[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-272835-105":59,"doc-detail-272835-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","affine-transformation-rotation-and-translation-object-vs-coordinate-axes","Affine Transformation - Rotation and Translation: Object vs Coordinate Axes","","This document provides an overview of affine transformations, focusing on their application in aligning 3D images from different time points or modalities, particularly in the context of oral and maxillofacial radiology. It details two primary methods of transformation: Method 1, which involves transforming the object itself through rotation, translation, and comparison; and Method 2, which focuses on transforming the coordinate system. The document includes a definition of affine transformation from Wikipedia, highlighting its properties of preserving dimension of affine subspaces and the ratio of lengths of parallel line segments. It specifies that in practical applications like image superposition for dental implants, only rotation and translation are utilized, with scaling being excluded. The method of using DICOM tag positioning information and Excel sheets for calculation is explained, with a crucial note on the reversed direction of rotation when switching between object transformation and coordinate axis rotation, and between right-handed and left-handed systems.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/affine-transformation-rotation-and-translation-object-vs-coordinate-axes/272835/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/affine-transformation-rotation-and-translation-object-vs-coordinate-axes/272835.png","ImageObject",300,407,{"name":92,"@type":93},"Graffin","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-28","2026-09-15",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is an affine transformation?","Question",{"text":113,"@type":114},"An affine transformation is a function that maps an affine space onto itself, preserving the dimension of affine subspaces and the ratio of lengths of parallel line segments. This means points are mapped to points, lines to lines, and planes to planes.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What are the common applications of affine transformations in this context?",{"text":118,"@type":114},"Affine transformations are used for superimposing 3D images taken at different times or by different methods, such as superimposing images on dental implants and comparing images before and after treatment.",{"name":120,"@type":111,"acceptedAnswer":121},"Which specific affine transformations are used in the described application, and why is scaling excluded?",{"text":122,"@type":114},"In the context of superimposing 3D images, only rotation and translation are used. Scaling is omitted to maintain the original proportions of the objects or structures.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},272835,1789467841,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":56,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":144,"preview_page_count":4},3573418547284,"https://eur-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","About affine transformation  \nDivision of Oral and Maxillofacial Radiology, Niigata University Graduate School of Medical and Dental Sciences  \nH.Nishiyama  \nFor example…  \nFirst time.  \nFor example, an image before treatment.  \nSecond time. For example, the image after treatment.  \nMethod 1. A method to transform an object.  \n[1] Rotate an object in 3D space…  \nMethod 1. A method to transform an object.  \n[2]  \n[2] Translate, and    \n[3] Compare (overlap)  \nMethod 2. A method to transform the coordinates.  \nNote! Opposite direction of rotation  \n~~ ~~ [1]  \n[1] Rotate the coordinates in 3D space    \nWhat is affine transformation?  \n(Difficult mathematical expression)  \n• On Wikipedia…  \n– [https://en.wikipedia.org/wiki/Affine_transforma](https://en.wikipedia.org/wiki/Affine_transforma)tion  \n– It is a function mapping an affine space onto itself that preserves the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and also preserves the ratio of the lengths of parallel line segments.  \nWhat do you use it for?  \n• Superposition of 3D images taken at different times (or by another method) .  \n• Superimposing images on implants.  \n• Of the affine transformations, only rotation and translation are used.  \n• No scaling is used.  \nHow to use?  \n• Uses the 3D position information embedded inside the DICOM tag.  \n• You can easily calculate (convert) using an Excel sheet.  \n• Note that the direction of rotation is reversed by \"Right-handed / Left-handed\"and \"Rotate objects / Rotate coordinate axes\"! !","cbCaigLZYOPHqxmx","https://ap.wps.com/l/cbCaigLZYOPHqxmx","pdf",591088,"English","# About Affine Transformation\n## Method 1. A method to transform an object.\n### [1] Rotate an object in 3D space…\n### Method 1. A method to transform an object.\n### [2]\n### [2] Translate, and\n### [3] Compare (overlap)\n## Method 2. A method to transform the coordinates.\n### Note! Opposite direction of rotation\n### [1] Rotate the coordinates in 3D space\n## What is affine transformation?\n## What do you use it for?\n## How to use?","[{\"question\":\"What is an affine transformation?\",\"answer\":\"An affine transformation is a function that maps an affine space onto itself, preserving the dimension of affine subspaces and the ratio of lengths of parallel line segments. This means points are mapped to points, lines to lines, and planes to planes.\"},{\"question\":\"What are the common applications of affine transformations in this context?\",\"answer\":\"Affine transformations are used for superimposing 3D images taken at different times or by different methods, such as superimposing images on dental implants and comparing images before and after treatment.\"},{\"question\":\"Which specific affine transformations are used in the described application, and why is scaling excluded?\",\"answer\":\"In the context of superimposing 3D images, only rotation and translation are used. Scaling is omitted to maintain the original proportions of the objects or structures.\"}]","Affine Transformation - Rotation and Translation: Object vs Coordinate Axes | PDF",48]