Interactive NURBS Tutorial in Virtual Reality

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1 Inteacte NURBS Ttoal n Vtal Realty Eng 477 Poect Fa Octobe Edmond Nadle enadle@mch.ed Eng 477 Poect Fa Octobe Eng 477 Poect Fa Octobe

2 NURBS Ces and Sfaces = Non-Unfom Ratonal B-Splnes Fndamental Pease Standad Enable ceaton of wde aety of fee-fom ces and sfaces bt Often not well ndestood. Inteacte Vtal Realty ttoal llstatng feates and popetes of NURBS ces and sfaces wold be nteestng and sefl. Eng 477 Poect Fa Octobe

3 What ae NURBS? A conenent effcent and geometc epesentaton fo pecewse polynomal and atonal ces and sfaces. atonal = polynomal / polynomal needed fo example fo exact epesentaton of conc sectons Ce: p 1 t pescbed contnty p 2 t p n t Sface: p 11 p 12 p 1n p 21 p 22 p 23 p m1 p 32 p mn Eng 477 Poect Fa Octobe

4 Polynomal NURBS Ce c t = ab t wth a Contol Ponts etces of the Contol Polygon B t B-Splne Bass Fnctons Ratonal: Add scala weghts w fo each Contol Pont: wa B t c t = w B t Redces to polynomal case f all w eqal. a B t defned by Polynomal Degee and Knot Seqence: Mltplcty of knots contols contnty Eng 477 Poect Fa Octobe t

5 Eng 477 Poect Fa Octobe NURBS Sface Now 2 sets of B-Splne Bass Fnctons { B } { C } each goened by ts own Polynomal Degee & Knot Seqence Polynomal = C B a s wth a a 2-D aay of Contol Ponts whch ae the etces of the Contol Net Ratonal: As fo ces add scala weghts w fo each Contol Pont: = C B w C B a w s Agan edces to polynomal case f all w eqal.

6 Inteacte NURBS Ttoal n VR: Some thngs t cold nclde Vsal ceaton and/o manplaton of NURBS ces and sfaces n VR o Contol Ponts a a o Weghts w w o Bass Fnctons B as detemned by Degee and Knot Seqence show effect on contnty when knots coalesce Ealaton of NURBS o Ce: Moe t on [01] See pont c t on ce o Sface: Moe on [01]µ[01] See pt s Moe on [01] See ce s Moe on [01] See ce s o Vsalze geometc algothm de Boo: teated conex combnatons of Contol Ponts on sface on sface on sface Demonstate NURBS Popetes o Localty egon of nflence of each Contol Pont o Conex Hll pont n C.H. of eleant Contol Ponts o Vaaton Dmnshng fo Ces Othe deas Eng 477 Poect Fa Octobe

7 Refeences and comments I wold be happy to help nteested stdents to lean abot NURBS. I ecommend "The NURBS Book" by Les Pegl and Wayne Tlle Spnge The fst fo chaptes pode good ntodcton. Tlle s a leadng expet on NURBS and has deeloped a NURBS softwae lbay Nlb fo whch many of the algothms ae descbed n the late chaptes of the book. I hae sed ths book and elated softwae lbay n deelopng geomety softwae n ndsty. Othe possble books ae: "Fndamentals of Compte Aded Geometc Desgn" by J. Hoschek and D. Lasse A.K. Petes 1993 "Ces and Sfaces fo Compte Aded Geometc Desgn" by Geald Fan Academc Pess 1990 thee ae moe ecent edtons Also many nmecal analyss textbooks sch as "N.A.: Math. of Scentfc Comptng" by D. Kncad and W. Cheney Books/Cole 2002 wll hae mateal on splnes n the sectons on ntepolaton and appoxmaton. Note: Each book has a slghtly dffeent appoach and notaton. Fo MATLAB ses the Splne Toolbox fo MATLAB wtten by splne ponee Cal de Boo wnne of the pestgos Natonal Medal of Scence s a good hands-on way to lean and play wth these thngs. Note: the Splne Toolbox does not se the tems NURBS no does t hae atonals o sfaces at least dectly at least n an olde eson. * * * The poect shold nclde at least the sal ceaton and/o manplaton of NURBS ces and sfaces wth the ablty to dag contol ponts and see the effect on the ce o sface. Adstment of weghts wold be good. Tme pemttng t wold also be nce to be able to adst the knots and see the effect. In the case of sfaces thee wold of cose be two sch sets. Fo ths pehaps the set of B-Splne Bass Fnctons coespondng to the cent knot seqence cold be shown on plots. Addtonal tems of nteest wold elate to ealaton of NURBS and demonstaton NURBS popetes as otlned on the peos page. The Vaaton Dmnshng Popety mentoned thee states that any plane wll ntesect a NURBS ce no moe tmes than t does ts Contol Polygon. Ed Nadle Eng 477 Poect Fa Octobe