyle \varepsilon }\) signatur. Ea velut mensura proximitatis a sectione ad circulum verum considerare potest.

Etiam duae sectiones conicae sunt geometricalibus similes si et solum si eae eccentricitatem eandem habent.

Index

Definitio


Si punctum F, linea L, et parametrum \({\displaystyle e>0}\) dantur, sectio conica est omnes puncti M ubi spatium inter M et F est \({\displaystyle e}\) multiplicatum a spatium inter M et M' (linea M-M' est perpendicularis a linea L). Tum F est focus sectionis conicae, L est directrix, et \({\displaystyle e}\) est eccentricitas.

Etiam si duplex conus verticale oriens et planus eum secans dantur, tum eccentricitas sectionis est \({\displaystyle e={\sin \alpha }/{\sin \beta }}\) ubi \({\displaystyle \alpha }\) est angulus inter planum et libratum, et \({\displaystyle \beta }\) est angulus inter conum et libratum.

Eccentricitas linearis sectionis conicae, quae a littera \({\displaystyle c}\) aut \({\displaystyle e}\) signatur, est spatium inter centrum et focum (aut unum ex duobus focis).

Alia nomina


Aliquando eccentricitas appellatur eccentricitas prima ut ab eccentricitate secunda et tertia distinguatur quae in ellipsibus definiuntur (vide infra). Aliquando eccentricitas appellatur eccentricitas numericalis.

In ellipsibus et hyperbolis, aliquando eccentricitas linearis appellatur semiseparatio focorum.

Notatio


Sunt duae doctrinae usitatae notationis:

Hic res doctrinam primam utitur.

Aequationes


sectio conica aequatio eccentricitas (\({\displaystyle e}\)) eccentricitas linearis (\({\displaystyle c}\))
circulus \({\displaystyle x^{2}+y^{2}=r^{2}}\) \({\displaystyle 0}\) \({\displaystyle 0}\)
ellipsis \({\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}\) \({\displaystyle {\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}\) \({\displaystyle {\sqrt {a^{2}-b^{2}}}}\)
parabola \({\displaystyle y^{2}=4ax}\) \({\displaystyle 1}\) \({\displaystyle a}\)
hyperbola \({\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}\) \({\displaystyle {\sqrt {1+{\frac {b^{2}}{a^{2}}}}}}\) \({\displaystyle {\sqrt {a^{2}+b^{2}}}}\)

Eccentricitas ellipsium


Si longitudinem axis semimaioris ellipseos \({\displaystyle a}\) et longitudinem axis semiminoris ellipseos \({\displaystyle b}\) habemus, definiamus:

nomen symbolus aequatio ex \({\displaystyle a}\) et \({\displaystyle b}\) aequatio ex \({\displaystyle \alpha }\)
eccentricitas angularis \({\displaystyle \alpha }\) \({\displaystyle \arccos \left({\frac {b}{a}}\right)}\) \({\displaystyle \alpha }\)
eccentricitas prima \({\displaystyle e\,}\) \({\displaystyle {\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}\) \({\displaystyle \sin(\alpha )}\)
eccentricitas secunda \({\displaystyle e'\,}\) \({\displaystyle {\sqrt {{\frac {a^{2}}{b^{2}}}-1}}}\) \({\displaystyle \tan(\alpha )}\)
eccentricitas tertia \({\displaystyle e''={\sqrt {m}}}\) \({\displaystyle {\frac {\sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}+b^{2}}}}}\) \({\displaystyle {\frac {\sin(\alpha )}{\sqrt {2-\sin ^{2}(\alpha )}}}}\)

Eccentricitas ut mensura in astronomia


In mechanica caelesti omnis orbita „normalis“ formam sectionis conicae (i.e. ellipseos, parabolae aut hyperbolae) habet. Eccentricitas orbitae a sectione conica, eccentricitas orbitalis dicta, est parametrum magni momenti ad formam eius definiendam, quae mensura adhibetur, qua deviatio orbitae describitur.

Notae


  1. Iohannes Keplerus (1635). Epitome astronomiae Copernicanae  
  2. Isaacus Newtonus (1714). Philosophiae naturalis principia mathematica  
  3. Kraus, L.A. (1844). Kritisch-etymologisches medicinisches Lexikon (Dritte Auflage). Göttingen: Verlag der Deuerlich- und Dieterichschen Buchhandlung.



Categoriae: Geometria | Mechanica caelestis



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