Bezier curves. 1. Defining a Bezier curve. A closed Bezier curve can simply be generated by closing its characteristic polygon

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1 Curve represetato YZU Optmal Desg Laboratory. All rghts reserved. Last updated: Yeh-Lag Hsu (--). Note: Ths s the course materal for ME55 Geometrc modelg ad computer graphcs, Yua Ze Uversty. art of ths materal s adapted from CAD/CAM Theory ad ractce, by Ibrahm Zed, McGraw-Hll, 99. Ths materal s be used strctly for teachg ad learg of ths course. ezer curves. Defg a ezer curve ezer curve was developed by the Frech egeer erre ezer (9-999) 96 for use the desg of Reault automoble bodes. erre ezer the wet o to develop the UNISURF CAD/CAM system. As show Fgure, the ezer curve s defed terms of the locatos of + pots. These pots are called data or cotrol pots. They form the vertces of what s called the cotrol or ezer characterstc polygo. Fgure shows cubc ezer curves for varous cotrol pots. I a ezer curve, oly the frst ad the last cotrol pots or vertces of the polygo actually le o the curve. The curve s also always taget to the frst ad last polygo segmets. I addto, the curve shape teds to follow the polygo shape. These three observatos should eable the user to sketch or predct the curve shape oce ts cotrol pots are gve. A closed ezer curve ca smply be geerated by closg ts characterstc polygo or choosg ad to be cocdet. Fgure also shows a example of closed curve.

2 Curve represetato u = k Cotrol pots (vertces) ---- Characterstc polygo u = 4 Fgure. Cubc ezer curve 5 4 Fgure. Cubc ezer curves for varous cotrol pots. Assgmet Costruct ezer curves smlar to those Fgure usg your CAD software. Descrbe the procedure for costructg these curves.. arametrc equatos of the ezer curve I geeral, a ezer curve secto ca be ftted to ay umber of cotrol pots. The umber of cotrol pots to be approxmated determes the degree of the ezer curve. For + cotrol pots, the ezer curve s defed by the followg polyomal of degree : u u u (),,

3 where u s ay pot o the curve ad s a cotrol pot, Curve represetato, are the erste polyomals. The erste polyomal serves as the bledg or bass fucto for the ezer curve ad s gve by where u C, u u, () C, s the bomal coeffcet C,! ()!! Equato () ca be expaded to gve ( ( C(,) u( C(, ) u C(,) u ( u, ( u (4) From Equato () ad (), we ca get the followgs: =, (, =, ( ) (, u (, u =,, ( ( (, u( ( u, =,,( (,( u(,( u ( ( u,

4 Curve represetato Therefore for =, ( ( u( u ( u. (5) Assgmet Derve ezer curve equatos smlar to Equato (5) for =,,. Expla what happes whe =,. Derve erste polyomals for = 4, ad costruct the correspodg ezer curve equato. Assgmet Assume the coordates of 4 cotrol pots, wrte a Matlab program to draw the correspodg cotrol polygo. Use u. to geerate the termedate pots of the ezer curve usg Equato (5). Use straght les to coect these pots to geerate the curve. Usg proper trasformato, create the frot, top, sde, ad sometrc vews of the -dmesoal curve ad ts cotrol polygo. Show your Matlab program too. From the dscusso above, we ca see that the major dffereces betwee the ezer curve ad the cubc sple curve are: () The shape of ezer curve s cotrolled by ts defg pots oly. Frst dervatves are ot used the curve developmet as the case of the cubc sple. Ths allows the desger a much better feel for the relatoshp betwee put (pots) ad output (curve). () The order or the degree of ezer curve s varable ad s related to the umber of pots defg t; + pots defed ad th degree curve whch permts hgher-order cotuty. Ths s ot the case for cubc sples where the degree s always cubc for a sple segmet. () The ezer curve s smoother tha the cubc sple because t has hgher-order dervatves.. ropertes of the ezer curves A very useful property of a ezer s that t always passes through the frst ad last cotrol pots. If we substtute u = ad Equato (4), the boudary codtos at the two ed pots are ( ) (6), () 4

5 Curve represetato The curve s taget to the frst ad last segmets of the characterstc polygo. From Equato (5), the frst dervatves whe there are 4 cotrol pots ( = ) s gve by ( ( (( 6u( ) (6u( u ) u (7) Therefore the taget vectors at the startg ad edg pots are ( ) (8) ( ) (9) Smlarly, t ca be show that the secod dervatve at s determed by,, ad ; or, geeral, the r-th dervatve at a edpot s determed by ts r eghborg vertces. Assgmet 4 rove from Equato (4) that geeral, the frst dervatves at the startg ad edg pots are gve by Equato () ad (), respectvely: where ( ) () ( ) () ad defe the frst ad last segmets of the curve polygo. Aother useful property of the ezer curve s that the curve s symmetrc wth respect to u ad (-. Ths meas that the sequece of cotrol pots defg the curve ca be reversed wthout chage of the curve shape; that s, reversg the drecto of parametrzato does ot chage the curve shape. Ths ca be acheved by substtutg u v Equato (4) ad otcg that C, C,. Ths s a result of the fact that, u ad, u The terpolato polyomal are symmetrc f they are plotted as fuctos of u., u has a maxmum value of C, / / occurrg at u / whch ca be obtaed from the equato d / du., Ths mples that each cotrol pot s most fluetal o the curve 5

6 Curve represetato shape at u = /. For example, for a cubc ezer curve,,,, ad are most fluetal whe u,,, ad respectvely. Therefore, each cotrol pot s weghed by ts bledg fucto for each u value. Assgmet 5, Derve the maxmum values of these 4 fuctos u u,, u,, u,, ad, ad the values of u at the maxmum. lot these 4 fuctos ad check whether your dervato s correct. The curve shape ca be modfed by ether chagg oe or more vertces of ts polygo or by keepg the polygo fxed ad specfyg multple cocdet pots at a vertex, as show Fgure. I Fgure (a), the vertex s pulled to the ew posto *. I Fgure (b), s assged a multplcty k, that s, or cotrol pots are placed o the same posto. The hgher the multplcty, the more the curve s pulled toward. * k= k= k= (a) Chagg a vertex (b) Specfyg multple cocdet pots at a vertex Fgure. Modfcatos of cubc ezer curve. Assgmet 6 Use your CAD software to geerate fgures smlar to Fgure. Aother mportat property of ay ezer curve s that t les wth the covex polygo boudary of the cotrol pots. Ths s called the covex hull property. Ths follows from the propertes of ezer bledg fucto: they are all postve ad for ay vald value of u the sum of the, fuctos s always equal to for ay degree of ezer curve. Ay curve posto s smply a weghted sum of the cotrol pot postos. 6

7 Curve represetato If the polygo defg a curve segmet degeerates to a straght le, the resultg segmet must therefore be lear. Also, the sze of the covex hull s a upper boud o the sze of the curve tself; that s, the curve always les sde ts covex hull. Ths s a useful property for graphcs fuctos such as dsplayg or clppg the curve. A ezer curve stll has some dsadvatages. Frst, the curve does ot pass through the cotrol pots, whch may be coveet to some desgers. Secod, the curve lacks local cotrol. It oly has the global cotrol ature. If oe cotrol pot s chaged, the whole curve chages. The degree of the ezer curve depeds o the umber of cotrol pots. Oly 4 cotrol pots are eeded for a cubc ezer Curve. Hgh order curves may result f there are may cotrol pots. Whe curves of may cotrol pots are to be geerated, they ca be formed by pecg several ezer sectos of lower degree together. ecg together smaller sectos also gves us better cotrol over the shape of the curve small regos. Sce ezer curves passes through edpots, t s easy to match curve sectos wth C cotuty. Also, ezer curve have the mportat property that t tagets to the cotrol polygo at the ed pots. Therefore we ca obta C cotuty by makg the two les jog the edpot to the adjacet cotrol pots collear. 7