Bezier curves. Figure 2 shows cubic Bezier curves for various control points. In a Bezier curve, only

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1 Edited: Yeh-Liag Hsu (998--; recommeded: Yeh-Liag Hsu (--9; last updated: Yeh-Liag Hsu (9--7. Note: This is the course material for ME55 Geometric modelig ad computer graphics, Yua Ze Uiversity. art of this material is adapted from CAD/CAM Theory ad ractice, by Ibrahim Zeid, McGraw-Hill, 99. This material is be used strictly for teachig ad learig of this course. ezier curves. Defiig a ezier curve ezier curve was developed by the Frech egieer ierre ezier (9-999 i 96 for use i the desig of Reault automobile bodies. ierre ezier the wet o to develop the UNISURF CAD/CAM system. As show i Figure, the ezier curve is defied i terms of the locatios of + poits. These poits are called data or cotrol poits. They form the vertices of what is called the cotrol or ezier characteristic polygo. Figure shows cubic ezier curves for various cotrol poits. I a ezier curve, oly the first ad the last cotrol poits or vertices of the polygo actually lie o the curve. The curve is also always taget to the first ad last polygo segmets. I additio, the curve shape teds to follow the polygo shape. These three observatios should eable the user to sketch or predict the curve shape oce its cotrol poits are give. A closed ezier curve ca simply be geerated by closig its characteristic polygo or choosig ad to be coicidet. Figure also shows a example of closed curve.

2 u = k Cotrol poits (vertices ---- Characteristic polygo u = 4 Figure. Cubic ezier curve 5 4 Figure. Cubic ezier curves for various cotrol poits. Assigmet Costruct ezier curves similar to those i Figure usig your CAD software. Describe the procedure for costructig these curves.. arametric equatios of the ezier curve I geeral, a ezier curve sectio ca be fitted to ay umber of cotrol poits. The umber of cotrol poits to be approximated determies the degree of the ezier curve. For + cotrol poits, the ezier curve is defied by the followig polyomial of degree : ( = u ( i= i i,, where u is ay poit o the curve ad is a cotrol poit, are the erstei polyomials. The erstei polyomial serves as the bledig or basis fuctio for the ezier curve ad is give by i i,

3 where C (, i i i = C(, i u ( ezier curves i, ( is the biomial coefficiet C (, i! = ( i!( i! Equatio ( ca be expaded to give ( u = ( C(, u( C(, u + C(, u ( + u, ( u (4 From Equatio ( ad (, we ca get the followigs: =, u, ( = =, u = (,( u u = u,( =,, = (, = u( u ( = u, u =,, = (, = u(,( u = u ( ( = u, u Therefore for =, ( u = ( + u( + u ( + u. (5

4 Assigmet Derive ezier curve equatios similar to Equatio (5 for =,,. Explai what happes whe =,. Derive erstei polyomials for = 4, ad costruct the correspodig ezier curve equatio. Assigmet Assume the coordiates of 4 cotrol poits, write a Matlab program to draw the correspodig cotrol polygo. Use Δu =. to geerate the itermediate poits of the ezier curve usig Equatio (5. Use straight lies to coect these poits to geerate the curve. Usig proper trasformatio, create the frot, top, side, ad isometric views of the -dimesioal curve ad its cotrol polygo. Show your Matlab program too. From the discussio above, we ca see that the major differeces betwee the ezier curve ad the cubic splie curve are: ( The shape of ezier curve is cotrolled by its defiig poits oly. First derivatives are ot used i the curve developmet as i the case of the cubic splie. This allows the desiger a much better feel for the relatioship betwee iput (poits ad output (curve. ( The order or the degree of ezier curve is variable ad is related to the umber of poits defiig it; + poits defied ad th degree curve which permits higher-order cotiuity. This is ot the case for cubic splies where the degree is always cubic for a splie segmet. ( The ezier curve is smoother tha the cubic splie because it has higher-order derivatives.. roperties of the ezier curves A very useful property of a ezier is that it always passes through the first ad last cotrol poits. If we substitute u = ad i Equatio (4, the boudary coditios at the two ed poits are ( = = (6, ( The curve is taget to the first ad last segmets of the characteristic polygo. From Equatio (5, the first derivatives whe there are 4 cotrol poits ( = is give by ( u = ( + (( 6u( + (6u( u + u (7 4

5 Therefore the taget vectors at the startig ad edig poits are ( ( = (8 ad vertices. ( ( = (9 Similarly, it ca be show that the secod derivative at is determied by,, ; or, i geeral, the r-th derivative at a edpoit is determied by its r eighborig Assigmet 4 rove from Equatio (4 that i geeral, the first derivatives at the startig ad edig poits are give by Equatio ( ad (, respectively: ( ( = ( = ( ( ( where ad ( defie the first ad last segmets of the curve polygo. ( Aother useful property of the ezier curve is that the curve is symmetric with respect to u ad (-u. This meas that the sequece of cotrol poits defiig the curve ca be reversed without chage of the curve shape; that is, reversig the directio of parametrizatio does ot chage the curve shape. This ca be achieved by substitutig i Equatio (4 ad oticig that C(, i = C(, i i, ( u i, i, has a maximum value of C (, i( i / i u = v. This is a result of the fact that ad are symmetric if they are plotted as fuctios of u. The iterpolatio polyomial i ( i / occurrig at u = i / which ca be obtaied from the equatio d (, / du =. i This implies that each cotrol poit is most ifluetial o the curve shape at u = i /. For example, for a cubic ezier curve,,,, ad are most ifluetial whe u =,,, ad respectively. Therefore, each cotrol poit is weighed by its bledig fuctio for each u value. 5

6 Assigmet 5, Derive the maximum values of these 4 fuctios, u, u, ad, ad the values of u at the maximum. lot these 4 fuctios ad check whether your derivatio is correct.,, (, ( The curve shape ca be modified by either chagig oe or more vertices of its polygo or by keepig the polygo fixed ad specifyig multiple coicidet poits at a vertex, as show i Figure. I Figure (a, the vertex * is pulled to the ew positio. I Figure (b, is assiged a multiplicity k, that is, or cotrol poits are placed o the same positio. The higher the multiplicity, the more the curve is pulled toward. * k= k= k= (a Chagig a vertex (b Specifyig multiple coicidet poits at a vertex Figure. Modificatios of cubic ezier curve. Assigmet 6 Use your CAD software to geerate figures similar to Figure. Aother importat property of ay ezier curve is that it lies withi the covex polygo boudary of the cotrol poits. This is called the covex hull property. This follows from the properties of ezier bledig fuctio: they are all positive ad for ay valid value of u the sum of the fuctios is always equal to for ay degree of i, ezier curve. Ay curve positio is simply a weighted sum of the cotrol poit positios. If the polygo defiig a curve segmet degeerates to a straight lie, the resultig segmet must therefore be liear. Also, the size of the covex hull is a upper boud o the size of the curve itself; that is, the curve always lies iside its covex hull. This is a useful property for graphics fuctios such as displayig or clippig the curve. A ezier curve still has some disadvatages. First, the curve does ot pass through the cotrol poits, which may be icoveiet to some desigers. Secod, the curve lacks 6

7 local cotrol. It oly has the global cotrol ature. If oe cotrol poit is chaged, the whole curve chages. The degree of the ezier curve depeds o the umber of cotrol poits. Oly 4 cotrol poits are eeded for a cubic ezier Curve. High order curves may result if there are may cotrol poits. Whe curves of may cotrol poits are to be geerated, they ca be formed by piecig several ezier sectios of lower degree together. iecig together smaller sectios also gives us better cotrol over the shape of the curve i small regios. Sice ezier curves passes through edpoits, it is easy to match curve sectios with cotiuity. Also, ezier curve have the importat property that it tagets to the cotrol polygo at the ed poits. Therefore we ca obtai C joiig the edpoit to the adjacet cotrol poits colliear. C cotiuity by makig the two lies 7