Reading. Parametric curves. Mathematical curve representation. Curves before computers. Required: Angel , , , 11.9.

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1 Readig Required: Agel.-.3,.5.,.6-.7,.9. Optioal Parametric curves Bartels, Beatty, ad Barsky. A Itroductio to Splies for use i Computer Graphics ad Geometric Modelig, 987. Fari. Curves ad Surfaces for CAGD: A Practical Guide, 4th ed., 997. Curves before computers Mathematical curve represetatio he loftsma s splie : log, arrow strip of wood or metal shaped by lead weights called ducks gives curves with secod-order cotiuity, usually Used for desigig cars, ships, airplaes, etc. Explicit y=f(x) what if the curve is t a fuctio, e.g., a circle? Implicit g(x,y) = But curves based o physical artifacts ca t be replicated well, sice there s o exact defiitio of what the curve is. Aroud 96, a lot of idustrial desigers were workig o this problem. oday, curves are easy to maipulate o a computer ad are used for CAD, art, aimatio, Parametric (x(u),y(u)) For the circle: x(u) = cos πu y(u) = si πu 3 4

2 Parametric polyomial curves We ll use parametric curves, Q(u)=(x(u),y(u)), where the fuctios are all polyomials i the parameter. de Casteljau s algorithm Recursive iterpolatio: x( u) = y( u) = k k= k k= k a u k b u Advatages: easy (ad efficiet) to compute ifiitely differetiable We ll also assume that u varies from to. What if u=? What if u=? 5 6 de Casteljau s algorithm, cot d Recursive otatio: Fidig Q(u) Let s solve for Q(u): V = (- u) V + uv V = (- u) V + uv V = (- u) V + uv 3 V = (- u) V + uv V = (- u) V + uv V What is the equatio for? Qu ( ) = (- uv ) + uv = (- u)[(- u) V + uv ] + u[(- u) V + uv ] = (- u)[(- u){(- u) V + uv } + u{(- u) V + uv }] = (- u) V + 3 u(- u) V + 3 u (- u) V + u V 3 7 8

3 Fidig Q(u) (cot d) I geeral, i i Qu ( ) = u( u) Vi i= i where choose i is:! = i ( i)! i! his defies a class of curves called Bézier curves. What s the relatioship betwee the umber of cotrol poits ad the degree of the polyomials? Berstei polyomials he coefficiets of the cotrol poits are a set of fuctios called the Berstei polyomials: For degree 3, we have: b ( u) = ( u) b u u u b u u u b u u 3 ( ) = 3 ( ) ( ) = 3 ( ) 3 3( ) = Qu ( ) = bi( uv ) i i= 9 Useful properties o the iterval [,]: each is betwee ad sum of all four is exactly (a.k.a., a partitio of uity ) hese together imply that the curve lies withi the covex hull of its cotrol poits. Displayig Bézier curves Subdivide ad coquer How could we draw oe of these thigs? It would be ice if we had a adaptive algorithm, that would take ito accout flatess. DisplayBezier( V, V, V, V3 ) begi if ( FlatEough( V, V, V, V3 ) ) Lie( V, V3 ); else somethig; ed; DisplayBezier( V, V, V, V3 ) begi if ( FlatEough( V, V, V, V3 ) ) Lie( V, V3 ); else Subdivide(V[]) L[], R[] DisplayBezier( L, L, L, L3 ); DisplayBezier( R, R, R, R3 ); ed;

4 estig for flatess Curve desiderata Bézier curves offer a fairly simple way to model parametric curves. But, let s cosider some geeral properties we would like curves to have Compare total legth of cotrol polygo to legth of lie coectig edpoits: V V + V V + V V 3 V V 3 < + ε 3 4 Local cotrol Oe problem with Béziers is that every cotrol poit affects every poit o the curve (except the edpoits). Movig a sigle cotrol poit affects the whole curve! Iterpolatio Bézier curves are approximatig. he curve does ot (ecessarily) pass through all the cotrol poits. Each poit pulls the curve toward it, but other poits are pullig as well. We d like to have a curve that is iterpolatig, that is, that always passes through every cotrol poit. We d like to have local cotrol, that is, have each cotrol poit affect some well-defied eighborhood aroud that poit. 5 6

5 Cotiuity We wat our curve to have cotiuity: there should t be ay abrupt chages as we move alog the curve. th order cotiuity would mea that curve does t jump from oe place to aother. st ad d Derivative Cotiuity First order cotiuity implies cotiuous first derivative: dq( u) Q'( u) = du Let s thik of u as time ad Q(u) as the path of a particle through space. What is the meaig of the first derivative, ad which way does it poit? We ca also look at derivatives of the curve to get higher order cotiuity. Secod order cotiuity meas cotiuous secod derivative: d Q( u) Q''( u) = du What is the ituitive meaig of this derivative? 7 8 C (Parametric) Cotiuity I geeral, we defie C cotiuity as follows: Qu ( ) is C cotiuous i ( i ) dqu ( ) Q ( u) = i is cotiuous for i du Note: these are ested degrees of cotiuity: C - : C : iff Reparameterizatio We have so far bee cosiderig parametric cotiuity, derivatives w.r.t. the parameter u. his form of cotiuity makes sese particularly if we really are describig a particle movig over time ad wat its motio (e.g., velocity ad acceleratio) to be smooth. But, what if we re thikig oly i terms of the shape of the curve? Is the parameterizatio actually itrisic to the shape, i.e., is it the case that a shape has oly oe parameterizatio? C, C : C 3, C 4, : 9

6 Arc legth parameterizatio We ca reparameterize a curve so that equal steps i parameter space (we ll call this ew parameter s ) map to equal distaces alog the curve: We call this a arc legth parameterizatio. We ca re-write the equal step requiremet as: Lookig at very small steps, we fid: [ ] Qs ( ) Δ s= s s= arclegthqs ( ), Qs ( ) s s arclegth Q s s s [ ( ), Q( s) ] = [ ] arclegth Q( s ), Q( s ) dq( s) lim = = s s ds G (Geometric) Cotiuity Now, we defie geometric G cotiuity as follows: Qs ( ) is G cotiuous iff i () i dqs ( ) Q ( s) = i is cotiuous for i ds where Q(s) is parameterized by arc legth he first derivative still poits alog the taget, but its legth is always. he secod derivative poits to the ceter of the radius of curvature. G cotiuity is usually a weaker costrait tha C cotiuity (e.g., speed alog the curve does ot matter). We ll focus o C (i.e., parametric) cotiuity of curves for the remaider of this lecture. Bézier curves splies Bézier curves have C-ifiity cotiuity o their iteriors, but we saw that they do ot exhibit local cotrol or iterpolate their cotrol poits. It is possible to defie poits that we wat to iterpolate, ad the solve for the Bézier cotrol poits that will do the job. But, you will eed as may cotrol poits as iterpolated poits -> high order polyomials -> wiggly curves. (Ad you still wo t have local cotrol.) Esurig C cotiuity Suppose we have a cubic Bézier defied by (V,V,V,V 3 ), ad we wat to attach aother curve (W,W,W,W 3 ) to it, so that there is C cotiuity at the joit. C : Q () = Q () V What costrait(s) does this place o (W,W,W,W 3 )? W Istead, we ll splice together a curve from idividual Béziers segmets, i particular, cubic Béziers. We call these curves splies. he primary cocer whe splicig cuves together is gettig good cotiuity at the edpoits where they meet 3 4

7 he C Bezier splie st derivatives at the edpoits How the could we costruct a curve passig through a set of poits P P? For degree 3 (cubic) curves, we have already show that we get: 3 3 Qu ( ) = (- u) V + 3 u(- u) V+ 3 u(- uv ) + uv 3 We ca expad the terms i u ad rearrage to get: 3 Qu ( ) = ( V + 3V 3 V + V3) u + (3V 6V + 3 V ) u + ( 3V + 3 V ) u + V What the is the first derivative whe evaluated at each edpoit, u= ad u=? Q () = We call this curve a splie. he edpoits of the Bezier segmets are called joits. Q () = I the aimator project, you will costruct such a curve by specifyig all the Bezier cotrol poits directly. 5 6 Esurig C cotiuity Suppose we have a cubic Bézier defied by (V,V,V,V 3 ), ad we wat to attach aother curve (W,W,W,W 3 ) to it, so that there is C cotiuity at the joit. he C Bezier splie How the could we costruct a curve passig through a set of poits P P? C : Q () = Q () V W ' ' C : Q () = Q () V W What costrait(s) does this place o (W,W,W,W 3 )? We ca specify the Bezier cotrol poits directly, or we ca devise a scheme for placig them automatically 7 8

8 Catmull-Rom splies If we set each derivative to be oe half of the vector betwee the previous ad ext cotrols, we get a Catmull-Rom splie. his leads to: V = P V= P+ 6 ( P - P) V = P - 6 ( P3 - P) V = P 3 Catmull-Rom to Beziers We ca write the Catmull-Rom to Bezier trasformatio as: V 6 P V 6 = P V 6 6 P V3 6 P3 V=MCatmull-RomP 9 3 Edpoits of Catmull-Rom splies We ca see that Catmull-Rom splies do t iterpolate the first ad last cotrol poits. By repeatig those cotrol poits, we ca force iterpolatio. esio cotrol We ca give more cotrol by exposig the derivative scale factor as a parameter: V = P V = + τ P 3 ( P - P) V = τ P - 3 ( P3 - P) V = P 3 he parameter τ cotrols the tesio. Catmull-Rom uses τ = /. Here s a example with τ =3/. 3 3

9 d derivatives at the edpoits Fially, we ll wat to develop C splies. o do this, we ll eed secod derivatives of Bezier curves. akig the secod derivative of Q(u) yields: Q () = 6( V - V+ V) = -6[( V- V) + ( V- V)] Q () = 6( V- V + V3) = -6[( V - V ) + ( V - V )] 3 Esurig C cotiuity Suppose we have a cubic Bézier defied by (V,V,V,V 3 ), ad we wat to attach aother curve (W,W,W,W 3 ) to it, so that there is C cotiuity at the joit. C : QV() = QW() ' ' C : QV() = QW() '' '' C : Q () = Q () V What costrait(s) does this place o (W,W,W,W 3 )? W Buildig a complex splie Istead of specifyig the Bézier cotrol poits themselves, let s specify the corers of the A-frames i order to build a C cotiuous splie. B-splies Here is the completed B-splie. What are the Bézier cotrol poits, i terms of the de Boor poits? hese are called B-splies. he startig set of poits are called de Boor poits. V = [ B + B ] + [ B + B ] = B + B + B V = B + B V = B + B V = B + B + B

10 B-splies to Beziers We ca write the B-splie to Bezier trasformatio as: V 4 B 4 V B = V 4 6 B V 4 3 B3 Edpoits of B-splies As with Catmull-Rom splies, the first ad last cotrol poits of B-splies are geerally ot iterpolated. Agai, we ca force iterpolatio by repeatig the edpoits twice. V=MB-splieB Closig the loop What if we wat a closed curve, i.e., a loop? With Catmull-Rom ad B-splie curves, this is easy: Curves i the aimator project I the aimator project, you will draw a curve o the scree: Q( u) = ( x( u), y( u) ) You will actually treat this curve as: θ ( u) = y( u) tu ( ) = xu ( ) Where θ is a variable you wat to aimate. We ca thik of the result as a fuctio: θ( t) I geeral, you have to apply some costraits to make sure that θ(t) actually is a fuctio. 39 4

11 Wrappig Oe of the extra credit optios i the aimator project is to implemet wrappig so that the aimatio restarts smoothly whe loopig back to the begiig. his is a lot like makig a closed curve: the calculatios for the θ -coordiate are exactly the same. he t-coordiate is a little trickier: you eed to create phatom t-coordiates before ad after the first ad last coordiates. Summary What to take home from this lecture: Geometric ad algebraic defiitios of Bézier curves. Basic properties of Bézier curves. How to display Bézier curves with lie segmets. Meaigs of C k cotiuities. Geometric coditios for cotiuity of cubic splies. Properties of B-splies ad Catmull-Rom splies. Geometric costructio of B-splies ad Catmull-Rom splies. How to costruct closed loop splies. 4 4