Understanding the Deltoid Phenomenon in the Perspective 3-Point (P3P) Problem

Understanding the Deltoid Phenomenon in the Perspective 3-Point (P3P) Problem | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Understanding the Deltoid Phenomenon in the Perspective 3-Point (P3P) Problem Michael Rieck This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4391421/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 31 Jan, 2025 Read the published version in Journal of Mathematical Imaging and Vision → Version 1 posted 9 You are reading this latest preprint version Abstract Concerning the Perspective 3-Point (P3P) Problem, Grunert's system of three quadratic equations has a repeated solution if and only if the cubic polynomial introduced by Finsterwalder has a repeated root. This polynomial is here shown to be obtainable from a particularly simple cubic polynomial with complex coefficients via a simple Mobius transformation. This provides surprising geometric insight into the P3P problem. In particular, (1) the discriminant of Finsterwalder's polynomial can be written using the formula for the standard deltoid curve, and (2) this discriminant, when regarded as a function of camera position, vanishes on a surface that approaches a deltoid shape when the camera is moved infinitely far from the control points in a direction perpendicular to the control points plane (the ''limit case"). These two facts have been previously reported, but obscure reasoning was required to establish them. In contrast, the present article uses the newly discovered cubic polynomial to easily produce the first fact, which then provides a basis for better understanding the second fact. Also presented are quartic polynomials whose real roots are the P3P solution point coordinates. A detailed geometric description of the P3P solution points in the ''limit case" is also supplied. P3P perspective pose camera Grunert system deltoid orthocentric system Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 31 Jan, 2025 Read the published version in Journal of Mathematical Imaging and Vision → Version 1 posted Editorial decision: Revision requested 14 Jul, 2024 Reviews received at journal 08 Jul, 2024 Reviews received at journal 04 Jun, 2024 Reviewers agreed at journal 30 May, 2024 Reviewers agreed at journal 29 May, 2024 Reviewers invited by journal 29 May, 2024 Editor assigned by journal 11 May, 2024 Submission checks completed at journal 10 May, 2024 First submitted to journal 08 May, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4391421","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":304164886,"identity":"239ff8b3-7208-4eb1-894f-c32fd4ca05b0","order_by":0,"name":"Michael Rieck","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA90lEQVRIie3NsYrCQBCA4VkWttqQdoKiT3CwchCV+DCCRUo7sbDINbHZB8jBce9gY60MpIrmFUxjZaGN2HiYyFlYJLG02L8Ydpb9WACT6R3jIPLZKw7Agmk+ZXEtagn+k+QVAg+Sx77CF8jHnO93lylCd87Xp+9fb2xrWsFxQqXEJdHt6AShSWLkLJZ+P9qEQxZtq4gUaIUIyKXLsiUpSKXiVlhNnOtfQewzy35ItVP7yK81pGEF918EWwSk1EYDZ5VEuI1mjBK5+HSi2FedJFZrvfXLSUp75zAbtNCm7KRnnmolo2x3mXil5JF82la1700mk8lU2Q1zU0vb3rPS+AAAAABJRU5ErkJggg==","orcid":"","institution":"Drake University","correspondingAuthor":true,"prefix":"","firstName":"Michael","middleName":"","lastName":"Rieck","suffix":""}],"badges":[],"createdAt":"2024-05-08 21:08:19","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4391421/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4391421/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s10851-024-01228-4","type":"published","date":"2025-01-31T15:57:35+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":75351294,"identity":"8a766238-4dbc-4234-a91b-3717f6339afa","added_by":"auto","created_at":"2025-02-03 16:09:04","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":385562,"visible":true,"origin":"","legend":"","description":"","filename":"JMIVRieck2024.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4391421/v1_covered_58910e63-3b53-4fa7-9881-c8cd33a674d9.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Understanding the Deltoid Phenomenon in the Perspective 3-Point (P3P) Problem","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-mathematical-imaging-and-vision","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jmiv","sideBox":"Learn more about [Journal of Mathematical Imaging and Vision](http://link.springer.com/journal/10851)","snPcode":"10851","submissionUrl":"https://submission.nature.com/new-submission/10851/3","title":"Journal of Mathematical Imaging and Vision","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"P3P, perspective, pose, camera, Grunert system, deltoid, orthocentric system","lastPublishedDoi":"10.21203/rs.3.rs-4391421/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4391421/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Concerning the Perspective 3-Point (P3P) Problem, Grunert's system of three quadratic equations has a repeated solution if and only if the cubic polynomial introduced by Finsterwalder has a repeated root. 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