{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T10:40:57Z","timestamp":1762252857557,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2018,10,25]],"date-time":"2018-10-25T00:00:00Z","timestamp":1540425600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"The National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No.51875454, No.11626185, and No.11571275"],"award-info":[{"award-number":["No.51875454, No.11626185, and No.11571275"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The modeling of free-form engineering complex curves is an important subject in product modeling, graphics, and computer aided design\/computer aided manufacturing (CAD\/CAM). In this paper, we propose a novel method to construct free-form complex curves using shape-adjustable generalized B\u00e9zier (or SG-B\u00e9zier, for short) curves with constraints of geometric continuities. In order to overcome the difficulty that most of the composite curves in engineering cannot often be constructed by using only a single curve, we propose the necessary and sufficient conditions for G1 and G2 continuity between two adjacent SG-B\u00e9zier curves. Furthermore, the detailed steps of smooth continuity for two SG-B\u00e9zier curves, and the influence rules of shape parameters on the composite curves, are studied. We also give some important applications of SG-B\u00e9zier curves. The modeling examples show that our methods in this paper are very effective, can easily be performed, and can provide an alternative powerful strategy for the design of complex curves.<\/jats:p>","DOI":"10.3390\/sym10110545","type":"journal-article","created":{"date-parts":[[2018,10,26]],"date-time":"2018-10-26T03:16:16Z","timestamp":1540523776000},"page":"545","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Modeling of Free-Form Complex Curves Using SG-B\u00e9zier Curves with Constraints of Geometric Continuities"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4916-3460","authenticated-orcid":false,"given":"Gang","family":"Hu","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Xi\u2019an University of Technology, Xi\u2019an 710054, China"}]},{"given":"Cuicui","family":"Bo","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Xi\u2019an University of Technology, Xi\u2019an 710054, China"}]},{"given":"Junli","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Xi\u2019an University of Technology, Xi\u2019an 710054, China"}]},{"given":"Guo","family":"Wei","sequence":"additional","affiliation":[{"name":"Department of Mathematics &amp; Computer Science, University of North Carolina at Pembroke, Pembroke, NC 28372, USA"}]},{"given":"Fei","family":"Hou","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Xi\u2019an University of Technology, Xi\u2019an 710054, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,25]]},"reference":[{"key":"ref_1","unstructured":"Farin, G. 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