Lecture 24: Bezier Curves and Surfaces. thou shalt be near unto me Genesis 45:10

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1 Lecture 24: Bezier Curves ad Surfaces thou shalt be ear uto me Geesis 45:0. Iterpolatio ad Approximatio Freeform curves ad surfaces are smooth shapes ofte describig ma-made objects. The hood of a car, the hull of a ship, the fuselage of a airplae are all examples of freeform shapes. Freeform surfaces differ from the classical surfaces we ecoutered i earlier lectures such as spheres, cyliders, coes, ad tori. Classical surfaces are typically easy to describe with a few simple parameters. A sphere ca be represeted by a ceter poit ad a scalar radius; a coe by a vertex poit, a vertex agle, ad a axis vector. The hood of a car or the hull of a ship are ot so easy to describe with a few simple parameters. The goal of the ext few chapters is to develop mathematical techiques for describig freeform curves ad surfaces. Scietists ad egieers use freeform curves ad surfaces to iterpolate data ad to approximate shape. But iterpolatio ad approximatio are ot always compatible operatios. Cosider the data i Figure (a). If we use a low degree polyomial as i Figure (b) to iterpolate this data, the polyomial oscillates about the x-axis, eve though there are o such oscillatios i the data. Thus the shape of the iterpolatig polyomial does ot reflect the shape of the data. Moreover, eve providig more ad more data poits o the desired curve may ot elimiate these uwated oscillatios. (a) Data Poits (b) Polyomial Iterpolatio Figure : Polyomial iterpolatio. Notice that there are oscillatios i the iterpolatig polyomial curve, eve though there are o oscillatios i the origial data poits. The goal of approximatio is to capture the shape of a desired curve or surface from a few data poits without ecessarily iterpolatig the poits. Ulike iterpolatio, i approximatio the data poits themselves are ot sacred. Rather the data poits are cotrol poits; these poits cotrol the shape of the curve or surface ad ca be adjusted i order to provide a better represetatio of the desired shape. The curve i Figure 2(b) approximates the shape of the data i Figure 2(a), eve

2 though the curve does ot pass through all the data poits. The curve i Figure 2(b) is called a Bezier curve. Bezier curves have may practical applicatios, ragig from the desig of ew fots to the creatio of mechaical compoets ad assemblies for idustrial desig ad maufacture. The goal of this lecture is to develop some of the theory uderlyig Bezier curves ad surfaces. (a) Cotrol Poits (b) Bezier Approximatio Figure 2: Polyomial approximatio. A Bezier curve approximates the shape described by the cotrol poits, but the Bezier curve does ot iterpolate all the cotrol poits. The height of the curve ca be adjusted by chagig the height of the middle cotrol poit. Compare to Figure (b). 2. The de Casteljau Evaluatio Algorithm The easiest curve to represet with cotrol poits is a straight lie. Give two poits P 0,P, the lie P(t) joiig P 0 ad P ca be expressed parametrically by settig P(t) = P 0 + t(p P 0 ) = ( t) P 0 + tp. (2.) It is easy to check that P(t) represets a straight lie, ad that P(0) = P 0 ad P() = P. Thus the curve P(t) passes through the poit P 0 at time t = 0 ad through the poit P at time t =. Suppose, however, that you wat the straight lie to pass through the poit P 0 at time t = a ad through the poit P at time t = b. The mimickig Equatio (2.), you might write P(t) = ( f (t))p 0 + f (t)p (2.2) with the requiremet that f (a) = 0 ad f (b) =. (2.3) To fid a simple explicit expressio for the fuctio f (t), you would eed to fid the lie i the coordiate plae iterpolatig the data (a,0) ad (b,). Usig stadard techiques from aalytic geometry, you ca write the equatio of this lie as () f (t) = (b a), (2.4) ad you ca easily verify that f (a) = 0 ad f (b) =. Substitutig Equatio (2.4) ito Equatio 2

3 (2.2) yields P(t) = b a P 0 + b a P. (2.5) Equatio (2.5) is called liear iterpolatio because the curve P(t) iterpolates the poits P 0,P with a straight lie. You have ecoutered liear iterpolatio may times before i Computer Graphics; for example, Gouraud ad Phog shadig are both based o liear iterpolatio. Equatio (2.5) is so importat that we are goig to represet this equatio by a simple graph. I Figure 3(a) the cotrol poits P 0,P are placed i the two odes at the base of the diagram ad the coefficiets of the cotrol poits P 0,P i Equatio (2.5) are placed alog the arrows emaatig from these odes. The values i the odes are multiplied by the values alog the arrows; these products are the added ad the result is placed i the ode at the apex of the diagram. Thus Figure 3(a) is a graphical represetatio of Equatio (2.5). Figure 3(b) is the same as Figure 3(a), except that to avoid clutterig the diagram, we have removed the ormalizig costat b a i the deomiator of the fuctios alog the arrows. We ca easily retrieve this ormalizig costat, sice the deomiator is simply the sum of the umerators: b a = () + (). Thus we shall iterpret Figure 3(b) to mea Figure 3(a), which i tur is equivalet to Equatio (2.5). P(t) P(t) b t b a b a b t P 0 P P 0 P (a) Normalized (b) Uormalized Figure 3: Diagrams represetig the lie i Equatio (2.5). A Bezier curve is a curve geerated by a algorithm where the steps i Figure 3 are repeated over ad over agai. Figure 4(a) with three cotrol poits at the base represets a quadratic Bezier curve, ad Figure 4(b) with four cotrol poits at the base represets a cubic Bezier curve. The correspodig curves are illustrated i Figures 5(a) ad 5(b). The piecewise liear curve cosistig of the lies coectig the cotrol poits with cosecutive idices is called the cotrol polygo. Notice how the Bezier curves i Figure 5 mimic the shape of their cotrol polygos. This algorithm for geeratig Bezier curves is called the de Casteljau evaluatio algorithm. A curve geerated by the de Casteljau algorithm with + cotrol poits at the base is called a degree Bezier curve. Notice that i the de Casteljau algorithm all the left poitig arrows are labeled with the fuctio ad all the right poitig arrows are labeled with the fuctio. 3

4 B(t) P 0 P P 2 B(t) P 0 P P2 P 3 (a) Quadratic de Casteljau Algorithm (b) Cubic de Casteljau Algorithm Figure 4: The de Casteljau evaluatio algorithm for (a) a quadratic Bezier curve, ad for (b) a cubic Bezier curve. The label o each edge must be ormalized by dividig by b a. (a) Quadratic Bezier Curve (b) Cubic Bezier Curve Figure 5: A quadratic Bezier curve (left) ad a cubic Bezier curve (right). Notice how the shape of the Bezier curve (dark) mimics the shape of the cotrol polygo (light). Itermediate odes of the de Casteljau algorithm represet Bezier curves of lower degree. Thus the de Casteljau algorithm is a dyamic programmig procedure for computig poits o a Bezier curve. Usually, for reasos that will become clear i the ext sectio, Bezier curves are restricted to the parameter iterval [a, b] -- that is, usually we shall isist that a t b. Typically, we shall take a = 0 ad b =, though there are cases where we will eed to take other values for a,b. Notice that whe a = 0 ad b =, o ormalizatio is required because b a = 0 =. Sice each ode i the de Casteljau algorithm represets the equatio of a straight lie joiig the poits i the odes immediately below to the left ad the right, each ode symbolizes a poit o the lie segmet joiig the two poits whose arrows poit ito the ode. Drawig all these lie segmets geerates the trellis i Figure 6. 4

5 Figure 6: Geometric costructio algorithm for a poit o a cubic Bezier curve based o a geometric iterpretatio of the de Casteljau evaluatio algorithm. If the labels alog the edges i the de Casteljau algorithm are () ad (), the at the parameter t, each lie segmet i the trellis is split i the ratio ()/(). 3. The Berstei Represetatio Bezier curves are polyomial curves. The de Casteljau algorithm proceeds by addig ad multiplyig the liear fuctios ad (see Figure 4). But addig ad multiplyig polyomials geerates polyomials of higher degree. Therefore the de Casteljau algorithm geerates polyomial curves. We ca fid a explicit polyomial represetatio for Bezier curves. Let B(t) deote the Bezier curve with cotrol poits P 0,K,P. Sice B(t) is a degree polyomial curve, we could try to express B(t) relative to the stadard polyomial basis,t, t 2,K,t -- that is, we could ask: what are the costat coefficiets of the basis fuctios,t, t 2,K,t for the polyomial B(t)? Ufortuately, these coefficiets are umerically ustable, so i practice these values are ot very useful. A better, more isightful questio to ask is: what are the polyomial coefficiets of the cotrol poits P 0,K,P? Let B k (t) deote the coefficiet of the cotrol poit Pk i the fuctio B(t). From the de Casteljau algorithm (Figure 4) it is easy to see that B k (t) = the sum over all paths from Pk at the base to B(t) at the apex of the graph where a path = the product of all the labels alog the arrows i the path. Sice P k lies i the kth positio at the base of the diagram, to reach the apex a path must take exactly k left turs ad exactly k right turs. But each left poitig arrow carries the label 5

6 () / (b a) ad each right poitig arrow carries the label () / (b a). Therefore, all the paths from P k at the base to B(t) at the apex of the diagram geerate the same product, so where B k (t) = P(,k) () k () k (b a), P(,k) = the umber of paths from P k to B(t). To fid a closed formula for P(,k), observe that P(,k) satisfies the recurrece P(,k) = P(,k ) + P(,k), because the oly way to reach the kth positio o the th level is from either the (k ) st positio o the ( ) st level or from the kth positio o the ( ) st level (see Figure 7). Sice P(0,0) =, the values i Pascal s triagle (biomial coefficiets) ad the values i the path triagle are ( ) idetical: k so ad P(,k) start at the same value ad satisfy the same recurrece. Therefore P(,k) = ( k ) =! k!( k)!, B k (t) = ( k ) ()k () k (b a) k = 0,K,. 2 P(0, 0) P(, 0) P(,) P(2, 0) P(2,) P(2, 2) 3 3 N M O L ( ) L ( k ) k P(3, 0) P(3,) P(3, 2) P(3, 3) N M O P(,0) L P(,k ) P(,k) L P(, ) L L ( k ) (a) Pascal s Triagle L P(, k) L P(, 0) P(, ) (b) Paths Triagle Figure 7: Pascal s triagle ad the paths triagle start at the same value ad satisfy the same recurrece. Therefore P(,k) = k ( ). The fuctios B 0 (t),k, B (t) are called the Berstei basis fuctios. We shall show i Lecture 26 that the Berstei basis fuctios form a basis for the polyomials of degree ; every 6

7 polyomial of degree ca be expressed i terms of the Berstei basis fuctios. From this perspective, the cotrol poits P 0,K,P of a Bezier curve B(t) are simply the coefficiets of the Bezier curve relative to the Berstei basis B 0 (t),k, B (t) -- that is, B(t) = B k (t)pk k=0 B k (t) = ( k ) ()k () k (3.) (b a) k = 0,K,. Equatio (3.) is called the Berstei represetatio of the Bezier curve B(t). The Berstei basis fuctios B 0 (t),k, B (t) are also called bledig fuctios, sice these fuctios bled the discrete cotrol poits P 0,K,P to form a smooth curve. 4. Geometric Properties of Bezier Curves Bezier curves have the followig key geometric features: they are affie ivariat, lie i the covex hull of their cotrol poits, satisfy the variatio dimiishig property (that is, they do ot oscillate more tha their cotrol polygo), ad iterpolate their first ad last cotrol poits. There are two ways to derive these properties: either we ca appeal to the de Casteljau evaluatio algorithm or we ca ivoke the Berstei represetatio of a Bezier curve. Here we shall derive most of these properties simply by readig these attributes off the graph represetig the de Casteljau algorithm. For simplicity we shall typically illustrate our argumets o cubic Bezier curves, but our proofs are completely geeral ad apply to Bezier curves of arbitrary degree. Alterative proofs based o the Berstei represetatio of a Bezier curve are provided i Exercises 4-7 at the ed of this lecture. 4.. Affie Ivariace. A curve scheme is said to be affie ivariat if applyig a affie trasformatio to the cotrol poits trasforms every poit o the curve by the same affie trasformatio. For example, a curve scheme is traslatio ivariat if traslatig each cotrol poit by a vector v traslates the etire curve by the vector v. Traslatio ivariace is a easy cosequece of the de Casteljau algorithm. If we traslate each cotrol poit by the vector v, the each ode i the de Casteljau algorithm traslates by the same vector v because the coefficiets alog the two arrows eterig each ode sum to oe (see Figure 8). More geerally, a affie trasformatio ca be represeted by a affie matrix. If we multiply each cotrol poit by a affie trasformatio matrix M, the each ode i the de Casteljau algorithm is multiplied by the same trasformatio matrix M because matrix multiplicatio distributes through both additio ad scalar multiplicatio (see Figure 9). 7

8 v +v B(t)+ v + v +v + v P 0 + v P + v P 2 + v P 3 + v Figure 8: Traslatio ivariace. Traslatig each cotrol poit by a vector v traslates each poit o the curve by the same vector v. ( B(t),) M (P 0,) M (,) M (,) M (,) M (,) M (,) M (P,) M (P 2,) M (P 3,) M Figure 9: Affie ivariace. Trasformig each cotrol poit by a affie trasformatio matrix M trasforms every poit o the curve by the same affie trasformatio matrix M. Traslatio ivariace is equivalet to assertig that the curve is idepedet of the choice of the origi of the coordiate system. Affie ivariace isures that the curve is also idepedet of the choice of the coordiate axes. Both of these properties are essetial for a good approximatio scheme. I Computer Graphics a curve should be completely determied by its cotrol poits; durig desig we do ot wat to worry about the locatio of the coordiate origi or the orietatio of the coordiate axes. Bezier curves deped oly o their cotrol poits; the locatio of the coordiate origi ad the orietatio of the coordiate axes affect either the locatio or the shape of a Bezier curve. 4.2 Covex Hull Property. A set of poits S is said to be covex if wheever P ad Q are poits i S the etire lie segmet from P to Q lies withi S (see Figure 0). The itersectio S of a 8

9 collectio of covex sets {S i } is a covex set because if P ad Q are poits i S, they must also be poits i each of the sets S i. Sice, by assumptio, the sets S i are covex, the etire lie segmet from P to Q lies i each set S i. Hece the etire lie segmet from P to Q lies i the itersectio S, so S too is covex. P Q P Q (a) Covex Set (b) No-Covex Set Figure 0: I a covex set (a) the lie segmet joiig ay two poits i the set lies etirely withi the set. I a o-covex set (b) part of the lie segmet joiig two poits i the set may lie outside the set. The covex hull of a collectio of poits {P k } is the itersectio of all the covex sets cotaiig the poits {P k }. Sice the itersectio of covex sets is a covex set, the covex hull is the smallest covex set cotaiig the poits {P k }. For two poits, the covex hull is the lie segmet joiig the poits. For three o-colliear poits, the covex hull is the triagle whose vertices are the three poits. The covex hull of a fiite collectio of poits i the plae ca be foud mechaically by placig a ail at each poit, stretchig a rubber bad so that its iterior cotais all the ails, ad the releasig the rubber bad. Whe the rubber bad comes to rest o the ails, the iterior of the rubber bad is the covex hull of the poits. Bezier curves always lie i the covex hull of their cotrol poits. Oce agai we ca prove this assertio directly from the de Casteljau algorithm. First recall that, by covetio, we always restrict the Bezier curve to the parameter iterval [a,b]. With this covetio, the poits o the first level of the de Casteljau algorithm certaily lie i the covex hull of the cotrol poits o the zeroth level because, by costructio, the poits o the first level lie o the lie segmets joiig adjacet cotrol poits (see Figure 6). For the same reaso, the poits o the ( +) st level of the de Casteljau algorithm lie i the covex hull of the poits o the th level of the de Casteljau algorithm. Hece, by iductio, the poit at the apex of the de Casteljau algorithm lies i the covex hull of the cotrol poits at the base of the de Casteljau algorithm. Thus each poit o a Bezier curve lies i the covex hull of the cotrol poits. 9

10 The covex hull property is importat i Computer Graphics because the covex hull property guaratees that if all the cotrol poits are visible o the graphics termial, the the etire curve is visible as well. The restrictio a t b o the parameter t is there precisely to guaratee the covex hull property. 4.3 Variatio Dimiishig Property. Ituitively, a curve is said to be variatio dimiishig if the curve does ot oscillate more tha its cotrol poits. I Sectio, we observed that iterpolatig polyomials may oscillate eve if the data does ot (see Figure ). We abadoed iterpolatio i favor of approximatio precisely i order to avoid uecessary oscillatios. Therefore we eed to be sure that Bezier curves are variatio dimiishig. But how do we measure oscillatios? I Figure we cosidered oscillatios with respect to the x-axis. The oscillatios of the curve i Figure are liked to the umber of times this curve crosses the x-axis. But i affie ivariat schemes, there is othig special about the x-axis; ay lie will do. Therefore we say that a curve is variatio dimiishig if the umber of itersectios of the curve with each lie i the plae (or each plae i 3-space) is less tha or equal to the umber of itersectios of the lie (or the plae) with the cotrol polygo (see Figure ). I this defiitio, we igore lies that coicide with a edge of the cotrol polygo. (a) Variatio Dimiishig (b) Not Variatio Dimiishig Figure. (a) A variatio dimiishig curve. A arbitrary lie L itersects both the curve ad the cotrol polygo twice. (b) A curve that is ot variatio dimiishig. The lie L itersects the curve three times, but the cotrol polygo oly twice. Bezier curves are ideed variatio dimiishig, but it is ot so clear how to derive this fact from the de Casteljau evaluatio algorithm. Therefore, we shall defer our proof of the variatio dimiishig property for Bezier curves till Lecture 25, where we shall derive the variatio dimiishig property for Bezier curves from de Casteljau s subdivisio algorithm. 0

11 Iterpolatio of the First ad Last Cotrol Poits. Bezier curves do ot geerally iterpolate all their cotrol poits. But Bezier curves always iterpolate their first ad last cotrol poits. I fact, if B(t) represets the Bezier curve over the iterval [a,b] with cotrol poits P 0,...,P, the B(a) = P 0 ad B(b) = P. Thus B(t) starts at the iitial cotrol poit P 0 ad eds at the fial cotrol poit P. As usual, we ca establish these results by examiig the de Casteljau algorithm. If we set t = a i de the Casteljau algorithm, the all the right poitig arrows become oe ad all the left poitig arrows become zero (see Figure 2(a)). Now if we follow the de Casteljau algorithm from the base to the apex, the P 0 appears at each ode alog the left edge of the diagram. Therefore at the apex, B(a) = P 0. Similarly, if we set t = b i de the Casteljau algorithm, the all the right poitig arrows become zero ad all the left poitig arrows become oe (see Figure 2(b)). Now if we follow the algorithm from the base to the apex, the P appears at each ode alog the right edge of the diagram. Therefore at the apex, B(b) = P. Iterpolatio at the ed poits is importat because we ofte wat to coect two Bezier curves. To isure that two Bezier curves joi at their ed poits, all we eed to do is to make sure that the iitial cotrol poit of the secod curve is the same as the fial cotrol poit of the first curve. This device guaratees cotiuity. I the ext sectio, we shall develop techiques for guarateeig higher order smoothess betwee adjacet Bezier curves, but before we ca perform this aalysis we eed to kow how to compute the derivative of a Bezier curve. P 0 0 P 3 0 P 0 0 P 0 0 P 2 0 P 3 P 0 P 0 0 P P P P 2 P 3 P 0 P 0 0 P 2 P 3 0 P P 2 P 3 (a) B(a) = P 0 (b) B(b) = P 3 Figure 2: Iterpolatio of the first ad last cotrol poits: B(a) = P 0 ad B(b) = P Differetiatig the de Casteljau Algorithm We ca compute the derivative of a Bezier curve directly from its Berstei represetatio. Let B(t) be a Bezier curve with cotrol poits P 0,K,P. Recall from Equatio (3.) that

12 0 2 where Thus B(t) = B k (t)pk k=0 B k (t) = ( k ) ()k () k (b a) k = 0,K,. B (t) = db k (t) P dt k. (5.) k=0 Therefore to compute the derivative of a Bezier curve, we eed oly differetiate the Berstei basis fuctios B k (t), k = 0,K,. But it follows easily from the product rule that (see Exercise 8) db k (t) dt = B k (t) Bk (t) b a. (5.2) Isertig Equatio (5.2) ito Equatio (5.) ad collectig the coefficiets of B k (t) yields B P (t) = B k (t) k+ P k. (5.3) k=0 b a Thus the derivative of the degree Berstei polyomial with coefficiets P 0,K,P is times the Berstei polyomial of degree with coefficiets P P 0 b a,k, P P. b a Oe importat cosequece of this observatio is that we ca compute the derivative of a Bezier curve usig the de Casteljau algorithm: Simply place the coefficiets P P 0 b a,k, P P b a at the base of the diagram, ru levels of the algorithm, ad multiply the output by (see Figure 3). B (t) = 3 _ P P 0 b a P 2 P b a P 3 P 2 b a Figure 3: The derivative of a cubic Bezier curve with cotrol poits {P k } is, up to a costat P multiple, a quadratic Berstei polyomial with coefficiets k+ P k b a. 2

13 0 2 We ca also compute the derivative of a Bezier curve by differetiatig the de Casteljau algorithm. Differetiatig the de Casteljau algorithm for a degree Bezier curve is actually quite easy: simply differetiate the labels -- that is, replace ad -- o the first level of the de Casteljau algorithm ad multiply the output by (see Figure 4). The validity of this differetiatio algorithm follows directly from Equatio (5.3) because if we differetiate the labels o the first level ad ru the algorithm, the the values i the odes o the first level become P P 0 b a,k, P P (remember the ormalizig factor of b a i the deomiator of every b a label), so the result of the algorithm is exactly the Berstei polyomial of degree with coefficiets P P 0 b a,k, P P. b a B (t) = 3 _ P 0 P P 2 P 3 Figure 4: Computig the first derivative of a cubic Bezier curve with cotrol poits P 0,P, P 2,P 3 by differetiatig the labels o the first level of the de Casteljau algorithm. To get the correct derivative B (t), we must multiply the output of this algorithm by = 3 ad we must remember to ormalize the labels o each arrow -- eve the costats alog the arrows o the first level -- by dividig by b a. Compare to Figure 3. I geeral, up to a costat multiple, the derivative of a Bezier curve with cotrol poits {P k } is a Berstei polyomial of oe lower degree with cotrol vectors P k+ P k b a. Therefore, by iductio, we ca express the kth derivative of a degree Bezier curve as a Berstei polyomial of degree k. Moreover, to compute the kth derivative of a Bezier curve, we ca simply differetiate the labels o the first k levels the de Casteljau algorithm ad multiply the output by Figure 5).! ( k)! (see 3

14 0 2 B (t) = 6 _ P 0 P P 2 P 3 Figure 5: Computig the secod derivative of a cubic Bezier curve by differetiatig the labels o the bottom two levels of the de Casteljau algorithm. Here to get the correct secod derivative B (t), we must multiply the output of this algorithm by 6 = 3 2. Also, as i Figure 4, we must remember to ormalize the labels alog all the arrows, eve the costats alog the arrows o the bottom two levels, by dividig by b a. 5. Smoothly Joiig Two Bezier Curves. We prefaced our discussio of differetiatio by sayig that we wated to be able to joi two Bezier curves together smoothly. To do so, we eed to calculate the derivatives at their ed poits. Let B(t) be a Bezier curve with cotrol poits P 0,K,P. Substitutig t = a or t = b ito the differetiatio algorithm ad recallig the iterpolatio property at the ed poits, we see that: B(a) = P 0 B(b) = P B (a) = (P P 0 ) / (b a) B (b) = (P P ) / (b a) B (a) = ( )(P 2 2P + P 0 ) / (b a) 2 B (b) = ( )(P 2P + P 2 ) / (b a) 2 I geeral, it follows by iductio o k that the kth derivative at t = a depeds oly o the first k + cotrol poits, ad the kth derivative at t = b depeds oly o the last k + cotrol poits. Suppose the that we are give a Bezier curve P(t) with cotrol poits P 0,K,P ad we wat to costruct aother Bezier curve Q(t) with cotrol poits Q 0,K,Q so that Q(t) meets P(t) ad matches the first r derivatives of P(t) at its ed poit. From the results i the previous paragraph, we fid that the cotrol poits Q 0,K,Q must satisfy the followig costraits: r = 0 : Q 0 = P r = : Q Q 0 = P P Q = P + (P P ) r = 2 : Q 2 2Q + Q 0 = P 2P + P 2 Q 2 = P 2 + 4(P P ) ad so o for higher values of r. Each additioal derivative allows us to solve for oe additioal cotrol poit. Clearly we could go o i this maer solvig for oe cotrol poit at a time. A 4

15 alterative approach for fidig a explicit formula for the cotrol poits {Q k } that avoids all this tedious computatio will be preseted i Lecture 25. Figure 6 illustrates two cubic Bezier curves that meet with matchig first derivatives at their joi. Figure 6: Two cubic Bezier curves -- oe with cotrol poits P 0,P,P 2,P 3 ad the other with cotrol poits Q 0,Q,Q 2,Q 3 -- that meet with matchig first derivatives at their joi. Here Q 0 = P 3 ad Q Q 0 = P 3 P Uiqueess of the Bezier Cotrol Poits. Aother cosequece of the differetiatio algorithm for Bezier curves is the uiqueess of the Bezier cotrol poits of a degree Bezier curve over a fixed parameter iterval. For suppose that P 0,K,P ad P 0,K,P are two distict sets of cotrol poits for the Bezier curve B(t) over the parameter iterval [a,b]. Substitutig t = a ito the differetiatio algorithm ad recallig the iterpolatio property at the ed poits, we have the followig formulas for the derivatives of B(t) at t = a i terms of P 0,K,P ad P 0,K,P : B(a) = P 0 B(a) = P 0 B (a) = (P P 0 ) / (b a) B (a) = (P P0 ) / (b a) B (a) = ( )(P 2 2P + P 0 ) / (b a) 2 B (a) = ( )(P 2 2P + P0 ) / (b a) 2 ad so o up to the th derivative B () (a). I geeral, it follows by iductio o k that B (k) (a) depeds oly o the first k + cotrol poits P 0,K,P k or P 0,K,Pk, ad the formulas for these derivatives are idetical with respect to P 0,K,P k ad P 0,K,Pk. From the formula for the zeroth derivative, we coclude that P 0 = P0 ; from this result ad the formula for the first derivative we coclude that P = P, ad so o. Thus from the formulas for first derivatives, we coclude that P k = Pk, k = 0,K,. Hece the cotrol poits of a degree Bezier curve over a fixed parameter iterval are uique. 5

16 6. Tesor Product Bezier Patches Bezier techology -- the de Casteljau algorithm ad Berstei polyomials -- ca be used to create surfaces as well as curves. A tesor product Bezier patch B(s,t) is a rectagular parametric surface patch -- the image of a plaar rectagular domai [a,b] [c, d] -- that approximates the shape of a rectagular array of cotrol poits {P ij }, i = 0,K,m, j = 0,K,. Coectig cotrol poits with adjacet idices by straight lies geerates a cotrol polyhedro that cotrols the shape of the Bezier patch i much the same way that a Bezier cotrol polygo cotrols the shape of a Bezier curve (see Figures 7). I particular, draggig a cotrol poit pulls the surface patch i the same geeral directio as the cotrol poit. Figure 7: A tesor product Bezier surface with its cotrol polyhedro, formed by coectig cotrol poits with adjacet idices. To costruct a tesor product Bezier patch B(s,t) from a rectagular array of cotrol poits {P ij }, i = 0,K,m, j = 0,K,, let P i (t) be the Bezier curve with cotrol poits P i,0,k,p i,. For each fixed value of t, let B(s,t) be the Bezier curve for the cotrol poits P 0 (t),k, P m (t). The as s varies from a to b ad t varies from c to d, the curves B(s,t) sweep out a surface (see Figures 8,9). This surface is called a tesor product Bezier patch of bidegree (m, ). This costructio suggests the followig evaluatio algorithm for tesor product Bezier patches: first use the de Casteljau algorithm m + times to compute the poits at the parameter t alog the degree Bezier curves P 0 (t),..., P m (t) ; the use the de Casteljau algorithm oe more time to compute the poit at the parameter s alog the degree m Bezier curve with cotrol poits P 0 (t),..., P m (t) (see Figure 20). 6

17 P 3 P 23 P 33 P 03 P 2 P 22 P 02 P 0 P 0 (t) P 00 P P 0 B(s, t) P 2 (t) P (t ) P 2 P 20 P 30 P 3 (t ) Figure 8: Costructio of poits o a bicubic tesor product Bezier surface B(s, t). First the Bezier curves P i (t), i = 0,..., 3 are costructed from the cotrol poits P i0, P i, P i2, P i3. The for a fixed value of t, the Bezier curve B(s, t) is costructed usig the poits P 0 (t), P (t), P 2 (t), P 3 (t) as cotrol poits. As s varies from a to b ad t varies from c to d, these curves sweep out a surface patch -- see Figure 9. P 32 P 3 Figure 9: The bicubic Bezier patch i Figure 7 alog with its cubic Bezier cotrol curves P 0 (t),...,p 3 (t). Notice that oly the boudary cotrol curves are iterpolated by the surface. 7

18 B(s,t) b s s a b s s a b s s a d t P 0 (t) P (t) P 2 (t) d t d t d t d t d t d t d t d t P 00 P 0 P 02 P 0 P P2 P 20 P 2 P 22 Figure 20: The de Casteljau evaluatio algorithm for a biquadratic Bezier patch. The three lower triagles represet Bezier curves i the t directio; the upper triagle bleds these curves i the s directio. d t B (s,t) d t d t b s P 0 (s) P (s) P 2 (s) s a b s s a b s s a b s s a b s s a b s s a b s s a b s s a b s s a P 00 P 0 P 20 P 0 P P2 P 02 P 2 P 22 Figure 2: A alterative versio of the de Casteljau evaluatio algorithm for a biquadratic Bezier patch with the same cotrol poits as i Figure 20. The three lower triagles represet Bezier curves i the s directio, ad the upper triagle bleds these curves i the t directio. Compare to Figure 20. 8

19 Alteratively, istead of startig with the Bezier curves P 0 (t),k, P m (t), we could begi with the Bezier curves P 0 (s),k,p (s), where Pj (s) is the degree m Bezier curve with cotrol poits P 0, j,k,p m, j. Now for each fixed value of s, let B (s,t) be the Bezier curve for the cotrol poits P 0 (s),k,p (s). Agai as s varies from a to b ad t varies from c to d, the curves B (s,t) sweep out a surface, ad we ca use the de Casteljau algorithm to evaluate poits alog this surface (see Figure 2). The surface B (s,t) is exactly the same as the surface B(s,t) because both of these surfaces have the same Berstei represetatio. To compute the Berstei represetatio of the Bezier patch B(s,t), recall that for a fixed value of t, B(s,t) is the Bezier curve with cotrol poits P 0 (t),..., P m (t). Therefore m m B(s,t) = B i (s)pi (t). (6.) i=0 Moreover, P i (t) is the Bezier curve with cotrol poits P i,0,k,p i,, so P i (t) = B j (t) Pi, j. (6.2) j=0 Substitutig Equatio (6.2) ito Equatio (6.) yields the Berstei represetatio m m B(s,t) = B i (s)b j (t)pi, j. (6.3) i=0 j=0 Similarly, B (s,t) = B j (t)pj (s) (6.4) j=0 where m P j (s) = m Bi (s)pi, j, (6.5) i=0 so B m m (s,t) = B i (s)b j (t)pi, j. (6.6) j=0 i=0 Hece B (s,t) = B(s,t) Thus it does ot matter which versio of the de Casteljau algorithm we apply; both costructios geerate the same surface. However, if m <, the the de Casteljau algorithm for B(s,t) is more efficiet, whereas if < m the the de Casteljau algorithm for B (s,t) is more efficiet. 9

20 0 2 2 Tesor product Bezier patches iherit may of the characteristic properties of Bezier curves: they are affie ivariat, lie i the covex hull of their cotrol poits, ad iterpolate their corer cotrol poits (see Exercises 2,3). These properties follow easily from the de Casteljau algorithm for Bezier patches (Figures 20, 2) ad the correspodig properties of Bezier curves. Moreover the boudaries of a tesor product Bezier patch are the Bezier curves determied by their boudary cotrol poits, sice B(a,t) = P 0 (t) B(s,c) = P ad 0 (s) B(b,t) = P m (t) B(s,d) = P (s). It follows that although tesor product Bezier patches do ot geerally iterpolate their cotrol poits, they always iterpolate the four corer cotrol poits P 00,P m0, P 0,P m. There is o kow aalogue of the variatio dimiishig property for tesor product Bezier patches. Thus although Bezier patches typically follow the shape of their cotrol polyhedra, there is o theorem which guaratees that Bezier surfaces do ot oscillate more tha their cotrol poits. To compute the partial derivatives of a Bezier patch, we ca apply our procedure for differetiatig the de Casteljau algorithm for Bezier curves. Cosider Figure 20. We ca compute B / t simply by differetiatig the de Casteljau algorithm for each of the Bezier curves P 0 (t),k, P m (t) at the base of the diagram. Similarly, we ca compute B / s by differetiatig the first upper s level of the diagram ad multiplyig the result by the degree i s (see Figure 22). Symmetric results hold for differetiatig the algorithm i Figure 2: simply reverse the roles of s ad t. The ormal vector N to a Bezier patch B(s,t) is give by settig N = B s B t. b s P 0 (t) b s B t s a b s s a s a P (t ) P 2 (t) 0 P 0 (t) b s B s = 2 _ P (t ) s a P 2 (t) Figure 22: Computig the partial derivatives of a biquadratic Bezier patch by applyig the procedure for differetiatig the de Casteljau algorithm for Bezier curves. To fid B / t (left) simply differetiate the de Casteljau algorithm for each of the Bezier curves P 0 (t), P (t), P 2 (t). To fid B / s (right) simply differetiate the first level of the de Casteljau algorithm i s ad multiply the result by the degree i s. 20

21 Summary The fudametal algorithm for Bezier curves ad surfaces is the de Casteljau evaluatio algorithm, a algorithm based o repeated liear iterpolatio (Figure 23(a)). The decompositio of evaluatio ito successive idetical liear iterpolatio steps makes the evaluatio algorithm easy to differetiate (Figure 23(b)), allowig us to compute derivatives as well as poits alog a Bezier curve usig a variat of the de Casteljau algorithm. We applied the differetiatio algorithm to derive costraits o the cotrol poits to guaratee that two Bezier curves meet smoothly at their joi. We also used the differetiatio algorithm to establish the uiqueess of the degree Bezier cotrol poits over a fixed parameter iterval. b t b t B(t) b t b t b t P 0 P P2 P 3 b t B (t ) = 3 _ b t P 0 P P2 P 3 (a) Evaluatio (b) Differetiatio Figure 23: The de Casteljau algorithm for (a) evaluatio ad (b) differetiatio. The Berstei represetatio provides a alterative explicit polyomial represetatio for Bezier curves. If B(t) is the Bezier curve over the iterval [a,b] with cotrol poits P 0,K,P, the the Berstei represetatio is give by settig B(t) = B k (t)pk a t b k=0 B k (t) = k ( ) ()k () k (b a) k = 0,K,. Bezier curves have the followig importat geometric properties:. Affie Ivariace 2. Covex Hull Property 3. Variatio Dimiishig Property 4. Iterpolate their First ad Last Cotrol Poits Tesor product Bezier patches are a extesio of the Bezier represetatio from curves to surfaces. Bezier patches ca be evaluated either by extedig the de Casteljau algorithm to surfaces (Figure 24) or by ivokig the Berstei represetatio i two variables (Equatio 7.2)). 2 (7.)

22 B(s,t) b s s a b s s a b s s a d t P 0 (t) P (t) P 2 (t) d t d t d t d t d t d t d t d t P 00 P 0 P 02 P 0 P P2 P 20 P 2 P 22 Figure 24: The de Casteljau evaluatio algorithm for a biquadratic Bezier patch. The three lower triagles represet Bezier curves i the t directio; the upper triagle bleds these curves i the s directio. The Berstei represetatio of a tesor product Bezier surface B(s,t) of bidegree (m, ) with cotrol poits {P ij } ad domai [a,b] [c, d] is give by m m B(s,t) = B i (s)b j (t)pi, j. (7.2) i=0 j=0 where B i m (s),b j (t) are the Berstei basis fuctios over the itervals [a,b] ad [c, d]. Bezier patches are affie ivariat, lie i the covex hull of their cotrol poits, ad iterpolate the boudary Bezier curves defied by their boudary cotrol poits. There is, however, o kow aalogue of the variatio dimiishig property for Bezier surfaces. Exercises:. Give a example to show that Bezier curves deped o the order of their cotrol poits P 0,..., P -- that is, if we chage the order but ot the locatio of the cotrol poits, we may geerate a differet Bezier curve. 2. Prove that the Bezier curve for the cotrol poits P,...,P 0 is the same as the Bezier curve with the cotrol poits P 0,..., P but with opposite orietatio. 22

23 3. Show that a Bezier curve collapses to a sigle poit P if ad oly if all the Bezier cotrol poits are located at P. 4. Cosider the Berstei basis fuctios B k (t) = ( k ) ()k () k (b a) k = 0,K,. Show that: a. B k (t) k=0 b. 0 B k (t) for a t b c. B k (a) =δk,0 ad B k (b) = δk, 5. Usig the results i Exercise 4, show that: a. Bezier curves are traslatio ivariat. b. Bezier curves iterpolate their first ad last cotrol poits. 6. Prove that CovexHull( P 0,K,P )= c k P k c k ad c k 0 k=0 k=0. 7. Usig the results of Exercises 4 ad 6, show that Bezier curves lie i the covex hull of their cotrol poits. 8. Cosider a Bezier curve B(t) = B k (t)pk k=0 where B k (t) = ( k ) ()k () k (b a) k = 0,K, are the Berstei basis fuctios. Show that: a db k (t) dt = B k (t) Bk (t) b a. b. B (t) = P B k (t) k+ P k. k=0 b a 23

24 9. Let P i 0 Li (t) deote the degree polyomial curve that iterpolates the cotrol poits P i 0,...,P i at the parameter values t i0,...,t i -- that is, P i 0 Li (t ik ) = P ik, k = 0,K,. Show that: a. P 0 (t) = t t t t 0 P 0 + t t 0 t t 0 P b. P 02 (t) ad P 023 (t) ca be built usig the dyamic programmig algorithm (Neville s algorithm) preseted i Figure 25. c. Explai how to exted the dyamic programmig algorithm i Figure 25 to P 0L (t). P 023 (t) t 3 t t t 0 P 02 (t) P 02 (t) P 23 (t) t 2 t t t 0 t 2 t t t 0 t 3 t t t t t P 0 (t) t t 0 t 2 t P 2 (t) t t P 0 (t) t t t t 0 t 2 t P 2 (t) t t t 3 t P 23 (t) t t 2 P 0 P P 2 P 0 P P2 P 3 Figure 25: Neville s algorithm for computig poits o the polyomial curve that iterpolates the poits at the base of the diagram. The value at each ode must be ormalized i the usual maer by dividig by the sum of the labels alog the arrows that eter the ode. 0. Cosider a Bezier curve with cotrol poits P 0,...,P. Show that: a. the lie segmet P 0 P is taget to the curve at P 0 ; b. the lie segmet P P is taget to the curve at P.. Give poit ad derivative data (R 0,v 0 ),...,(R,v ), explai how to place Bezier cotrol poits to geerate a smooth piecewise cubic curve to iterpolate this data. 2. Show that tesor product Bezier patches a. are affie ivariat; b. lie i the covex hull of their cotrol poits. 3. Show that every tesor product Bezier patch iterpolates a. the four Bezier curves defied by the boudary cotrol poits; b. the four corer cotrol poits. 24