Parametric curves. Reading. Parametric polynomial curves. Mathematical curve representation. Brian Curless CSE 457 Spring 2015

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1 Readig Required: Agel , 0.5., , 0.9 Parametric curves Bria Curless CSE 457 Sprig 05 Optioal Bartels, Beatty, ad Barsy. 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. Mathematical curve represetatio Parametric polyomial curves Explicit: y =f (x ) what if the curve is t a fuctio, e.g., a circle? We ll use parametric curves, Q (u )=(x (u ), y (u )), where the fuctios are all polyomials i the parameter. Implicit: g (x, y ) = 0 Advatages: easy (ad efficiet) to compute ifiitely differetiable (all derivatives above the th derivative are zero) Parametric: Q (u ) = (x (u ), y (u )) For the circle: x (u ) = cos u y (u ) = si u We ll also assume that u varies from 0 to. Note that we ll focus o D curves, but the geeralizatio to 3D curves is completely straightforward. 3 4

2 de Casteljau s algorithm de Casteljau s algorithm, cot d We will ow build a curve geometrically, ad the show how it is a parametric polyomial curve. Recursive otatio: We start with cotrol poits {V, V, V 3, V 4 } ad coect them together to mae a cotrol polygo. We the recursively subdivide: V 0 What is the equatio for? What if u = 0? What if u =? 5 6 Fidig Q(u) Fidig Q(u) (cot d) Let s solve for Q(u): 0 (- ) 0 (- ) (- ) 3 V u V uv V u V uv V u V uv I geeral, i i Qu ( ) u ( u) Vi i 0 i where choose i is: 0 (- ) 0 (- ) V u V uv V u V uv ( ) (- ) 0 Qu uv uv! i ( i)! i! This defies a class of curves called Bézier curves. We ca also write this as: (- u)[(- u) V0 uv] u[(- u) V uv] (- u)[(- u){(- u) V0 uv} u{(- u) VuV}] (- u) V0 3 u(- u) V3 u (- u) V u V3 where the are the Berstei polyomials: 3 V0 ( 3V0 3 V) u(3v0 6V3 V) u ( V0 3V3 V V3) u 3 V0, x ( 3V0, x 3 V, x) u(3v0, x 6V, x 3 V, x) u ( V0, x 3V, x 3 V, x V3, x) u 3 V0, y ( 3V0, y 3 V, y ) u(3v0, y 6V, y 3 V, y ) u ( V0, y 3V, y 3V, y V3, y ) u 7 bi ( u) u ( u) i i i Q: If we have cotrol poits, what is the polyomial order of the curve? 8

3 Berstei polyomials Displayig Bézier curves For degree 3, the Berstei polyomials are: How could we draw oe of these thigs? Useful properties (for Berstei polyomials of ay degree) o the iterval [0,]: The sum of all four is exactly for ay u. (We say the curves form a partitio of uity ). Each polyomial has value betwee 0 ad. These together imply that the curve is geerated by covex combiatios of the cotrol poits ad therefore lies withi the covex hull of those cotrol poits. The covex hull of a poit set is the smallest covex polygo (i D) or polyhedro (i 3D) eclosig the poits. I D, thi of a strig looped aroud the outside of the poit set ad the pulled tightly aroud the set. 9 0 Curve desiderata Local cotrol Bézier curves offer a fairly simple way to model parametric curves. But, let s cosider some geeral properties we would lie curves to have 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! We d lie to have local cotrol, that is, have each cotrol poit affect some well-defied eighborhood aroud that poit.

4 Iterpolatio Cotiuity Bézier curves are approximatig. The 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 wat our curve to have cotiuity: there should t be ay abrupt chages as we move alog the curve. 0 th order cotiuity would mea that curve does t jump from oe place to aother. We d lie to have a curve that is iterpolatig, that is, that always passes through every cotrol poit. We ca also loo at derivatives of the curve to get higher order cotiuity. 3 4 st ad d Derivative Cotiuity C (Parametric) Cotiuity First order cotiuity implies cotiuous first derivative: dq( u) Q'( u) du Let s thi 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? I geeral, we defie C cotiuity as follows: Qu ( ) is C cotiuous i ( i ) dqu ( ) Q ( u) i is cotiuous for 0 i du Note: these are ested degrees of cotiuity: iff C - : C 0 : Secod order cotiuity meas cotiuous secod derivative: dqu ( ) Q''( u) du What is the ituitive meaig of this derivative? C, C : C 3, C 4, : 5 6

5 Bézier curves splies Esurig C 0 cotiuity 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.) Suppose we have a cubic Bézier defied by (V 0,V,V,V 3 ), ad we wat to attach aother curve (W 0,W,W,W 3 ) to it, so that there is C 0 cotiuity at the joit. What costrait(s) does this place o (W 0,W,W,W 3 )? Istead, we ll splice together a curve from idividual Béziers segmets, i particular, cubic Béziers. We call these curves splies. The primary cocer whe splicig cuves together is gettig good cotiuity at the edpoits where they meet 7 8 The C 0 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: We ca expad the terms i u ad rearrage to get: What the is the first derivative whe evaluated at each edpoit, u = 0 ad u =? We call this curve a splie. The edpoits of the Bezier segmets are called joits. All other Bezier poits (i.e., ot edpoits) are called ier Bezier poits; these poits are geerally ot iterpolated. I the aimator project, you will costruct such a curve by specifyig all the Bezier cotrol poits directly. 9 0

6 Esurig C cotiuity The C Bezier splie Suppose we have a cubic Bézier defied by (V 0,V,V,V 3 ), ad we wat to attach aother curve (W 0,W,W,W 3 ) to it, so that there is C cotiuity at the joit. How the could we costruct a curve passig through a set of poits P 0 P? What costrait(s) does this place o (W 0,W,W,W 3 )? We ca specify the Bezier cotrol poits directly, or we ca devise a scheme for placig them automatically Catmull-Rom splies Catmull-Rom to Beziers If we set each derivative to be oe half of the vector betwee the previous ad ext cotrols, we get a Catmull-Rom splie. We ca write the Catmull-Rom to Bezier trasformatio as: This leads to: 3 4

7 Edpoits of Catmull-Rom splies Tesio cotrol We ca see that Catmull-Rom splies do t iterpolate the first ad last cotrol poits. We ca give more cotrol by exposig the derivative scale factor as a parameter: By repeatig those cotrol poits, we ca force iterpolatio. The parameter cotrols slacess. Catmull-Rom uses = /. Here s a example with =3/. 5 6 d derivatives at the edpoits Esurig C cotiuity Fially, we ll wat to develop C splies. To do this, we ll eed secod derivatives of Bezier curves. Taig the secod derivative of Q (u ) yields: Suppose we have a cubic Bézier defied by (V 0,V,V,V 3 ), ad we wat to attach aother curve (W 0,W,W,W 3 ) to it, so that there is C cotiuity at the joit. What costrait(s) does this place o (W 0,W,W,W 3 )? 7 8

8 Buildig a complex splie B-splies 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. Here is the completed B-splie. What are the Bézier cotrol poits, i terms of the de Boor poits? These are called B-splies. The startig set of poits are called de Boor poits B-splies to Beziers Edpoits of B-splies We ca write the B-splie to Bezier trasformatio as: 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. 3 3

9 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: ( u) y( u) tu ( ) xu ( ) Where is a variable you wat to aimate. We ca thi of the result as a fuctio: ( t) I geeral, you have to apply some costraits to mae sure that (t ) actually is a fuctio Closig the loop What if we wat a closed curve, i.e., a loop? With Catmull-Rom ad B-splie curves, this is easy: Drawig Bézier curves, revisited Let s retur to the questio of how to draw Bezier curves, the buildig bloc for splies. Cosider a set of Bézier cotrol poits are arraged as follows: 35 How may lie segmets do you really eed to draw? It would be ice if we had a adaptive algorithm, that would tae ito accout flatess. DisplayBezier( V0, V, V, V3 ) begi if ( FlatEough( V0, V, V, V3 ) ) Lie( V0, V3 ); else somethig; ed; 36

10 Subdivide ad coquer Testig for flatess DisplayBezier( V0, V, V, V3 ) begi if ( FlatEough( V0, V, V, V3 ) ) Lie( V0, V3 ); else Subdivide(V[ ]) L[ ], R[ ] DisplayBezier( L0, L, L, L3 ); DisplayBezier( R0, R, R, R3 ); ed; 37 Compare total legth of cotrol polygo to legth of lie coectig edpoits: 38 Reparameterizatio Arc legth parameterizatio We have so far bee cosiderig parametric cotiuity, derivatives w.r.t. the parameter u. This form of cotiuity maes sese particularly if we really are describig a particle movig over time ad wat its motio (e.g., velocity ad acceleratio) to be smooth. 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: But, what if we re thiig 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? We call this a arc legth parameterizatio. We ca re-write the equal step requiremet as: Looig at very small steps, we fid: 39 40

11 G (Geometric) Cotiuity Now, we defie geometric G cotiuity as follows: Qs ( ) is G cotiuous iff i () i dqs ( ) Q ( s) is cotiuous for 0 i i ds Where Q (s ) is parameterized by arc legth. The first derivative still poits alog the taget, but its legth is always. G cotiuity is usually a weaer costrait tha C cotiuity (e.g., speed alog the curve does ot matter). G Cotiuity (cot d) The secod derivative ow has a specific geometric iterpretatio. First, the osculatig circle at a poit o a curve ca be defied based o the limit behavior of three poits movig toward each other: c r The secod derivative Q (s ) the has these properties: Q( s) ( s) Q( s) c( s) Q( s) rs ( ) where r (s ) ad c(s ) are the radius ad ceter of O (s ), respectively, ad (s ) is the curvature of the curve at s. 4 4 Ratioal polyomial curves Remarably, parametric polyomial curves caot represet somethig as simple as a circle! BUT, ratios of polyomials ca. We ca write these i terms of homogeeous coordiates, which we the ormalize: Ratioal polyomial curves (cot d) What do we get for the followig curve? u Q D( u) u u xu ( ) au 0 yu 0 Q D( u) ( ) b u wu ( ) cu 0 Normalize cu 0 Q D ( u ) au cu 0 0 bu cu 0 0 The equatios above describe a ratioal Bézier curve. It ca be represeted i terms of cotrol poits, but ow we add the homogeous dimesio. So for a D curve, we have cotrol poits with three compoets (lofted up ito 3D), where the homogeous compoet ca be somethig other tha. Q: How does Illustrator represet a circle? 43 44

12 NURBS I geeral, we ca splie together ratioal Bézier curves, to get thigs lie ratioal B-splies. Aother thig we ca do is vary the rage of u so that it is ot always [0..] i each Bézier segmet of a splie. E.g, it could be [0..] i oe segmet ad the [0..] i the ext. Summary What to tae 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 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. The u-rage affects placemet of cotrol poits. The result is a o-uiform splie. A very commo type of splie is a No-Uiform Ratioal B-Splie or NURBS. (The B i B-splie techically stads for Basis. ) 45 46