4: Parametric curves and surfaces

Transcription

1 4: Parametric curves and surfaces Lecture 1: Euclidean, similarity, affine and projective transformations omogeneous coordinates and matrices Coordinate frames Perspective projection and its matri representation Lecture : Vanishing points orizons Applications of projective transformations Lecture 3: Conveity of point-sets, conve hull and algorithms Conics and quadrics, implicit and parametric forms, computation of intersections Lecture 4: Interpolating and Approimating Splines: Cubic splines, Bezier curves, B-splines 1

2 Introduction 41 In the last lecture we saw that curves can be represented conveniently in parametric form Eg in D and we saw some regular curves like ellipses, and so on In this lecture we introduce how to build irregular shaped curves in a piecewise fashion out of splines Splines are widely used in graphics and CAD indeed one of the pioneers of their computational use, Pierre Bézier, developed his ideas while working for Renault in the 1970 s Eample of an interpolating spline 4 An idea of what were are about can be gained by considering the following problem ow should one specify the path of a road through a number of villages Roads made up of separate functions Continuous Smooth! " Cont nd d? # $% # $% These are interpolating splines, because they actually go through the points Later we will see approimating splines, where the points are control points, which define the shape of the curve, but do not necessary lie on the curve Other constraints might be applied Eg, need to have particular orientation to meet other roads, rivers at right angles

3 Parameterization 43 Convenient to use as a global parameter along the entire curve, and as a local parameter running from 0 to 1 between points That is, within spline &, the local parameter is )( $ ' $*( $ Polynomials provide usable functions for the splines But what degree to use? Linear? a Roman Road spline, so not smooth! Quadratic? let s see P 0 P 1 P i i P p0 p1 p i p i+1 i+1 P m Quadratic splines? 44 Think just with the + part of, Using the global parameterization $5;:< / > !5;:< 758?@:< This suggests: Choose gradient C at D Use gradient and, to fit quadratic This fies gradient at D Use gradient and >, to fit net and so on Then repeat for part - / Using the local parameterization 3 4 > 465A7B5A: So for a continuous, smooth quadratic spline simply choose of gradient at the first (or any one) point, and the entire curve is fied y y 0 y 1 Too sensitive to choice of gradient Likely to oscillate 1st choice nd choice p p p Gradients equal 3

4 Cubic splines 45 Now consider just the local parameterization of the & th cubic spline sequence Again, we just look at similar equations are written for but you should realize that the 4, 7, etc are different 4 5A7 E5;: 5GF $ I KJL 1N 4O95A7P95A:Q95GFO KJL RST7P 1N RSBÜ7P95V?@:QW58XYFO F Re-arranging, and writing 3Z 7P R] :Q X/ %@ ^(_@% (8?ORS/(ARS (`?$ % (a!5vr 5VR We make the derivative equal at the junction of two splines The is called ermite interpolation So, for a b segment spline, there are b 5c degrees of freedom ow are the derivatives specified? 46 1 By eternal constraints (eg, the perpendicular road) Automatically: For eample, fit parabola to ed$,,, and use value of gradient at as the gradient R Also need to choose end point gradients: eg, set R cj and RSfGTJ 3 But, we could ask the nd derivatives to match as well This is a natural cubic spline 4

5 g b f p g b I g b Natural cubic splines 47 Try a counting argument to show that we have sufficient dof Again we think only about the cubic: the cubic is similiar Each of the b segments between D and D requires 4 parameters to specify the cubic in UNKNOWNS At each of b (U internal points on [\ Bḧi iz jjc Bḧi ed$ [\ Bḧ jjc ed$ # [\ kül jjc I g (h\ CONSTRAINTS At the end points Now, g b jjc ḧ@ so we can do it! f [\ Bḧ f I? CONSTRAINTS The etra two degrees of freedom are setting the second derivatives at the end points: l jjc BTJ l f 1N TJ I? CONSTRAINTS Computing Natural Cubic Splines 48 Don t solve for the cubic splines by inverting a g b b matri Instead Basic equations Leads to 4 5G7 E5A: 5AF $ ?@: E58XYF / 7 R #?i:q]58qyfor :Qh X/ %@(ti- (A?ORS/(8RS Put in s J and, into the first two FOh (`?$ %i(t@% 58RS<5uRS on the left Now equate the nd derivs ( v for the & (U\ th section, wcj for the & th I I g?@:qed$"5gqyfoed$?i:q?$yx/ %@<(tied$z (G?OR]{d$^(8RS{ 5 q} (`?< -i<(t@ed$z!58r]ed$"58r]~?<rx/ %i!(ti- (A?ORS/(8R] R]ed$5 R]W5uRS X} -i(tied$z 5

6 ˆ Computing Natural Cubic Splines, ctd 49 To repeat At the first point, C # KJL TJ R ed$ 5 g R 5uR TX/ % (a ed$ : So TJ and X/ % (a (8?OR (AR cj, so?or 5uR ÜX/ % (t Similarly for the end section l f [\ TJ, so?@: f 5GqYF f TJ from which R f d$!5u?or f ÜX/ % f (a f d$[ Gather all this together? g g g? ƒ R] R R R]f d$ RSf ƒ X/ %>*(t@0 X/ % (t@0 X/ % (tc1 X} -if,(t@f d X} -if,(t@f d$ ƒ Computing Natural Cubic Splines, ctd 410 Use row ops to simplify further to Ĉ ˆ$ ˆLf d$ ƒ R] R R R]f d$ RSf ƒ f ƒ R f R f 4 7 R Then back substitute to find, d$ and so in reverse order aving got the s substitute in to :Q X/ %@ ^(_@% (8?ORS/(ARS FO (`?$ %i*(ai-!5vrs95vrs 6

7 Approimating curves as opposed to interpolating 411 In D %Š" 1" %Š each a polynomial in Š Local control of curve Important eamples are Bézier curves and B-splines Used in graphics, CAD, drawing packages Bézier curves 41 Bezier curves were one of the earliest spline curves The curves were originally devised using the de Casteljau construction First consider a Bezier curve of degree 1, between D and D, [ (tš Œ 5;Š/ Now consider three control points, D and set Then Ž [ (tš Œ!5;Š/ Ž [ (tš Œ"5;Š/ [ (tš Ž 5tŠ Ž 1#Œ follows a quadratic Bezier curve s [(aš!5u?išk [ (_Š" Œ5GŠ 0 a 0 0 u (1 u) 1 a 1 1 7

8 Ž Carrying on 413 Now consider four control points, D l and set Ž 1(_Š" Œ 5tŠ} Ž 1(_Š" Œ!5tŠ} 1(_Š" Œ 5tŠ} 1(_Š" Ž 5GŠ Ž 1(_Š" Ž "5GŠ Ž 1(_Š" 5GŠ Then u 0 follows a cubic Bezier curve 0 1 u 3 u 1 %Š" B - %Š %Š" 1 z [(oš" Œ 5 X3Š [(Š 5 X3Š 1/(Š 5)Š/z T % Y J Šš Bézier curves of degree %Š [(aš 58X@Šk [ (aš ZZ 5AX3Š N [ (tš Œ 5;Š % Y J Šš Properties jjc B! and [\, so that a Bézier curve interpolates its end control points Ÿ EœŒiž cx/ ^( œœ Geometric significance? The curve lies inside the conve hull of its control points The control points need not be planar Is it always smooth It can have inflections It can self-cross 8

9 Blending functions actually Bernstein basis functions 415 %Š" B [(aš 58X3Š [ (aš ZZ 58X@Š N [ (tš Œ 5;Š One can think of the coefficients of the control points as blending functions These are symmetric under Š 1 (tš" 1 ( 1 - u ) 3 u 3 3 u ( 1 - u ) 3 u ( 1 - u ) Bézier curves of degree 416 The de Casteljau construction can be epressed as eg for ² %Š +ª < TX -Š Œ < (cubic Bézier): %Š ± ²³ µ [(aš -Š 1(_Š" -Š X/ [ (tš" Š -Š X/ [ (tš" Š -Š Š Note that (what is the geom significance?) d Š Y $¹ J Šš +ª -Š Šo5º [ (tš 1 Y $¹ -Š»hJ % Y J Šš 9

10 ± ± µ µ ± µ 5 µ ± µ atri form 417 Another representation is %Š" B %Š %Š" %Š" -Š 1! % %Š [ (tš ¾Š Š Š/[ (ÀX 5`X (E % Collecting these together: %Š ¾ Š Š Š/ ÃTÄWÅYÆ%Ç-ÈlÆ-ÉÊ J J J (ÁX X J J X (Áq X J ( X (ÀX ƒ ± % Ä ÅYÆ%Ç-ÈlÆ-É is the Bezier basis matri for order 3 Eample 418 Where is jj<ëìo?? Í3 Î? g Î/Ï z XÀXEN J J J (ÀX X J J X (Àq X J (E X (ÀX k Î!5 X Î ƒ "5! X Î Î So, it is a centre-weighted centroid 10

11 Ò Parametric surface (Tensor Product) 419 We would like to represent a surface using control points and blending functions, in the same manner as a curve %Š - -Š %Š" PÐ/ %Š" 1 z %Š For a surface, we need two parameters: %Š*Ñ< - %Š^1Ñ$ 1" %Š*Ñ< QÐ/ %Š*1Ñ< 1 Z f Ò f %Š*Ñ< Œ f with f a grid of 3D points Natural choice f %Š^1Ñ$ k fá %Š This is simply a product of Bézier curves -Š*Ñ< k f %Š" Í f %Ñ< %Ñ< "f Ï The 16 control points for a Bézier bicubic patch 40 u 0 v u 0 v 1 %Š^1Ñ$ k 10 Properties jj<pjc B 1 +^ KJ9m\ k! Conve hull property f ª +ª Ò v ËNË+Ë f 1 1 f -Š u %Ñ$ Œ"f u 1 v so that corners are pinned f -Š*Ñ< k ± f ª < f -Š µ ± +ª < u 1 v 1 -Ñ< µ, 11

12 Another way of picturing the process 41 You might prefer to look at the epression %Š^1Ñ< Ó f f %Š"! %Ó %Ñ$ Œ f in a rather different way For b TJ<m@0?WPXÂ Ó %Ñ< Œ"f for Then construct a Bezier curve out of J Ñv defines four separate Bezier curves, each parallel to the Ñ these for J Ñ ie ais ÓÔf f -Š! Ó %Ñ< Œ f u v 13 3 u v Choose a value of Ñ and you have four points on these curves 03 Then pick a point on this curve by choosing Š atri form 4 Another representation is using matrices Recall -Š ÃSÄÕÊ %Š*Ñ< kvãâäõö6ä E $ g`ø W YÙ}Ú where Ö 1 1#T! kœ kœ#t #T #! ƒ ¾ Ñ`Ñ>Ñ [ 1

13 Graphics application of Bézier tensor product surface 43 ore teapots 44 13

14 ² & ² ² Other properties of Bezier curves 45 Switching to higher degree A Bezier curve of degree ² is also a Bezier curve of degree 5U ² This means that the 5c ² control points can be replaced by 5V? ² control points in different positions and the curve shape remains unchanged, and hence one can increase the degree arbitrarily ere are the epressions for moving the control points to increase the degree by 1! & 5T ² & Ï!{d$"5 Í3( 5T 3+ËNË+Ë0 " Degree 7 Degree 4 Piecewise Bezier curves 46 Suppose we require many control points Piecewise Bézier curves Need an etra point inserted midway This will be a point of infleion 14

15 b b ± µ Ü f B-Splines 47 Like draftsperson s spline, but approimating A B-spline consists of several curve segments, each represented by a polynomial Controlled smoothness: eg cubic B-splines are Û, or can be made to be less: Û or Û Key advantage: Local control: moving a control point only affects part of the curve This isn t the case with a Bezier curve of high degree But often it is said not to be the case with, say, cubic Bezier curves But introducing the etra points does provide decoupling Eample - a cubic B-spline 48 5c (8? control points, [ NË{Ë~Ë{[ f, with b cubic polynomial curve segments Ü»hX ÜYÝ +Ë{Ë{Ë~ Each curve segment is defined as "Z %Š" B ¾ ÞŠŠ Š Ö ed ed "ed$! ß àp Ö) q g J (ÁX J X J X (Áq X J ( X (ÁX ƒ If the spline is closed then Ö is the same for all curve segments and ḧ f If the spline is open then the matrices Ö can be modified at the curve ends so that the curve segments interpolate the end control points 15

16 Ý B-splines, ctd 49 1 join 5 0 influenced by 1 1 influenced by 1 1 Conve hull property 1 1á 3 join 4 Spline demo: Tracking application of B-splines 430 Used in computer vision as Active Contours ere the local control is a bonus See Blake and Isard Active Contours Springer