{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,13]],"date-time":"2023-09-13T19:10:53Z","timestamp":1694632253856},"reference-count":17,"publisher":"Association for Computing Machinery (ACM)","issue":"3","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[1994,7]]},"abstract":"Quaternions play a vital role in the representation of rotations in computer graphics, primarily for animation and user interfaces. Unfortunately, quaternion rotation is often left as an advanced topic in computer graphics education due to difficulties in portraying the four-dimensional space of the quaternions. One tool for overcoming these obstacles is the quaternion demonstrator, a physical visual aid consisting primarily of a belt. Every quaternion used to specify a rotation can be represented by fixing one end of the belt and rotating the other. Multiplication of quaternions is demonstrated by the composition of rotations, and the resulting twists in the belt depict visually how quaternions interpolate rotation.<\/jats:p>\n This article introduces to computer graphics the exponential notation that mathematicians have used to represent unit quaternions. Exponential notation combines the angle and axis of the rotation into concise quaternion expression. This notation allows the article to present more clearly a mechanical quaternion demonstrator consisting of a ribbon and a tag, and develop a computer simulation suitable for interactive educational packages. Local deformations and the belt trick are used to minimize the ribbon's twisting and simulate a natural-appearing interactive quaternion demonstrator.<\/jats:p>","DOI":"10.1145\/195784.197480","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:29:00Z","timestamp":1027769340000},"page":"256-276","update-policy":"http:\/\/dx.doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":66,"title":["Visualizing quaternion rotation"],"prefix":"10.1145","volume":"13","author":[{"given":"John C.","family":"Hart","sequence":"first","affiliation":[{"name":"Washington State Univ., Pullman"}]},{"given":"George K.","family":"Francis","sequence":"additional","affiliation":[{"name":"Univ. of Illinois, Urbana"}]},{"given":"Louis H.","family":"Kauffman","sequence":"additional","affiliation":[{"name":"Univ. of Illinois, Chicago"}]}],"member":"320","published-online":{"date-parts":[[1994,7]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/142920.134086"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/964965.808573"},{"key":"e_1_2_1_3_1","first-page":"9","article-title":"The spinor spanner","volume":"80","author":"I~K R, E","year":"1973","unstructured":"B() I~K }: R, E . 1973 . The spinor spanner . Am. Math. Month. 80 , 9 (Nov.), 977 984. B()I~K}:R, E. 1973. The spinor spanner. Am. Math. Month. 80, 9 (Nov.), 977 984.","journal-title":"Am. Math. Month."},{"key":"e_1_2_1_4_1","volume-title":"The Collected Mathematical Papers af Arthur Cayley","author":"CAYIJ~ Y, A","unstructured":"CAYIJ~ ; Y, A . 1897. The Collected Mathematical Papers af Arthur Cayley . Cambridge University Press , Cambridge, U.K. CAYIJ~;Y, A. 1897. The Collected Mathematical Papers af Arthur Cayley. Cambridge University Press, Cambridge, U.K."},{"key":"e_1_2_1_5_1","unstructured":"FItnN('Is G.K. 1987. A Topological Picturebook. Springer-Verlag New York. FItnN('Is G.K. 1987. A Topological Picturebook. Springer-Verlag New York."},{"key":"e_1_2_1_6_1","unstructured":"FICAN4'IS G. K. ANI) KAUFFMAN L.H. 1994. Air on the Dirae strings. In Mathematical Legacy of Wilhelm Magnus. AMS Providence R.I. FICAN4'IS G. K. ANI) KAUFFMAN L.H. 1994. Air on the Dirae strings. In Mathematical Legacy of Wilhelm Magnus. AMS Providence R.I."},{"key":"e_1_2_1_7_1","unstructured":"HAMII 'I'{)N W.R. 1866. Elements af Quaternions. Longmans Green Lond()n. HAMII 'I'{)N W.R. 1866. Elements af Quaternions. Longmans Green Lond()n."},{"key":"e_1_2_1_8_1","volume-title":"Vol. 3","author":"HANSON A.","unstructured":"HANSON , A. 1992. The rolling ball: Applications of a method for controlling three degrees of freedom using two-dimensional input devices . In Graphics Gems, D. Kirk, Ed. Vol. 3 . Academic Press , San Diego , Calif., 51-60. HANSON, A. 1992. The rolling ball: Applications of a method for controlling three degrees of freedom using two-dimensional input devices. In Graphics Gems, D. Kirk, Ed. Vol. 3. Academic Press, San Diego, Calif., 51-60."},{"key":"e_1_2_1_9_1","volume-title":"Wash. Appears in SIGGRAPH '93","author":"HART J. C.","year":"1993","unstructured":"HART , J. C. 1993 . Sphere tracing: Simple robust antialiased rendering of distance-based implicit surfaces. Tech. Pep. EECS-93-15, School of EECS, Washington State Univ., Pu}lman , Wash. Appears in SIGGRAPH '93 , Course Notes #25 \"Modeling, Visualizing and Animating Implicit Surfaces.\" HART, J. C. 1993. Sphere tracing: Simple robust antialiased rendering of distance-based implicit surfaces. Tech. Pep. EECS-93-15, School of EECS, Washington State Univ., Pu}lman, Wash. Appears in SIGGRAPH '93, Course Notes #25 \"Modeling, Visualizing and Animating Implicit Surfaces.\""},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/74334.74364"},{"key":"e_1_2_1_11_1","doi-asserted-by":"crossref","unstructured":"KAUF~'MAN L.H. 1991. Knots and Physics. World Scientific Teaneck N.J. KAUF~'MAN L.H. 1991. Knots and Physics. World Scientific Teaneck N.J.","DOI":"10.1142\/1116"},{"key":"e_1_2_1_12_1","volume-title":"Princeton University Press","author":"KAU FMAN","unstructured":"KAU r FMAN , L.H. 1987. On Knots. Princeton University Press , Princeton, N.J. KAUrFMAN, L.H. 1987. On Knots. Princeton University Press, Princeton, N.J."},{"key":"e_1_2_1_13_1","volume-title":"itation","author":"MISNER C.","unstructured":"MISNER , C. Wl, THORNE , K. S. , AND WHEELER , J.A. 1973. Grat , itation . Freeman , San Francisco , Calif. MISNER, C. Wl, THORNE, K. S., AND WHEELER, J.A. 1973. Grat, itation. Freeman, San Francisco, Calif."},{"key":"e_1_2_1_14_1","unstructured":"SANDm D. J. KAUFFMAN L. H. AND FRAr~CIS G.K. 1993. Air on the Dirac strings. SIGGRAPH Video Rev. 93. Animation. SANDm D. J. KAUFFMAN L. H. AND FRAr~CIS G.K. 1993. Air on the Dirac strings. SIGGRAPH Video Rev. 93. Animation."},{"key":"e_1_2_1_15_1","volume-title":"Proceedings of Graphics Interface '92 (May) 151-156","author":"SHOEMAKE K.","year":"1992","unstructured":"SHOEMAKE , K. 1992 . ARCBALL: A user interface for specifying three-dimensional orientation using a mouse . In Proceedings of Graphics Interface '92 (May) 151-156 . SHOEMAKE, K. 1992. ARCBALL: A user interface for specifying three-dimensional orientation using a mouse. In Proceedings of Graphics Interface '92 (May) 151-156."},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/325165.325242"},{"key":"e_1_2_1_17_1","volume-title":"Calculus on Manifolds. W. A","author":"SPIVAK M.","unstructured":"SPIVAK , M. 1965. Calculus on Manifolds. W. A . Benjamin, New York . SPIVAK, M. 1965. Calculus on Manifolds. W. A. Benjamin, New York."}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/195784.197480","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,29]],"date-time":"2022-12-29T18:27:17Z","timestamp":1672338437000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/195784.197480"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,7]]},"references-count":17,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1994,7]]}},"alternative-id":["10.1145\/195784.197480"],"URL":"http:\/\/dx.doi.org\/10.1145\/195784.197480","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":["Computer Graphics and Computer-Aided Design"],"published":{"date-parts":[[1994,7]]},"assertion":[{"value":"1994-07-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}