# Cartesian oval > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Cartesian_oval > Markdown URL: https://mediated.wiki/source/Cartesian_oval.md > Source: https://en.wikipedia.org/wiki/Cartesian_oval > Source revision: 1292957362 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{short description|Class of geometric plane curves}} thumb|138px|Example of Cartesian ovals. In [geometry](/source/geometry), a '''Cartesian [oval](/source/oval)''' is a [plane curve](/source/plane_curve) consisting of points that have the same [linear combination](/source/linear_combination) of distances from two fixed points ([foci](/source/Focus_(geometry))). These curves are named after French mathematician [René Descartes](/source/Ren%C3%A9_Descartes), who used them in [optics](/source/optics). ==Definition== Let {{mvar|P}} and {{mvar|Q}} be fixed points in the plane, and let {{math|d(''P'', ''S'')}} and {{math|d(''Q'', ''S'')}} denote the [Euclidean distance](/source/Euclidean_distance)s from these points to a third variable point {{math|''S''}}. Let {{math|''m''}} and {{math|''a''}} be arbitrary [real number](/source/real_number)s. Then the Cartesian oval is the [locus](/source/locus_(mathematics)) of points ''S'' satisfying {{math|1=d(''P'', ''S'') + ''m'' d(''Q'', ''S'') = ''a''}}. The two ovals formed by the four equations {{math|1=d(''P'', ''S'') + ''m'' d(''Q'', ''S'') = ± ''a''}} and {{math|1=d(''P'', ''S'') − ''m'' d(''Q'', ''S'') = ± ''a''}} are closely related; together they form a [quartic plane curve](/source/quartic_plane_curve) called the '''ovals of Descartes'''.{{MacTutor|class= Curves|id= Cartesian|title=Cartesian Oval}} ==Special cases== In the equation {{math|1=d(''P'', ''S'') + ''m'' d(''Q'', ''S'') = ''a''}}, when {{math|1=''m'' = 1}} and {{math|''a'' > d(''P'', ''Q'')}} the resulting shape is an [ellipse](/source/ellipse). In the [limiting case](/source/limiting_case_(mathematics)) in which ''P'' and ''Q'' coincide, the ellipse becomes a [circle](/source/circle). When m = a/\!\operatorname{d}(P, Q) it is a [limaçon](/source/lima%C3%A7on) of Pascal. If m = -1 and 0 < a < \operatorname{d}(P, Q) the equation gives a branch of a [hyperbola](/source/hyperbola) and thus is not a closed oval. ==Polynomial equation== The [set](/source/set_(mathematics)) of points {{math|(''x'', ''y'')}} satisfying the [quartic polynomial equation](/source/quartic_equation) : \left[(1-m^2)(x^2 + y^2) + 2m^2 cx + a^2 - m^2 c^2\right]^2 = 4a^2 (x^2+y^2) where {{math|''c''}} is the distance \text{d}(P,Q) between the two fixed [foci](/source/focus_(geometry)) {{math|1=''P'' = (0, 0)}} and {{math|1=''Q'' = (''c'', 0)}}, forms two ovals, the sets of points satisfying two of the following four equations : \operatorname{d}(P, S) \pm m \operatorname{d}(Q, S) = a \, : \operatorname{d}(P, S) \pm m \operatorname{d}(Q, S) = -a \, that have real solutions. The two ovals are generally [disjoint](/source/disjoint_sets), except in the case that {{mvar|P}} or {{mvar|Q}} belongs to them. At least one of the two perpendiculars to {{math|''PQ''}} through points {{mvar|''P''}} and {{mvar|''Q''}} cuts this quartic curve in four real points; it follows from this that they are necessarily nested, with at least one of the two points {{mvar|P}} and {{mvar|Q}} contained in the interiors of both of them.{{citation|title=An elementary treatise on the differential calculus founded on the method of rates or fluxions|first1=John Minot|last1=Rice|first2=William Woolsey|last2=Johnson|edition=4th|publisher=J. Wiley|year=1888|pages=295–299|url=https://books.google.com/books?id=KuM2AAAAMAAJ&pg=PA295}}. For a different parametrization and resulting quartic, see Lawrence.{{citation|last=Lawrence|first=J. Dennis|title=A Catalog of Special Plane Curves|publisher=Dover|year=1972|pages=[https://archive.org/details/catalogofspecial00lawr/page/155 155–157]|isbn=0-486-60288-5|url-access=registration|url=https://archive.org/details/catalogofspecial00lawr/page/155}}. ==Applications in optics== As Descartes discovered, Cartesian ovals may be used in [lens](/source/Lens_(optics)) design. By choosing the ratio of distances from {{mvar|P}} and {{mvar|Q}} to match the ratio of [sine](/source/sine)s in [Snell's law](/source/Snell's_law), and using the [surface of revolution](/source/surface_of_revolution) of one of these ovals, it is possible to design a so-called [aplanatic lens](/source/aplanatic_lens), that has no [spherical aberration](/source/spherical_aberration).{{citation|title=Lenses and waves: Christiaan Huygens and the mathematical science of optics in the seventeenth century|volume=9|series=Archimedes, New studies in the history and philosophy of science and technology|first=Fokko Jan|last=Dijksterhuis|publisher=Springer-Verlag|year=2004|isbn=978-1-4020-2697-3|pages=13–14|url=https://books.google.com/books?id=cPFevyomPUIC&pg=PA13}}. Additionally, if a spherical wavefront is refracted through a spherical lens, or reflected from a concave spherical surface, the refracted or reflected wavefront takes on the shape of a Cartesian oval. The [caustic](/source/Caustic_(optics)) formed by spherical aberration in this case may therefore be described as the [evolute](/source/evolute) of a Cartesian oval.{{citation|contribution=Chapter XVI. Contour of the refracted wave-front. Caustics|url=https://books.google.com/books?id=36cOAAAAYAAJ&pg=PA312|pages=312–327|title=Optics, a manual for students|first=Archibald Stanley|last=Percival|publisher=Macmillan|year=1899}}. ==History== The ovals of Descartes were first studied by René Descartes in 1637, in connection with their applications in optics. These curves were also studied by [Newton](/source/Isaac_Newton) beginning in 1664. One method of drawing certain specific Cartesian ovals, already used by Descartes, is analogous to a standard construction of an [ellipse](/source/ellipse) by a pinned thread. If one stretches a thread from a pin at one [focus](/source/focus_(geometry)) to wrap around a pin at a second focus, and ties the free end of the thread to a pen, the path taken by the pen, when the thread is stretched tight, forms a Cartesian oval with a 2:1 ratio between the distances from the two foci. However, Newton rejected such constructions as insufficiently [rigorous](/source/Rigour).{{citation|pages=49 & 104|title=Isaac Newton on mathematical certainty and method|volume=4|series=Transformations: Studies in the History of Science and Technology|publisher=MIT Press|first=Niccolò|last=Guicciardini|year=2009|isbn=978-0-262-01317-8}}. He defined the oval as the solution to a [differential equation](/source/differential_equation), constructed its [subnormals](/source/Subtangent), and again investigated its optical properties.{{citation|pages=139, 495, & 551|title=The Mathematical Papers of Isaac Newton, Vol. 3|first=Derek Thomas|last=Whiteside|publisher=Cambridge University Press|year=2008|isbn=978-0-521-04581-0}}. The French mathematician [Michel Chasles](/source/Michel_Chasles) discovered in the 19th century that, if a Cartesian oval is defined by two points {{mvar|P}} and {{mvar|Q}}, then there is in general a third point {{mvar|R}} on the same line such that the same oval is also defined by any pair of these three points. [James Clerk Maxwell](/source/James_Clerk_Maxwell) rediscovered these curves, generalized them to curves defined by keeping constant the weighted sum of distances from three or more foci, and wrote a paper titled ''Observations on Circumscribed Figures Having a Plurality of Foci, and Radii of Various Proportions''. An account of his results, titled ''On the description of oval curves, and those having a plurality of foci'', was written by [J.D. Forbes](/source/James_David_Forbes) and presented to the [Royal Society of Edinburgh](/source/Royal_Society_of_Edinburgh) in 1846, when Maxwell was at the young age of 14 (almost 15).{{citation|title=The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications|first=Martin|last=Gardner|authorlink=Martin Gardner|publisher=Springer-Verlag|year=2007|isbn=978-0-387-25827-0|pages=46–49}}.The Scientific Letters and Papers of James Clerk Maxwell, Edited by P.M. Harman, Volume I, 1846–1862, Cambridge University Press, pg. 35[https://mathshistory.st-andrews.ac.uk/Biographies/Maxwell/ MacTutor History of Mathematics - Biographies - Maxwell] ==See also== *[Cassini oval](/source/Cassini_oval) *[Two-center bipolar coordinates](/source/Two-center_bipolar_coordinates) ==References== {{reflist}} ==External links== {{commons category|Cartesian oval}} *{{mathworld|title=Cartesian Ovals|urlname=CartesianOvals}} *[https://books.google.com/books/about/An_elementary_treatise_on_the_differenti.html?id=PWO2yvV13c0C Benjamin Williamson, An Elementary Treatise on the Differential Calculus, Containing the Theory of Plane Curves (1884)] Category:Quartic curves --- Adapted from the Wikipedia article [Cartesian oval](https://en.wikipedia.org/wiki/Cartesian_oval) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Cartesian_oval?action=history)). 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