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[[File:Fokalkegelschnitte-def-papa.svg|thumb|upright=1.4|Focal conics: two parabolas<br> === Two parabolas === ...

{{Short description|Plane region bounded by three parabolas}} ...of the outer parabola is equal to the sum of the arc lengths of the inner parabolas. ...

...r advantage of this completion is that the plane geometry of the completed parabolas/lines is [[isomorphic]] to the geometry of the [[plane section]]s of a [[cy ...r plane and hence a circle/ellipse without point <math>(0,1,a)</math>. The parabolas/line <math>z=ax^2+a</math> are mapped onto (horizontal) circles. ...

...centroid and the [[orthocentre]] of the reference triangle; and the Artzt parabolas, which are [[parabola]]s touching two [[extended side|sidelines]] of the re ===Triangle parabolas=== ...

...set of points is <math> (\R\cup{\infty})\times\R</math>. This geometry of parabolas is called the classical Laguerre plane (Originally it was designed as the g ...

The above family of parabolas are enveloped by the parabola <math>4By=-ABx^2</math>, therefore this envel ...

...es on two confocal parabolas which intersect orthogonally (see [[#Confocal parabolas|below]]). == Confocal parabolas == ...

...e-12.svg|thumb|upright=1.2|Peano surface and its level curves for level 0 (parabolas, green and purple)]] ...math> on the surface that corresponds to the intersection point of the two parabolas, the surface has a [[saddle point]].<ref name="Fischer"/> The surface itsel ...

...|eccentricity]] {{nobr| {{math|''e'' : 0 ≤ ''e'' < 1}} )}} to open orbits (parabolas and hyperbolas, {{nobr|( {{math|''e'' ≥ 1}} ),}} with no singularities and ...

...ogram".<ref name="Eddy"/><ref>{{cite journal |last1=Sharp, J. |title=Artzt parabolas of a triangle |journal=The Mathematical Gazette |date=2015 |volume=99 |issu ...

.... Downs: ''Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas''. Courier Dover 2003, {{ISBN|978-0-486-42876-5}}, pp.&nbsp; ...

...ectively as '''higher parabolae''' and '''higher hyperbolae''' (or "higher parabolas" and "higher hyperbolas"). [[Pierre de Fermat]] also computed these areas ( ...ley |year=1998 |chapter=Section 3.4: "Cavalieri Calculates Areas of Higher Parabolas" |chapter-url=https://books.google.com/books?id=ubFg7DI83JoC&pg=PA125 |page ...

...he dispersion relations show conics of the free-electron energy dispersion parabolas for all possible reciprocal lattice vectors. This results in a very complic ...

...h ''D'' and ''E'' both equal to zero (with certain exceptions—for example, parabolas). The principal tool in this process is "completing the square."<ref>{{har ...

...2009|title=The most general planar transformations that map parabolas into parabolas|url=https://projecteuclid.org/euclid.involve/1513799118|journal=Involve: A ...

...e centroid and the [[orthocentre]] of the reference triangle and the Artzt parabolas which are [[parabola]]s touching two sidelines of the reference triangle at ArtztParabolas.png|Artz parabolas ...

...t al.''.<ref>Dan Joseph, Gregory Hartman & Caleb Gibson (2011) Generalized Parabolas, ''The College Mathematics Journal'', 42:4, 275-282, DOI: 10.4169/college.m ...