{"loaderData":{"0":null,"0-5":[{"type":"wordpress/fetchEditorialNotes/rejected","meta":{"arg":{"word":"mobius-transformation"},"requestId":"H-_BbpyuG0jsJ-951yzN7","rejectedWithValue":false,"requestStatus":"rejected","aborted":false,"condition":false},"error":{"name":"SyntaxError","message":"Unexpected token < in JSON at position 0","stack":"SyntaxError: Unexpected token < in JSON at position 0\n at JSON.parse ()\n at Response.json (node:internal/deps/undici/undici:1817:21)\n at process.processTicksAndRejections (node:internal/process/task_queues:95:5)\n at async /app/server.js:389:13314"}},{"type":"fetchPageData/fulfilled","payload":{"_id":"home-legacy","story":{"created_at":"2023-07-04T15:14:18.526Z","published_at":"2024-05-02T04:05:00.000Z","content":{"heroModule":[{"cta":[{"url":"https://www.dictionary.com/e/their-vs-there-vs-theyre/","text":"Read more.","target":"_self"}],"article":{"name":"\"Their\" vs. \"There\" vs. \"They're\": What's the Difference?","content":{"topicLabel":"Commonly confused"}},"imageUrl":"https://assets.dictionary.com/home/featured/hero/their-vs-there-vs-theyre-whats-the-difference-20241120.png","description":"Unsure about when to use \"their,\" \"there,\" or \"they're\"? Clear up the confusion with these handy tips.","subjectLabel":"Today"}],"quizModule":[{"cta":[{"url":"https://www.dictionary.com/learn/quizzes","text":"Go to all quizzes","target":"_self"}],"image":"https://assets.dictionary.com/home/daily-quiz.png","articleLink":[{"url":"https://www.dictionary.com/games/quizzes/parenting-buzzwords-quiz","text":"Take the full quiz.","target":"_self"}],"subjectLabel":"Quiz","wordpressSlug":"parenting-buzzwords-quiz","questionNumber":"3"}],"xotdModule":[],"gamesModule":[{"gamesList":[{"name":"Daily Crossword","created_at":"2023-07-04T15:24:23.318Z","published_at":"2024-02-07T05:05:00.000Z","id":336141299,"content":{"link":[{"url":"https://www.dictionary.com/e/crossword/","text":"Daily Crossword","target":"_self"}],"image":{"id":10215805,"alt":"","name":"","focus":"","title":"","source":"","filename":"https://assets.dictionary.com/home/daily-crossword.png","copyright":"","fieldtype":"asset","meta_data":{},"is_external_url":false},"subscriberLink":""},"slug":"daily-crossword","full_slug":"games/daily-crossword","sort_by_date":null,"position":0,"tag_list":[],"is_startpage":false,"parent_id":336140985,"meta_data":null,"group_id":"013f4974-3fad-4e2b-8aa2-c573ef31f263","first_published_at":"2023-07-04T15:32:55.972Z","release_id":null,"lang":"default","path":null,"alternates":[],"default_full_slug":null,"translated_slugs":null,"_stopResolving":true},{"name":"Hurdle","created_at":"2023-07-07T20:42:46.109Z","published_at":"2024-02-06T05:05:00.000Z","id":337835996,"content":{"link":[{"url":"https://play.dictionary.com/games/todays-hurdle","text":"Hurdle","target":"_self"}],"image":{"id":10215808,"alt":"","name":"","focus":"","title":"","source":"","filename":"https://assets.dictionary.com/home/hurdle.png","copyright":"","fieldtype":"asset","meta_data":{},"is_external_url":false},"subscriberLink":"https://www.dictionary.com/games/hurdle"},"slug":"hurdle","full_slug":"games/hurdle","sort_by_date":null,"position":-30,"tag_list":[],"is_startpage":false,"parent_id":336140985,"meta_data":null,"group_id":"3342f4e9-9403-4609-a956-7cd5eab50dc9","first_published_at":"2023-07-10T17:24:55.648Z","release_id":null,"lang":"default","path":null,"alternates":[],"default_full_slug":null,"translated_slugs":null,"_stopResolving":true},{"name":"Word Puzzle","created_at":"2023-07-04T15:31:56.964Z","published_at":"2023-07-10T17:24:59.422Z","id":336150578,"content":{"link":[{"url":"https://www.dictionary.com/games/word-puzzle","text":"Word Puzzle","target":"_self"}],"image":{"id":10215807,"alt":"","name":"","focus":"","title":"","source":"","filename":"https://assets.dictionary.com/home/word-puzzle.png","copyright":"","fieldtype":"asset","meta_data":{},"is_external_url":false}},"slug":"word-puzzle","full_slug":"games/word-puzzle","sort_by_date":null,"position":-20,"tag_list":[],"is_startpage":false,"parent_id":336140985,"meta_data":null,"group_id":"71792320-7229-4ab6-8558-b3af16f5e557","first_published_at":"2023-07-04T15:32:43.933Z","release_id":null,"lang":"default","path":null,"alternates":[],"default_full_slug":null,"translated_slugs":null,"_stopResolving":true},{"name":"Word Wipe","created_at":"2023-07-04T15:28:02.623Z","published_at":"2024-02-06T05:05:00.000Z","id":336141320,"content":{"link":[{"url":"https://play.dictionary.com/games/word-wipe","text":"Word Wipe","target":"_self"}],"image":{"id":10215806,"alt":"","name":"","focus":"","title":"","source":"","filename":"https://assets.dictionary.com/home/word-wipe.png","copyright":"","fieldtype":"asset","meta_data":{},"is_external_url":false},"subscriberLink":"https://www.dictionary.com/games/word-wipe"},"slug":"word-wipe","full_slug":"games/word-wipe","sort_by_date":null,"position":-10,"tag_list":[],"is_startpage":false,"parent_id":336140985,"meta_data":null,"group_id":"3789fc63-90b3-4407-99f4-fa263d1a3e39","first_published_at":"2023-07-04T15:32:50.947Z","release_id":null,"lang":"default","path":null,"alternates":[],"default_full_slug":null,"translated_slugs":null,"_stopResolving":true}],"description":"Calling all cruciverbalists! Sharpen your mind with crosswords and word games, or take a brain break with your favorite classic games.","subjectLabel":"Games"}],"commonModule":[],"ribbonModule":[],"trendingModule":[{"articles":[{"content":{"link":[{"url":"https://www.dictionary.com/e/onto-vs-on-to/","text":"\"Onto\" vs. \"On To\": What's the Difference?"}]}},{"content":{"link":[{"url":"https://www.dictionary.com/e/that-vs-which/","text":"\"That\" vs. \"Which\": When Do You Use Each?"}]}},{"content":{"link":[{"url":"https://www.dictionary.com/e/scrabble/","text":"What Was Scrabble's Original Name?"}]}},{"content":{"link":[{"url":"https://www.thesaurus.com/e/ways-to-say/list-of-vegetables/","text":"This List of Vegetables Is No Small Potatoes"}]}},{"content":{"link":[{"url":"https://www.dictionary.com/e/imbed-vs-embed/","text":"\"Imbed\" vs. \"Embed\": What's the Difference?"}]}}],"subjectLabel":"Trending"}],"newsletterModule":[{"title":"Salutations, logophile!","imageUrl":"https://assets.dictionary.com/home/newsletter.png","description":"Sign up to get everything a word lover could want: word origins, fun facts, and the latest language trends.","subjectLabel":"Newsletter"}],"largeRibbonModule":[],"recirculationModule":[{"articles":[{"content":{"link":[{"url":"https://www.dictionary.com/e/context-clues/","text":"What Are Context Clues?"}],"topicLabel":"🔎"}},{"content":{"link":[{"url":"https://www.dictionary.com/e/meteor-vs-asteroid-vs-comet/","text":"Meteor vs. Asteroid vs. Comet: What's the Difference?"}],"topicLabel":"Space ☄️"}},{"content":{"link":[{"url":"https://www.dictionary.com/e/octopuses-or-octopi/","text":"Should You Say \"Octopuses\" Or \"Octopi\"?"}],"topicLabel":"Plurals"}}],"subjectLabel":"Featured"}]},"slug":"shared-homepage","full_slug":"homepages/shared-homepage","sort_by_date":null,"position":0,"tag_list":[],"is_startpage":false,"parent_id":336140983,"meta_data":null,"group_id":"c7ed375b-9129-448c-aeec-9f52e3fef270","first_published_at":"2023-07-04T18:23:43.845Z","release_id":null,"lang":"default","path":null,"alternates":[],"default_full_slug":null,"translated_slugs":null}},"meta":{"requestId":"y89ic61KD6birQOySyCm2","requestStatus":"fulfilled"}},{"type":"fetchWordsNearby/fulfilled","payload":{"data":{"before":[{"slug":"mobility-housing","entry":"mobility housing"},{"slug":"mobilization","entry":"mobilization"},{"slug":"mobilize","entry":"mobilize"},{"slug":"mobius","entry":"Möbius"},{"slug":"mobius-strip","entry":"Möbius strip"}],"after":[{"slug":"moble","entry":"moble"},{"slug":"moblog","entry":"moblog"},{"slug":"mobo","entry":"MOBO"},{"slug":"mobocracy","entry":"mobocracy"},{"slug":"mobs","entry":"mobs"}]}},"meta":{"arg":{"word":"mobius-transformation"},"requestId":"7AtonvrB_9y41oeSzc6aG","requestStatus":"fulfilled"}},{"type":"luna/fetchExampleSentences/fulfilled","payload":{"data":{"nlp":[],"tdb":[],"gbg":[]}},"meta":{"arg":{"word":"mobius-transformation"},"requestId":"HwM6-88V7ywqEtg6UJcJq","requestStatus":"fulfilled"}},{"type":"fetchDictionaryResults/fulfilled","payload":{"slugs":[{"entries":[{"__typename":"Entry","audioPronunciations":[],"confusableNotes":[],"genderNotes":[],"grammarNotes":[],"headword":{"displayForm":"Möbius transformation","slug":"mobius-transformation"},"idioms":[],"orderKey":99999010,"ordinal":10,"origin":{"originChunks":"[{\"type\": \"cross_reference\", \"children\": [{\"text\": \"M\\u00f6bius strip\", \"type\": \"text\"}], \"slug\": \"mobius-strip\", \"trailingSpace\": true}]"},"partOfSpeechBlock":[{"definitions":[{"antonymsConnection":{"edges":[]},"definitionSetChunks":"[{\"type\": \"def\", \"children\": [{\"text\": \"a map of the complex plane to itself in which a point\", \"type\": \"text\"}, {\"text\": \"z\", \"type\": \"emphasis\"}, {\"text\": \"is mapped to a point\", \"type\": \"text\"}, {\"text\": \"w\", \"type\": \"emphasis\"}, {\"text\": \"by\", \"type\": \"text\"}, {\"text\": \"w\", \"type\": \"emphasis\"}, {\"text\": \"= (\", \"type\": \"text\"}, {\"text\": \"az\", \"type\": \"emphasis\"}, {\"text\": \"+\", \"type\": \"text\"}, {\"text\": \"b\", \"type\": \"emphasis\"}, {\"text\": \")/(\", \"type\": \"text\"}, {\"text\": \"cz\", \"type\": \"emphasis\"}, {\"text\": \"+\", \"type\": \"text\"}, {\"text\": \"d\", \"type\": \"emphasis\"}, {\"text\": \"), where\", \"type\": \"text\"}, {\"text\": \"a, b, c,\", \"type\": \"emphasis\"}, {\"text\": \"and\", \"type\": \"text\"}, {\"text\": \"d\", \"type\": \"emphasis\"}, {\"text\": \"are complex numbers and\", \"type\": \"text\"}, {\"text\": \"ad\", \"type\": \"emphasis\"}, {\"text\": \"\\u2212\", \"type\": \"text\"}, {\"text\": \"bc\", \"type\": \"emphasis\"}, {\"text\": \"does not equal zero.\", \"type\": \"text\"}]}]","examples":[],"ordinal":1,"subdefinitions":[],"synonymsConnection":{"edges":[]}}],"inflectionSets":[],"labelSet":{"labelSetText":"Mathematics."},"ordinal":1,"partOfSpeech":"noun"}],"pronunciationNotes":[],"pronunciationSet":null,"relatedFormSets":[],"sensitiveNotes":[],"spellingNotes":[],"synonymStudy":[],"usageAlerts":[],"usageNotes":[],"variants":[],"verbPhrases":[]}],"url":"mobius-transformation"}]},"meta":{"arg":{"slug":"mobius-transformation"},"requestId":"SghgsbPc4XbudefPaXgXf","requestStatus":"fulfilled"}},{"type":"fetchResultsData/fulfilled","payload":{"data":{"url":"mobius-transformation","displayForm":"Möbius transformation","canonicalSlug":"mobius-transformation","content":[{"source":"luna","entries":[{"id":"u3bb48a285cbafaf0.-3bc55cee.14d9e6f2547.-19d8","type":"normal","entry":"Möbius transformation","entrySyllabified":"Möbius transformation","entryVariants":[],"homograph":null,"pronunciation":null,"predefinitionContent":"","posBlocks":[{"definitions":[{"order":1,"type":"simple","predefinitionContent":"","postdefinitionContent":"","definition":"a map of the complex plane to itself in which a point z is mapped to a point w by w = (az + b)/(cz + d), where a, b, c, and d are complex numbers and ad − bc does not equal zero.","synonyms":"","antonyms":"","cites":[]}],"pos":"noun","posSupplementaryInfo":"Mathematics."}],"origin":"see origin at Möbius strip","originData":null,"supplementaryNotes":[],"variantForms":["Also called bilinear transformation, linear fractional transformation. "],"variantSpellings":[],"referenceData":[],"relatedForms":[],"confusables":[]}]}],"nearbyWordOrdinal":100295,"index-rank-01":["möbius transformation"],"index-rank-02":["mobius transformation"],"index-rank-03":["linear fractional transformation"],"index-rank-04":["mobius transformation's","mobius transformations","mobius transformations'","möbius transformation's","möbius transformations","möbius transformations'"],"index-rank-05":["linear fractional transformation's","linear fractional transformations","linear fractional transformations'"],"index-rank-06":[],"index-rank-07":[],"index-rank-08":[],"index-rank-09":[],"index-rank-10":[],"index-rank-11":[],"index-rank-12":[],"index-rank-13":[],"index-rank-14":[],"index-rank-15":[],"categoryId":"356","tunaSlug":null,"isFeaturedMetaSlug":false}},"meta":{"arg":{"word":"mobius-transformation"},"requestId":"Ixkon61nd9udTFhUiY0ae","requestStatus":"fulfilled"}},{"type":"fetchSpellSuggestions/fulfilled","payload":{"data":{"luna":["Möbius transformation","biotransformation","Lorentz transformation","linear transformation","shear transformation","transformation","Galilean transformation","bilinear transformation","similarity transformation","misinformation"],"tuna":["marked transformation","transformation","heat of transformation","air transportation","misinformation","public transportation","transformations","disinformation","request information","transformational"],"lunaWithSlug":[{"displayForm":"Möbius transformation","slug":"mobius-transformation"},{"displayForm":"biotransformation","slug":"biotransformation"},{"displayForm":"Lorentz transformation","slug":"lorentz-transformation"},{"displayForm":"linear transformation","slug":"linear-transformation"},{"displayForm":"shear transformation","slug":"shear-transformation"},{"displayForm":"transformation","slug":"transformation"},{"displayForm":"Galilean transformation","slug":"galilean-transformation"},{"displayForm":"bilinear transformation","slug":"bilinear-transformation"},{"displayForm":"similarity transformation","slug":"similarity-transformation"},{"displayForm":"misinformation","slug":"misinformation"}],"tunaWithSlug":[{"displayForm":"marked transformation","slug":"marked-transformation"},{"displayForm":"transformation","slug":"transformation"},{"displayForm":"heat of transformation","slug":"heat-of-transformation"},{"displayForm":"air transportation","slug":"air-transportation"},{"displayForm":"misinformation","slug":"misinformation"},{"displayForm":"public transportation","slug":"public-transportation"},{"displayForm":"transformations","slug":"transformations"},{"displayForm":"disinformation","slug":"disinformation"},{"displayForm":"request information","slug":"request-information"},{"displayForm":"transformational","slug":"transformational"}]}},"meta":{"arg":{"word":"mobius-transformation"},"requestId":"329XqETHYbWaMfIoxkjxp","requestStatus":"fulfilled"}}]},"actionData":null,"errors":null}

Möbius transformation

noun

Word History and Origins