{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T06:53:04Z","timestamp":1771656784439,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2012,3,9]],"date-time":"2012-03-09T00:00:00Z","timestamp":1331251200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In a curved surface design, the overall shape features that emerge from combinations of shape elements are important. However, controlling the features of the overall shape in curved profiles is difficult using conventional microscopic shape information such as dimension. Herein two types of macroscopic shape information, curvature entropy and quadrature curvature entropy, quantitatively represent the features of the overall shape. The curvature entropy is calculated by the curvature distribution, and represents the complexity of a shape (one of the overall shape features). The quadrature curvature entropy is an improvement of the curvature entropy by introducing a Markov process to evaluate the continuity of a curvature and to approximate human cognition of the shape. Additionally, a shape generation method using a genetic algorithm as a calculator and the entropy as a shape generation index is presented. Finally, the applicability of the proposed method is demonstrated using the side view of an automobile as a design example.<\/jats:p>","DOI":"10.3390\/e14030533","type":"journal-article","created":{"date-parts":[[2012,3,9]],"date-time":"2012-03-09T10:55:01Z","timestamp":1331290501000},"page":"533-558","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Curvature Entropy for Curved Profile Generation"],"prefix":"10.3390","volume":"14","author":[{"given":"Yoshiki","family":"Ujiie","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Takeo","family":"Kato","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Koichiro","family":"Sato","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yoshiyuki","family":"Matsuoka","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,3,9]]},"reference":[{"key":"ref_1","unstructured":"Metzger, W. (1953). Gesetze des Sehens, Kramer. [2nd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"300","DOI":"10.3758\/BF03207395","article-title":"Effect of orientation and of shape similarity on perceptual grouping","volume":"1","author":"Beck","year":"1966","journal-title":"Percept. Psychophys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"538","DOI":"10.1126\/science.154.3748.538","article-title":"Perceptual grouping produced by changes in orientation and shape","volume":"154","author":"Beck","year":"1966","journal-title":"Science"},{"key":"ref_4","unstructured":"Polanyi, M. (1966). The Tacit Dimension, Routledge & Kegan Paul Ltd."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1038\/scientificamerican0476-48","article-title":"Subjective contours","volume":"234","author":"Kanizsa","year":"1976","journal-title":"Sci. Am."},{"key":"ref_6","first-page":"35","article-title":"Measurements of feature-parameters of signs","volume":"45","author":"Kuyama","year":"1998","journal-title":"Science of Design"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1109\/TPAMI.1986.4767747","article-title":"The curvature primal sketch","volume":"8","author":"Asada","year":"1986","journal-title":"IEEE Trans. Pattern. Anal. Mach. Intell."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1109\/TPAMI.1986.4767750","article-title":"Scale-based description and recognition of planar curves and two-dimensional shapes","volume":"8","author":"Mokhtarian","year":"1986","journal-title":"IEEE Trans. Pattern. Anal. Mach. Intell."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1109\/34.149591","article-title":"A theory of multiscale, curvature-based shape representation for planar curves","volume":"14","author":"Mokhtarian","year":"1992","journal-title":"IEEE Trans. Pattern. Anal. Mach. Intell."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1109\/34.391387","article-title":"Silhouette-based isolated object recognition through curvature scale space","volume":"17","author":"Mokhtarian","year":"1995","journal-title":"IEEE Trans. Pattern. Anal. Mach. Intell."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Harada, T., Yoshimoto, F., and Moriyama, M. (1999, January 13\u201316). An aesthetic curve in the field of industrial design. Proceedings of the 1999 IEEE Symposium on Visual Language, Tokyo, Japan.","DOI":"10.1109\/VL.1999.795873"},{"key":"ref_12","first-page":"9","article-title":"Study of quantitative analysis of curve\u2019s character","volume":"40","author":"Harada","year":"1994","journal-title":"Science of Design"},{"key":"ref_13","first-page":"1","article-title":"Algorithm of creating the curve by controlling its characteristics","volume":"41","author":"Harada","year":"1994","journal-title":"Science of Design"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1109\/TPAMI.1984.4767507","article-title":"Entorpy based texture analysis in the spatial frequency domain","volume":"6","author":"Jernigan","year":"1984","journal-title":"IEEE Trans. Pattern. Anal. Mach. Intell."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1109\/81.473585","article-title":"Pattern formation properties of autonomous cellular neural networks","volume":"42","author":"Thiran","year":"1995","journal-title":"IEEE Trans. Circuits. Syst. I"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ujiie, Y., and Matsuoka, Y. (2000, January 5\u201310). Shape-generation method using curvature entropy. Proceedings of the 2000 ASME International Mechanical Engineering Congress and Exposition, Orlando, FL, USA.","DOI":"10.1115\/IMECE2000-1054"},{"key":"ref_17","unstructured":"Shannon, C.E., and Weaver, W. (1964). The Mathematical Theory of Communication, The University of Illinois Press."},{"key":"ref_18","unstructured":"Blahut, R.E. (1987). Principles and Practice of Information Theory, Addison-Wesley Publishing Company."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.aei.2008.10.004","article-title":"Macro-informatics of cognition and its application for design","volume":"23","author":"Ujiie","year":"2009","journal-title":"Adv. Eng. Informat."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Birkhoff, G.D. (1933). Aesthetic Measure, Harvard University Press.","DOI":"10.4159\/harvard.9780674734470"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1587\/transinf.E94.D.456","article-title":"GKJ: Group KJ method support system utilizing digital pens","volume":"E94-D","author":"Miura","year":"2011","journal-title":"IEICE Trans. Info. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Hosaka, M. (1992). Modeling of Curves and Surfaces in CAD\/CAM, Springer-Verlag.","DOI":"10.1007\/978-3-642-76598-8"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Yamaguchi, F. (1988). Curves and Surfaces in Computer Aided Geometric Design, Springer-Verlag.","DOI":"10.1007\/978-3-642-48952-5"},{"key":"ref_24","unstructured":"Suzuki, T., Asanuma, T., and Matsuoka, Y. (1999, January 30\u201331). The proposal of the macroscopic design information as a shape design index. Proceedings of the 4th Asian Design Conference, Nagaoka, Niigata, Japan."},{"key":"ref_25","first-page":"77","article-title":"A car form generation system based on evolutionary computation","volume":"44","author":"Tian","year":"1997","journal-title":"Science of Design"},{"key":"ref_26","first-page":"65","article-title":"A design support system based on co-evolutionary process","volume":"45","author":"Tian","year":"1998","journal-title":"Science of Design"},{"key":"ref_27","first-page":"21","article-title":"Aerodynamic design and inverse problem","volume":"1","author":"Obayashi","year":"1996","journal-title":"Science of Design"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/3\/533\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:49:19Z","timestamp":1760219359000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/3\/533"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3,9]]},"references-count":27,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2012,3]]}},"alternative-id":["e14030533"],"URL":"https:\/\/doi.org\/10.3390\/e14030533","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,3,9]]}}}