islerp: An Incremental Approach to Slerp
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1 isler: A Icremetal Aroach to Sler Xi Li Comuter Sciece Deartmet Digie Istitute of Techology xli@digie.edu Abstract I this aer, a icremetal uaterio iterolatio algorithm is itroduced. With the assumtio of a costat iterval betwee a air of uaterios, the cost of iterolatio algorithm is sigificatly reduced. Exesive trigoometric calculatios i Sler are relaced with simle liear combiatio arithmetic. The roud-off errors ad driftig behavior accumulated through icremetal stes are also aalyzed i the aer. Itroductio Quaterios are widely emloyed to rereset rotatios of objects i 3D grahics. Durig a aimatio, uaterios are ofte iterolated to geerate itermediate orietatios betwee a air of ey frames. A umber of uaterio iterolatio algorithms have bee ivestigated (Ler, Sler [], Ler [2], log-ler [3], olysler [4], etc.. Amog them, the sherical liear iterolatio (Sler iterolates a air of uaterios o the surface of a uit shere. The resultig uaterios follow a geodesic ath with costat agular velocity. Ufortuately, the alicatio of Sler i video games ad real-time simulatio software is limited due to its high comutatioal cost. This aer itroduces a icremetal aroach of uaterio iterolatio. With the assumtio of a costat iterval, for each iterolated uaterio (=,,, a additioal uaterio is also comuted. This auxiliary uaterio allows us to exress the ext uaterio + o the ath as additio of two scaled uaterios from the revious ste. Therefore the cost of iterolatio is sigificatly reduced. Sler We start with the origial Sler euatio t (si( t si( t, t - -
2 where ad are uit uaterios ad = cos(. If we assume that is iterolated icremetally through stes with a fixed iterval, the the agle betwee ay itermediate uaterios ad + ca be exressed by a costat = /. Whe t varies from to, we always have t=with =,,, ad (si( si(,,,..., (E- Note that (E- roduces exactly the same seuece of iterolated uaterios as Sler. isler Now we itroduce the icremetal aroach by resetig the followig euatios: (si( (cos( si( cos( =,,, (E-2 The first euatio recites the Sler defiitio (E-. It comutes betwee ad for ay. (The terms of ad are swaed for a cosmetic reaso. The secod euatio secifies aother uaterio, which is the derivative of (with resect to divided by to ormalize it. Therefore is always a uit uaterio eredicular to. We hece call as s taget uaterio Pluggig = i (E-2, we have (si( (cos( cos( ( cos( (E-3 The euatios ca also be exteded to the followig forms for +: (si(( cos( (cos(( cos( si( si( si( ( cos( ( (E-4-2 -
3 (E-3 ad (E-4 are recursive forms of (E-2. They idicate that, alog the geodesic ath of iterolated orietatios, the ext + (ad its couterart + ca be calculated based o two costats (cos( ad si(, the curret uaterio ad its taget. Figure illustrates the geometric ituitios. + + si( + cos( cos( -si( + =cos( + si( + =cos( - si( Figure : Comutig + ad + by ad Discussio From (E-4, + is a liear combiatio of ad with two scalars. Hece + must be o the same lae defied by ad. Moreover, sice ad are defied by liear combiatios of ad (E-2 ad hece must be o the same lae with ad. Coseuetly, all uaterios with differet defie a geodesic ath betwee ad. Obviously isler is faster tha most of uaterio iterolatio algorithms. Yet it reserves all geometric characteristics of Sler. The algorithm is simle to uderstad ad very easy to imlemet. No logarithmic, olyomial or slie fuctios ivolved, o arameter-tuig eeded. Sice the icremetal iterolatio oly use additios ad multilicatios, the calculatio is immue to umerical istability. However, sice the method is based o icremetal stes over a costat agle, the imlemetatio o ay hardware ad software latform would ievitably itroduce some cumulative floatig-oit roud-off errors. To measure accumulated errors ad examie the driftig behavior, a test is coducted o a PC i Widows eviromet. The - 3 -
4 rogram was writte i C++ to test, airs of uaterios with agles of 5 o to 9 o betwee each air. Those uaterio airs are iterolated through 2 stes by both isler ad origial Sler fuctios. The the results are comared ad the differeces are comuted. Two tyes of errors are recorded: ( the maximum discreacies of agles betwee the start ad iterolated uaterios, ad (2 the maximum differeces of x, y, z ad w comoets of two iterolated uaterios. Figures i ext age illustrate the agle ad comoets errors of each ste of, airs of uaterios. As exected, both agles ad comoets of isler-iterolated uaterios ted to drift away from results of covetioal Sler fuctio. The cumulative roud-off errors icrease ear-liearly with the umber of iterolatio stes betwee the start ad ed uaterios. However, the test also idicates that this driftig behavior is isigificatly trivial. Eve i the worst cases, all maximum roud-off errors are well below -5. Note that oly the sigle-recisio 32-bits floatig oit was used i the above test. The roud-off errors should be further reduced uder double-recisio 64-bits oeratios. I coclusio, sice the deviatio is relatively small, the accuracy of isler algorithm should be sufficiet to most alicatios i real-time simulatio ad video game software. 6.E-6 5.E-6 Agle Errors 4.E-6 3.E-6 2.E-6.E-6.E Iterolatio Stes Figure 2: Cumulative roud-off errors of agles - 4 -
5 7.E-6 6.E-6 Comoet Errors 5.E-6 4.E-6 3.E-6 2.E-6.E-6.E Iterolatio Stes x y z w Refereces: Figure 3: Cumulative roud-off errors of uaterio comoets [] Ke Shoemae, Aimatig Rotatio with Quaterio Curves, Comuter Grahics, Volume 9, Number 3, 985. [2] Joatha Blow, Hacig Quaterios, Game Develoer Magazie, March 22. [3] F. Sebastia Grassia, Practical arameterizatio of rotatios usig the exoetial ma, Joural of Grahics Tools, volume 3.3, 998. [4] Thomas Busser, PolySler, A Fast ad Accurate Polyomial Aroximatio of Sler, Game Develoer Magazie, February,
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