[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81808-en":3,"doc-seo-81808-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},81808,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","On Reconstructing a Convex Polygon from Partial Information","The reconstruction problem asks for a convex polygon realization that matches specified geometric features, such as an ordered set of edge lengths or an ordered set of polygon angles. The paper systematically explores scenarios where one or two feature sets are given, identifying which cases were already studied, where new testing algorithms and complexity hardness results are developed, and where open problems remain. It defines several star-shaped-from-the-origin features tied to edge normals, triangle areas, and corner rays.","arXiv :2607 .0 1423v 1 [ cs .CG] 1 Jul 2026  \nOn Reconstructing a Convex Polygon from Partial Information ∗ Alexander Baumann† Therese Biedl‡ Mahmoud Elashmawi† Simon D. Fink§  \nMaria Saumell¶ Andr´e Schulz ‖  \nJuly 3, 2026  \nAbstract  \nThe reconstruction problem asks to construct a (convex) polygon that has a specified set of features, such as an ordered set of edge-lengths or an ordered set of polygon-angles. In this paper, we do a systematic exploration of the reconstruction problem in all scenarios where one or two sets of features have been specified. Some of these scenarios were well-studied already, for some we develop testing-algorithms and/or hardness results, and many give rise to interesting open problems for future study.  \n1 Introduction  \nConsider a polygon P with n ≥ 3 corners. This polygon has a number of features, for example the lengths of the n edges, or the n angles at the corners, which can be computed from the coordinates of the corners. In this work, we study the reverse reconstruction question: Given some features of an unknown polygon P, is there a realization, i.e., a polygon P that has these features? Can we efficiently test whether there isone? If so, can we construct P efficiently and is it unique (at least up to some obvious transformations)?  \nQuestions of this type have been studied previously; we list many results in Appendix A and a few especially relevant ones here. The Minkowski problem asks whether a convex polygon exists that has the origin inside, has a given ordered set of edge lengths ℓ0 ,...,ℓn−1, and a given ordered set of (unit) edge normals n0 , . . . , nn−1, where ni should be the normal of the halfplane supporting the edge of length ℓi. See Huang, Yang, and Zhang [14] for a history and a short proof that this is possible if and only if P01 ℓini is the 0-vector.  \nOur research was inspired by the open question asked by J. O’Rourke at CCCG’25 [1](which in turn was inspired by [14]): Can we reconstruct a polygon if we are given the edge normals and the areas of the triangles spanned by the edges, i.e., the triangles formed by two consecutive corners and the origin? This problem was introduced by Stancu, who called it the L0-Minkowski problem [21] . Stancu showed that there always is a solution as long as no two edge normals are opposite to each other. Stancu’s proof is not constructive, and implies no algorithm for computing the desired polygon.  \nWe were not able to find a practical algorithm for the L0-Minkowski problem, but inspired by these questions, we decided to explore the problem of reconstructing a convex polygon from features more broadly. Specifically, we looked at six features that naturally arise in any convex polygon with the origin inside, and more generally in any polygon P that is star-shaped from the origin, i.e., all segments from the origin to a corner lie strictly inside P. (We simply call this “star-shaped” here.) Say polygon P has corners c0 , . . . , cn−1 in counter-clockwise order. For i ∈ {0, . . . , n − 1}, we define the following, treating all addition among indices modulo n (see also Figure 1):  \n∗ This work was initiated at the European Research Week on Geometric Graphs 2025, held in Chorin (Germany) . We thank the organizers for a fruitful atmosphere. We also thank Wolfgang Mulzer for helpful input.  \n†Institut f¨ur Informatik, Freie Universit¨at Berlin, Germany.  \n‡David R. Cheriton School of Computer Science, University of Waterloo, Canada. Research done while visiting Freie Universit¨at Berlin, Germany. Supported by NSERC.  \n§ Algorithms and Complexity Group, TU Wien, Austria.  \n¶ Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Czech Republic. Supported by the Czech Science Foundation, grant number 23-04949X.  \n‖FernUniversit¨at in Hagen, Germany.  \ntriangle area Ai−1  \ncorner distance di  \nedge ei−1  \nedge distance hi−1  \nci+1  \nedge  \nnormal ni  \no  \nli  \ncorner  \nvector ci−1  \nu","cbCaicBrBnDBLNgY","https://ap.wps.com/l/cbCaicBrBnDBLNgY","pdf",1977489,1,21,"English","en",105,"# Introduction\n## Polygon reconstruction problem and key questions\n## Minkowski and L0-Minkowski background\n## Star-shaped polygon features and definitions","[{\"question\":\"What does the polygon reconstruction problem ask for in this paper?\",\"answer\":\"It asks whether there exists a convex polygon whose realization matches a specified set of features (e.g., ordered edge lengths or polygon angles), and whether such a realization can be tested for efficiently and constructed efficiently.\"},{\"question\":\"How are the edge normals and triangle areas connected to reconstruction?\",\"answer\":\"The work is motivated by problems asking to reconstruct polygons when given edge normals and triangle areas formed by consecutive edges and the origin, and it builds broader reconstruction studies around these feature relationships.\"},{\"question\":\"Which features does the paper focus on for star-shaped polygons with the origin inside?\",\"answer\":\"It considers six types of features per vertex/edge, including edge length, triangle area spanned with the origin, unit edge normal, edge distance from the origin, unit corner vector (corner ray direction), and corner distance from the origin.\"}]",1784176285,53,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"on-reconstructing-a-convex-polygon-from-partial-information","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/on-reconstructing-a-convex-polygon-from-partial-information/81808/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the polygon reconstruction problem ask for in this paper?","Question",{"text":75,"@type":76},"It asks whether there exists a convex polygon whose realization matches a specified set of features (e.g., ordered edge lengths or polygon angles), and whether such a realization can be tested for efficiently and constructed efficiently.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the edge normals and triangle areas connected to reconstruction?",{"text":80,"@type":76},"The work is motivated by problems asking to reconstruct polygons when given edge normals and triangle areas formed by consecutive edges and the origin, and it builds broader reconstruction studies around these feature relationships.",{"name":82,"@type":73,"acceptedAnswer":83},"Which features does the paper focus on for star-shaped polygons with the origin inside?",{"text":84,"@type":76},"It considers six types of features per vertex/edge, including edge length, triangle area spanned with the origin, unit edge normal, edge distance from the origin, unit corner vector (corner ray direction), and corner distance from the origin.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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