{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:10:30Z","timestamp":1760029830785,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,27]],"date-time":"2025-02-27T00:00:00Z","timestamp":1740614400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSTC","award":["112-221-E-031-003"],"award-info":[{"award-number":["112-221-E-031-003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this study, we investigated circle design in a bubble chart. The bubble chart that we designed has a unit circle at the center and is surrounded by a series of layered circle rings. In order to add the next layer of circles, there is always a silhouette circle that is tangential to all circles in the newest layer. The extension of the circle is partitioned into two classes: one in which each layer has the same number of circles with the same radii, and another in which the new layer has circles with different radii. We solve this problem geometrically and\/or algebraically if the problem is simple and present a heuristic algorithm for solving more complex problems.<\/jats:p>","DOI":"10.3390\/axioms14030175","type":"journal-article","created":{"date-parts":[[2025,2,27]],"date-time":"2025-02-27T08:04:44Z","timestamp":1740643484000},"page":"175","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Algorithm for Designing Circles in a Bubble Chart"],"prefix":"10.3390","volume":"14","author":[{"given":"Ching-Shoei","family":"Chiang","sequence":"first","affiliation":[{"name":"Computer Science and Information Management, Soochow University, Taipei 100006, Taiwan"}]},{"given":"Hung-Chieh","family":"Li","sequence":"additional","affiliation":[{"name":"Computer Science and Information Management, Soochow University, Taipei 100006, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,27]]},"reference":[{"key":"ref_1","unstructured":"Heath, T.L. (1956). The Thirteen Books of Euclid\u2019s Element. Translated by Sir Thomas Heath, Dover Publication."},{"key":"ref_2","unstructured":"Stewart, I. (2009). The Foundations of Geometry, Dover Publications."},{"key":"ref_3","unstructured":"Tufte, E.R. (2020). Seeing with Fresh Eyes: Meaning, Space, Data, Truth, Graphics Press."},{"key":"ref_4","unstructured":"McKinney, W. (2013). Data Science from Scratch: First Principles with Python, O\u2019Reilly Media."},{"key":"ref_5","unstructured":"Few, S. (2009). Now You See It: Simple Visualization Techniques for Quantitative Analysis, Analytics Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1145\/1743546.1743567","article-title":"A tour through the visualization zoo","volume":"53","author":"Heer","year":"2010","journal-title":"Commun. Acm."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Wang, W., Wang, H., Dai, G., and Wang, H. (2006, January 22\u201327). Visualization of large hierarchical data by circle packing. Proceedings of the SIGCHI conference on Human Factors in computing systems, Montr\u00e9al, QC, Canada.","DOI":"10.1145\/1124772.1124851"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"15","DOI":"10.33697\/ajur.2004.010","article-title":"Apollonius\u2019 problem: A study of solutions and their connections","volume":"3","author":"Gisch","year":"2004","journal-title":"Am. J. Undergrad. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/j.comgeo.2010.06.005","article-title":"A generalized Malfatti Problem","volume":"45","author":"Chiang","year":"2012","journal-title":"Comput. Geom. Theory Appl."},{"key":"ref_10","first-page":"141","article-title":"Kontaktprobleme der Konformen Abbildung","volume":"88","author":"Koebe","year":"1936","journal-title":"Ber. S\u00a8Achs. Akad. Wiss. Leipz. Math.-Phys. Kl."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chiang, C.-S. (2024). Extended General Malfatti\u2019s Problem. Algorithms, 17.","DOI":"10.3390\/a17080374"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3390\/math12172733","article-title":"The various radii circle packing problem in a triangle","volume":"12","author":"Chiang","year":"2024","journal-title":"Mathematics"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/S0166-218X(01)00359-6","article-title":"Packing equal circles in a square: A deterministic global optimization approach","volume":"122","author":"Locatelli","year":"2002","journal-title":"Discret. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/S0012-365X(97)00050-2","article-title":"Dense packings of congruent circles in a circle","volume":"181","author":"Graham","year":"1998","journal-title":"Discret. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s00454-005-1172-4","article-title":"Compact packings of the plane with two sizes of discs","volume":"35","author":"Kennedy","year":"2006","journal-title":"Discret. Comput. Geom."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/S0925-7721(02)00099-8","article-title":"A circle packing algorithm","volume":"25","author":"Collins","year":"2003","journal-title":"Comput. Geom."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Hartshorne, R. (2000). Geometry: Euclid and Beyond, Springer.","DOI":"10.1007\/978-0-387-22676-7"},{"key":"ref_18","unstructured":"M\u00fcller, E., and Krams, J. (1929). Die Zyklographie, Franz Deuticke."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/175\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:43:39Z","timestamp":1760028219000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/175"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,27]]},"references-count":18,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030175"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030175","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,2,27]]}}}