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A fourth-order partial differential equation involving a sculpting force originating from elastic bending of thin plates is proposed to define physics-based deformations and achieve <jats:inline-formula><jats:alternatives><jats:tex-math>$$ C^1 $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> continuity at the boundary of deformation regions. In order to obtain real-time physics-based surface manipulation, we construct a mapping relationship between a deformation region in a 3D coordinate space and a unit circle on a 2D parametric plane, formulate corresponding <jats:inline-formula><jats:alternatives><jats:tex-math>$$ C^1 $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> continuous boundary conditions for the unit circle, and obtain a simple analytical solution to describe the physics-based deformation in the unit circle caused by a sculpting force. After that, the obtained physics-based deformation is mapped back to the 3D coordinate space, and added to the original surface to create a new surface shape with <jats:inline-formula><jats:alternatives><jats:tex-math>$$ C^1 $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> continuity at the boundary of the deformation region. We also develop an interactive user interface as a plug-in of the 3D modelling software package Maya to achieve real-time surface manipulation. The effectiveness, easiness, real-time performance, and better realism of our proposed method is demonstrated by testing surface deformations on several 3D models and comparing with other methods and ground-truth deformations.<\/jats:p>","DOI":"10.1007\/s00371-021-02169-4","type":"journal-article","created":{"date-parts":[[2021,6,17]],"date-time":"2021-06-17T12:02:31Z","timestamp":1623931351000},"page":"2741-2753","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Real-time surface manipulation with $$C^{1}$$ continuity through simple and efficient physics-based deformations"],"prefix":"10.1007","volume":"37","author":[{"given":"Shuangbu","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nan","family":"Xiang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8139-0554","authenticated-orcid":false,"given":"Yu","family":"Xia","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lihua","family":"You","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianjun","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,6,17]]},"reference":[{"issue":"1","key":"2169_CR1","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1007\/s00170-002-1442-8","volume":"22","author":"JM Zheng","year":"2003","unstructured":"Zheng, J.M., Chan, K.W., Gibson, I.: Constrained deformation of freeform surfaces using surface features for interactive design. 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