Pengguna:Dedhert.Jr/Uji halaman 01/22: Perbedaan antara revisi
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Baris 38:
Adapun [[pertidaksamaan Ptolemaus]], sebuah sifat lain yang melibatkan jarak Euklides di antara empat titik {{math|1=''p''}}, {{math|1=''q''}}, {{math|1=''r''}}, dan {{math|1=''s''}}, yang menyatakan bahwa
<math display=block>d(p,q)\cdot d(r,s)+d(q,r)\cdot d(p,s)\ge d(p,r)\cdot d(q,s).</math>
Untuk titik-titik di bidang tersebut, rumus di atas dapat dikatakan bahwa untuk setiap [[segi empat]], perkalian antara sisi yang berhadapan dari jumlah segi empat lebih besar dari perkalian dari sisi diagonalnya. Akan tetapi, pertidaksamaan Ptolemaus lebih umumnya berlaku untuk titik-titik yang ada di ruang Euklides untuk setiap dimensi, tidak peduli bentuk susunannya.<ref>{{citation|title=Rays, Waves, and Scattering: Topics in Classical Mathematical Physics|series=Princeton Series in Applied Mathematics|first=John A.|last=Adam|publisher=Princeton University Press|year=2017|isbn=978-1-4008-8540-4|pages=26–27|chapter-url=https://books.google.com/books?id=DnygDgAAQBAJ&pg=PA26|chapter=Chapter 2. Introduction to the "Physics" of Rays|doi=10.1515/9781400885404-004}}</ref> Untuk titik-titik di ruang metrik yang bukan ruang Euklides, pertidaksamaan ini tidak berlaku benar. [[Geometri jarak]] Euklides mempelajari sifat-sifat dari jarak Euklides seperti pertidaksamaan Ptolemaus,
== Jarak Euklides kuadrat ==
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Baris 52:
<math display=block>d^2(p,q) = (p_1 - q_1)^2 + (p_2 - q_2)^2+\cdots+(p_i - q_i)^2+\cdots+(p_n - q_n)^2.</math>
Selain penerapannya dalam membandingkan jarak, jarak Euklides kuadrat merupakan alat penting di bidang [[statistika]]
| last = Csiszár | first = I. | author-link = Imre Csiszár
| doi = 10.1214/aop/1176996454
Baris 61:
| title = {{mvar|I}}-divergence geometry of probability distributions and minimization problems
| volume = 3
| year = 1975}}</ref> The addition of squared distances to each other, as is done in least squares fitting, corresponds to an operation on (unsquared) distances called [[Pythagorean addition]].<ref>{{citation |author=Moler, Cleve and Donald Morrison |title=Replacing Square Roots by Pythagorean Sums |journal=IBM Journal of Research and Development |volume=27 |issue=6 |pages=577–581 |year=1983 |url=http://www.research.ibm.com/journal/rd/276/ibmrd2706P.pdf |doi=10.1147/rd.276.0577 | citeseerx = 10.1.1.90.5651 }}</ref>
== Perumuman ==
Dalam cabang matematika lebih lanjut, saat jarak Euklides dipandang sebagai [[ruang vektor]], jaraknya diiringi dengan [[Norma (matematika)|norma]] yang disebut sebagai [[Norma (matematika)#Norma Euklides|norma Euklides]], didefinisikan sebagai jarak dari masing-masing vektor yang berawal dari [[Titik asal (matematika)|titik asal]]. One of the important properties of this norm, relative to other norms, is that it remains unchanged under arbitrary rotations of space around the origin.<ref>{{citation|title=Relativistic Celestial Mechanics of the Solar System|first1=Sergei|last1=Kopeikin|first2=Michael|last2=Efroimsky|first3=George|last3=Kaplan|publisher=John Wiley & Sons|year=2011|isbn=978-3-527-63457-6|page=106|url=https://books.google.com/books?id=uN5_DQWSR14C&pg=PA106}}</ref> Menurut [[teorema Dvoretzky]], setiap [[ruang vektor bernorma]] dimensi terhingga mempunyai subruang dimensi tinggi dengan norma yang kira-kira dekat dengan norma Euklides, satu-satunya norma yang ada di sifat tersebut.<ref>{{citation|last=Matoušek|first=Jiří|author-link=Jiří Matoušek (mathematician)|isbn=978-0-387-95373-1|page=349|publisher=Springer|series=[[Graduate Texts in Mathematics]]|title=Lectures on Discrete Geometry|url=https://books.google.com/books?id=K0fhBwAAQBAJ&pg=PA349|year=2002}}</ref> Jarak Euklides dapat diperluas ke ruang vektor berdimensi tak terhingga sebagai [[Norma L2|norma L<sup>2</sup>]].<ref>{{citation|title=Linear and Nonlinear Functional Analysis with Applications|first=Philippe G.|last=Ciarlet|publisher=Society for Industrial and Applied Mathematics|year=2013|isbn=978-1-61197-258-0|page=173|url=https://books.google.com/books?id=AUlWAQAAQBAJ&pg=PA173}}</ref> Jarak Euklides memberikan ruang Euklides suatu struktur [[ruang topologis]], [[topologi Euklides]], dengan [[Bola pejal (matematika)|bola pejal terbuka]] (subhimpunan dari titik yang lebih sedikit daripada jarak yang berawal dari titik yang diketahui) sebagai [[Lingkungan (matematika)|lingkungannya]].<ref>{{citation|title=General Topology: An Introduction|publisher=De Gruyter|first=Tom|last=Richmond|year=2020|isbn=978-3-11-068657-9|page=32|url=https://books.google.com/books?id=jPgdEAAAQBAJ&pg=PA32}}</ref>
Jarak umum lainnya dalam ruang Euklides beserta ruang vektor berdimensi rendah melibatkan:<ref>{{citation|last=Klamroth|first=Kathrin|author-link=Kathrin Klamroth|contribution=Section 1.1: Norms and Metrics|doi=10.1007/0-387-22707-5_1|pages=4–6|publisher=Springer|series=Springer Series in Operations Research|title=Single-Facility Location Problems with Barriers|year=2002}}</ref>
* [[Jarak Chebyshev]], yang menghitung jarak yang hanya dengan mengasumsi bahwa dimensi yang sangat signifikan adalah penting.
* [[Jarak Manhattan]], yang menghitung jarak hanya dengan mengikuti arah pada sumbu.
* [[Jarak Minkowski]], suatu perumuman yang menyatukan jarak Euklides, jarak Manhattan, dan jarak Chebyshev.
For points on surfaces in three dimensions, the Euclidean distance should be distinguished from the [[geodesic]] distance, the length of a shortest curve that belongs to the surface. In particular, for measuring great-circle distances on the earth or other spherical or near-spherical surfaces, distances that have been used include the [[haversine distance]] giving great-circle distances between two points on a sphere from their longitudes and latitudes, and [[Vincenty's formulae]] also known as "Vincent distance" for distance on a spheroid.<ref>{{citation|title=Computing in Geographic Information Systems|first=Narayan|last=Panigrahi|publisher=CRC Press|year=2014|isbn=978-1-4822-2314-9|contribution=12.2.4 Haversine Formula and 12.2.5 Vincenty's Formula|pages=212–214|url=https://books.google.com/books?id=kjj6AwAAQBAJ&pg=PA212}}</ref>
== Asal-muasal ==
Jarak Euklides adalah jarak di dalam [[ruang Euklides]] yang dinamai dari seorang matematikawan Yunani kuno yang bernama [[Euklides]]. Konsep tersebut dijelaskan dalam bukunya, [[Euclid's Elements|''Elements'']], yang menjadi buku cetak standar selama bertahun.<ref>{{citation|title=Visualization for Information Retrieval|first=Jin|last=Zhang|publisher=Springer|year=2007|isbn=978-3-540-75148-9}}</ref> Konsep tentang [[panjang]] dan [[jarak]] yang tersebar luas di seluruh budaya, kemungkinan berawal dari <u>earliest surviving "protoliterate" bureaucratic documents from [[Sumer]] in the fourth millennium BC (far before Euclid)</u>,<ref>{{citation|last=Høyrup|first=Jens|author-link=Jens Høyrup|editor1-last=Jones|editor1-first=Alexander|editor2-last=Taub|editor2-first=Liba|editor2-link=Liba Taub|contribution=Mesopotamian mathematics|contribution-url=https://akira.ruc.dk/~jensh/Publications/2018%7Bj%7D_Mesopotamian%20Mathematics_S.pdf|pages=58–72|publisher=Cambridge University Press|title=The Cambridge History of Science, Volume 1: Ancient Science|year=2018}}</ref> and have been hypothesized to develop in children earlier than the related concepts of speed and time.<ref>{{citation|last1=Acredolo|first1=Curt|last2=Schmid|first2=Jeannine|doi=10.1037/0012-1649.17.4.490|issue=4|journal=[[Developmental Psychology (journal)|Developmental Psychology]]|pages=490–493|title=The understanding of relative speeds, distances, and durations of movement|volume=17|year=1981}}</ref> But the notion of a distance, as a number defined from two points, does not actually appear in Euclid's ''Elements''. Instead, Euclid approaches this concept implicitly, through the [[Congruence (geometry)|congruence]] of line segments, through the comparison of lengths of line segments, and through the concept of [[Proportionality (mathematics)|proportionality]].<ref>{{citation|last=Henderson|first=David W.|author-link=David W. Henderson|journal=[[Bulletin of the American Mathematical Society]]|pages=563–571|title=Review of ''Geometry: Euclid and Beyond'' by Robin Hartshorne|url=https://www.ams.org/journals/bull/2002-39-04/S0273-0979-02-00949-7|volume=39|year=2002|doi=10.1090/S0273-0979-02-00949-7|doi-access=free}}</ref>
The [[Pythagorean theorem]] is also ancient, but it could only take its central role in the measurement of distances after the invention of [[Cartesian coordinates]] by [[René Descartes]] in 1637. The distance formula itself was first published in 1731 by [[Alexis Clairaut]].<ref>{{citation|last=Maor|first=Eli|author-link=Eli Maor|isbn=978-0-691-19688-6|pages=133–134|publisher=Princeton University Press|title=The Pythagorean Theorem: A 4,000-Year History|url=https://books.google.com/books?id=XuWZDwAAQBAJ&pg=PA133|year=2019}}</ref> Because of this formula, Euclidean distance is also sometimes called Pythagorean distance.<ref>{{citation|last1=Rankin|first1=William C.|last2=Markley|first2=Robert P.|last3=Evans|first3=Selby H.|date=March 1970|doi=10.3758/bf03210143|issue=2|journal=[[Perception & Psychophysics]]|pages=103–107|title=Pythagorean distance and the judged similarity of schematic stimuli|volume=7|s2cid=144797925|doi-access=free}}</ref> Although accurate measurements of long distances on the earth's surface, which are not Euclidean, had again been studied in many cultures since ancient times (see [[history of geodesy]]), the idea that Euclidean distance might not be the only way of measuring distances between points in mathematical spaces came even later, with the 19th-century formulation of [[non-Euclidean geometry]].<ref>{{citation|last=Milnor|first=John|author-link=John Milnor|doi=10.1090/S0273-0979-1982-14958-8|issue=1|journal=[[Bulletin of the American Mathematical Society]]|mr=634431|pages=9–24|title=Hyperbolic geometry: the first 150 years|volume=6|year=1982|doi-access=free}}</ref> The definition of the Euclidean norm and Euclidean distance for geometries of more than three dimensions also first appeared in the 19th century, in the work of [[Augustin-Louis Cauchy]].<ref>{{citation|title=Foundations of Hyperbolic Manifolds|volume=149|series=[[Graduate Texts in Mathematics]]|first=John G.|last=Ratcliffe|edition=3rd|publisher=Springer|year=2019|isbn=978-3-030-31597-9|page=32|url=https://books.google.com/books?id=yMO4DwAAQBAJ&pg=PA32}}</ref>
== Referensi ==
{{Reflist|colwidth=30}}
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