Designing Parametric Cubic Curves. Prof. George Wolberg Dept. of Computer Science City College of New York

Transcription

1 Designing Parametric Cbic Crves Prof. George Wolberg Det. of Comter Science City College of New York

2 Objectives Introdce the tyes of crves - Interolating -Hermite - Bezier - B-Sline Analyze their erformance

3 Design Criteria Why we refer arametric olynomials of low degree: - Local control of shae, - Smoothness and continity, - Ability to evalate derivatives, - Stability, - Ease of rendering.

4 Smoothness Smoothness garanteed becase or olynomial eqations are differentiable. Difficlties arise at the join oints. 4

5 Control Points We refer local control for stability. - The most common interface is a gro of control oints. - In this examle, the crve asses throgh, or interolates, some of the control oints, bt only comes close to, or aroximates, others. 5

6 Parametric Cbic Polynomial Crves Choosing the degree: - High degree allows many control oints, bt comtation is exensive. - Low degree may mean low level of control. The comromise: se low-degree crves over short intervals. - Most designers work with cbic olynomial crves. 6

7 7 Matrix Notation, ) ( k T k k c c, c c c c c,. kz ky kx k c c c c where the coefficient matrix to be determined control oints

8 Interolation An interolating olynomial asses throgh its control oints. - Sose we have for controls oints k x y z k k k, for k -We let vary over the interval [,], giving s for eqally saced vales:, /, /,.. 8

9 Evalating the Control Points We seek coefficients c, c, c, c satisfying the for conditions: () c, ( ) c c ( ( ) c c ( () c c c c. ) ) c c ( ( ) ) c, c, 9

10 Matrix Notation In matrix notation Ac, where and A ( ( ) ) ( ( ) ). a colmn vector of row vectors nonsinglar: we will se its inverse

11 Interolating Geometry Matrix M I A The desired coefficients are c M. I

12 Interolating Mltile Segments Use the last control oint of one segment as the first control oint of the next segment. - To achieve smoothness in addition to continity, we will need additional constraints on the derivatives.

13 Blending Fnctions Sbstitting the interolating coefficients into or olynomial: Let T ( ) c M. T ( ) b( ), where b( ) M. The b() are the blending olynomials. T I T I

14 Visalizing the Crve Using Blending Fnctions The effect on the crve of an individal control oint is easier to see by stdying its blending fnction. 4

15 The Cbic Interolating Patch A bicbic srface atch: (, v) i j i v j c ij. 5

16 Matrix Notation In matrix form, the atch is defined by T (, v) Cv, - The colmn vector v = [ v v v ] T. - C is a 4 x 4 matrix of colmn vectors. 6 eqations in 6 nknowns. 6

17 Solving the Srface Eqations By setting v =, /, /, we can samle the srface sing crves in : T M P I - The coefficient matrix C is comted by T CA T. C M I PM T I. - The eqation for the srface becomes T (, v) M PM v. I T I 7

18 Blending Patches Extending or se of blending olynomials to srfaces: (, v) bi ( ) b j ( v) ij. i j - 6 simle atches form a srface. - Also known as tensor-rodct srfaces. - These srfaces are not very smooth. Bt they are searable, meaning they allow s to work with fnctions in and v indeendently. 8

19 Other Tyes of Crves and Srfaces How can we get arond the limitations of the interolating form - Lack of smoothness - Discontinos derivatives at join oints We have for conditions (for cbics) that we can aly to each segment - Use them other than for interolation - Need only come close to the data 9

20 Hermite Form () () () () Use two interolating conditions and two derivative conditions er segment Ensres continity and first derivative continity between segments

21 Hermite Crves and Srfaces Use the data at control oints differently in an attemt to get smoother reslts. - We insist that the crve interolate the control oints only at the two ends, and. () c, () c c c c.

22 Additional Conditions The derivative is a qadratic olynomial: dx d ( ) dy d c c c. dz d - We now can derive two additional conditions: () () c c, c c.

23 Matrix Form The desired coefficient matrix is - M H is the Hermite geometry matrix. q. M c H. c call this q

24 4 The Hermite Geometry Matrix The reslting olynomial is. M H. ) ( q M H T

25 5 Blending olynomials Using blending fnctions ()=b() T q, Althogh these fnctions are smooth, the Hermite form is not sed directly in Comter Grahics and CAD becase we sally have control oints bt not derivatives However, the Hermite form is the basis of the Bezier form. ) ( M T H b

26 Parametric and Geometric Continity We can reqire the derivatives of x, y,and z to each be continos at join oints (arametric continity) Alternately, we can only reqire that the tangents of the reslting crve be continos (geometry continity) The latter gives more flexibility as we have need satisfy only two conditions rather than three at each join oint 6

27 Examle Here the and q have the same tangents at the ends of the segment bt different derivatives Generate different Hermite crves This techniqes is sed in drawing alications 7

28 Parametric Continity Continity is enforced by matching olynomials at join oints. - C arametric continity: () x y z () () q() () q q q x y z () (). () 8

29 C Parametric Continity Matching derivatives at the join oints gives s C continity: () x () () y z () q() qx q y qz () (). () 9

30 Another Aroach: Geometric Continity If the derivatives are roortional, then we have geometric continity. same direction different magnitde - One extra degree of freedom. - Extends to higher dimensions.

31 Bezier Crves: Basic Idea In grahics and CAD, we sally don t have derivative data Bezier sggested sing the same 4 data oints as with the cbic interolating crve to aroximate the derivatives in the Hermite form

32 Bezier Crves and Srfaces Bezier added control oints to manilate derivatives. - The two derivative conditions become., c c c c

33 Bezier Geometry Matrix We solve c=m B, where The cbic Bezier olynomial is ths. 6 M B. ) ( M B T

34 Bezier Blending Fnctions - These fnctions are Bernstein olynomials: b kd ( ) d! k!( d k)! k ( ) d k. 4

35 Proerties of Bernstein Polynomials All zeros are either at = or =. - Therefore, the crve mst be smooth over (,) The vale of never exceeds. - () is a convex sm, so the crve lies inside the convex hll of the control oints. 5

36 Bezier Srface Patches Using a 4 x 4 array of control oints P, (, v) i j T M b B i ( ) b PM two blending fnctions T B j v. ( ) ij 6

37 Convex Hll Proerty in D The atch is inside the convex hll of the control oints and interolates the for corner oints,,,. 7

38 Bezier Patch Edges Partial derivatives in the and v directions treat the edges of the atch as D crves. (,) (,) v ( ( ), ). 8

39 Bezier Patch Corners The twist vector draws the center of the atch away from the lane. v (,) 9( ). 9

40 Cbic B-Slines Bezier crves and srfaces are widely sed. - One limitation: C continity at the join oints. B-Slines are not reqired to interolate any control oints. - Relaxing this reqirement makes it ossible to enforce greater smoothness at join oints. 4

41 The Cbic B-Sline Crve The control oints now reside in the middle of a seqence: { i, i, i, i}. - The crve sans only the distance between the middle two control oints. 4

42 Formlating the Geometry Matrix We are looking for a olynomial ( ) M, where is the matrix of control oints. - M can be made to enforce a nmber of conditions. - In articlar, we can imose continity reqirements at the join oints. T 4

43 4 Join Point Continity Constrct q from the same matrix as : - Now let q() = T Mq. - Constraints on derivates allow s to control smoothness. i i i i. i i i i q and

44 Symmetric Aroximations Enforcing symmetry at the join oints is a olar choice for M. Two conditions that satisfy symmetry are () q() ( 6 i 4 i i ), () q() ( i i ), 44

45 Additional Conditions We aly the same symmetry conditions to (), the other endoint. - We now have for eqations in the for nknowns c, c, c, c : T ( ) c. 45

46 46 The B-Sline Geometry Matrix Once we have the coefficient matrix, we can solve for the geometry matrix: M S

47 47 B-Sline Blending Fnctions The blending fnctions are 6 4 ) ( 6

48 Advantages of B-sline Crves In seqence, B-sline crve segments have C continity at the join oints. - They are also confined to their convex hlls. On the other hand, we need more control oints than we did for Bezier crves. 48

49 49 B-Slines and Bases Each control oint affects for adjacent intervals.., ) (, ) (, ) (, ) (, ) ( i i i b i i b i i b i i b i B i

50 Sline Basis Fnction A single exression for the sline crve sing basis fnctions: ( ) m i B i ( ) i. 5

51 Aroximating Slines Each B i is a shifted version of a single fnction. - Linear combinations of the B i form a iecewise olynomial crve over the whole interval. 5

52 Sline Srfaces The same form as Bezier srfaces: (, v) i j ( ) b ( ) - Bt one segment er atch, instead of nine! b i j ij. - However, they are also mch smoother. 5

53 General B-Slines Polynomials of degree d between n knots,..., n : ( ) c d j jk k - If d =, then each interval contains a cbic olynomial: 4n eqations in 4n nknowns. - A global soltion that is not well-sited to comter grahics. j, k 5

54 54 The Cox-deBoor Recrsion A articlar set of basis slines is defined by the Cox-deBoor recrsion: otherwise;, k k k B ). ( ) (,, B B B d k k d k d k d k k d k k kd

55 Recrsively Defined B-Slines Linear interolation of olynomials of degree k rodces olynomials of degree k +. 55

56 Uniform Slines Eqally saced knots. eriodic niform B-sline 56

57 Nonniform B-Slines Reeated knots ll the sline closer to the control oint. - Oen slines extend the crve by reeating the endoints. - Knot seqences: {,,,,,,, n, n, n, n, n} {,,,,,,,}. cbic Bezier crve often sed - Any sacing between the knots is allowed in the general case. 57

58 NURBS Use weights to increase or decrease the imortance of a articlar oint. - The weighted homogeneos-coordinate reresentation of a control oint i =[x i y i z i ] is q i w i xi y i. zi 58

59 59 The NURBS Basis Fnctions A 4D B-sline - Derive the w comonent from the weights: n i i i d i w B z y x,. ) ( ) ( ) ( ) ( ) q( n i i d i w B w,. ) ( ) (

60 Nonniform Rational B-Slines Each comonent of () is a rational fnction in. - We se ersective division to recover the D oints: n B ( ) w ( ) q( ) w( ) i i, d n B i i, d ( ) w - These crves are invariant nder ersective transformations. - They can aroximate qadrics one reresentation for all tyes of crves. i i i. 6

61 Rendering Crves and Srfaces Prof. George Wolberg Det. of Comter Science City College of New York

62 Objectives Introdce methods to draw crves - Aroximate with lines - Finite Differences Derive the recrsive method for evalation of Bezier crves and srfaces Learn how to convert all olynomial data to data for Bezier olynomials 6

63 Evalating Polynomials Simlest method to render a olynomial crve is to evalate the olynomial at many oints and form an aroximating olyline For srfaces we can form an aroximating mesh of triangles or qadrilaterals Use Horner s method to evalate olynomials ()=c +(c +(c +c )) - mltilications/evalation for cbic 6

64 Polynomial Evalation Methods Or standard reresentation: ( ) c n i Horner's method: i i, ( ) c ( c ( c ( cn))). - If the oints { i } are saced niformly, we can se the method of forward differences. 64

65 65 The Method of Forward Differences Forward differences defined iteratively: If k+ - k = h is constant, then (n) ( k ) is constant for all k. ). ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( () () k m k m k m k k k k k

66 Comting The Forward- Difference Table For the cbic olynomial ( ) we constrct the table as follows:, comte these 66

67 Using the Table Comte sccessive vales of ( k ) starting from the bottom: ( m) ( k ) ( m) ( k ) ( m) ( k ). 67

68 Sbdivision Crves and Srfaces A rocess of iterative refinement that rodces smooth crves and srfaces. 68

69 Recrsive Sbdivision of Bezier Polynomials: decastelja Algorithm Break the crve into two searate olynomials, l() and r(). - The convex hlls for l and r mst lie inside the convex hll for : the variation-diminishing roerty: 69

70 7 Efficient Comtation of the Sbdivision ), (, l l., ) ( r l r l l l Reqires only shifts and adds!

71 Every Crve is a Bezier Crve We can render a given olynomial sing the recrsive method if we find control oints for its reresentation as a Bezier crve Sose that () is given as an interolating crve with control oints q ()= T M I q There exist Bezier control oints sch that ()= T M B Eqating and solving, we find =M B - M I 7

72 7 Matrices Interolating to Bezier B-Sline to Bezier M M I B M M S B

73 Examle These three crves were all generated from the same original data sing Bezier recrsion by converting all control oint data to Bezier control oints Bezier Interolating B Sline 7

74 Srfaces Can aly the recrsive method to srfaces if we recall that for a Bezier atch crves of constant (or v) are Bezier crves in (or v) First sbdivide in - Process creates new oints - Some of the original oints are discarded original and discarded original and ket new 74

75 Second Sbdivision 6 final oints for of 4 atches created 75

76 Normals For rendering we need the normals if we want to shade - Can comte from arametric eqations n (, v) (, v) v - Can se vertices of corner oints to determine - OenGL can comte atomatically 76

77 Utah Teaot Most famos data set in comter grahics Widely available as a list of 6 D vertices and the indices that define Bezier atches 77

78 Algebraic Srfaces Qadric srfaces are described by imlicit eqations of the form T T A b c. - indeendent coefficients A, b, and c determine the qadric. - Ellisoids, araboloids, and hyerboloids can be created by different gros of coefficients. - Eqations for qadric srfaces can be redced to standard form by affine transformation. 78

79 Rendering Qadric Srfaces Finding the intersection of a qadric with a ray involves solving a scalar qadratic eqation. - We sbstitte ray = +d and se the qadratic formla. - Derivatives determine the normal at a given oint. 79

80 Qadric Objects in OenGL OenGL sorts disks, cylinders and sheres with qadric objects. GLUqadricObj *qobj; qobj = glnewqadric(); - Choose wire frame rendering with glqadricdrawstyle(qobj, GLU_LINE); - To draw an object, ass the reference: glshere(qobj, RADIUS, SLICES, STACKS); 8

81 Bezier Crves in OenGL Creating a D evalator: glmaf(tye, _min, _max, stride, order, oint_array); -tye: oints, colors, normals, textres, etc. -_min, _max: range. -stride: oints er crve segment. -order: degree +. -oint_array: control oints. 8

82 Drawing the Crve One evalator call takes the lace of vertex, color, and normal calls. - The ser enables them with glenable. tyedef float oint[]; oint data[] = {...}; glmaf(gl_map_vertex_,.,.,, 4, data); glenable(gl_map_vertex_); glbegin(gl_line_strip) for(i=; i<; i++) glevalcoordf(i/.); glend(); 8

83 Bezier Srfaces in OenGL Using a D evalator: glmaf(gl_map_vertex_,,,,4,,,,4,data);... for(j=; j<99; j++) { glbegin(gl_quad_strip); for(i=; i<=; i++) { glevalcoordf(i/., j/.); glevalcoordf((i+)/., j/.); } glend(); } 8

84 Examle: Bezier Teaot Vertex information goes in an array: GLfloat data[][4][4]; Initialize the grid for wireframe rendering: void myinit() { glenable(gl_map_vertex_); glmagridf(,.,.,,.,.); } 84

85 Drawing the Teaot for(k=; k<; k++) { glmaf(gl_map_vertex_,,,, 4,,,, 4, &data[k][][][]); for (j=; j<=8; j++) { glbegin(gl_line_strip); for (i=; i<=; i++) glevalcoordf((glfloat)i/., (GLfloat)j/8.); glend(); glbegin(gl_line_strip); for (i=; i<=; i++) glevalcoordf((glfloat)j/8., (GLfloat)i/.); glend(); } } 85