CAP 6736 Geometric Modeling Team Hamilton B-Spline Basis Functions. Presented by Padmavathi Siddi Vigneshkumar Suresh Jorge Medrano

Transcription

1 CAP 6736 Geometric Modeling Team Hamilton B-Spline Basis Functions Presented by Padmavathi Siddi Vigneshkumar Suresh Jorge Medrano 1

2 Team members contributions Member: Padmavathi Siddi Piecewise Polynomials Piecewise Polynomials in Bezier Form B Splines Examples Member: Vigneshkumar Suresh N 3,2 as Piecewise Polynomial Properties Derivative Formula B Spline Derivatives Member: Jorge Medrano Derivatives with respect to Knot Computational Algorithms Computational Algorithms : all Derivatives 2

3 Father of B-splines A draft-man s spline Isaac J. Schoenberg

4 From Bows to Boats the Mechanical Spline Figure : Evidence from Mesolithic settlements, as well as vernacular structures like the native American wigwam, show one of the simplest forms of shelter, formed by ramming poles into a circular formation, bending them inward, and lashing them together to form a sturdy dome-shaped latticework 4

5 Flexible strip of timber or steel - Spline Figure : Hooked weights, called ducks, accurately secure a spline here, no more than a thin strip of balsa for tracing the hull of a sailing vessel. Source: Edson International (with kind permission). 5

6 L Automobile Inventing the Computational Spline Figure: Citroe n s DS 19 was introduced at the 1955 at the Paris Motor Show. Within the first 15 minutes of the show, 743 orders were received. 6

7 Bezier Cubic Splines form the backbone of Desktop Figure: Bézier s cubic splines form the backbone of desktop publishing standards like PostScript and TrueType. 7

8 Aerospace Finding a New Language Figure: The Mustang P51 s geometry was stored numerically in tables, rather than vulnerable physical blueprints. Source: NASA 8

9 Polynomials Polynomials are incredibly useful mathematical tools as they are simply defined, can be calculated quickly on computer systems and represent a tremendous variety of functions. They can be differentiated and integrated easily, and can be pieced together to form spline curves that can approximate any function to any accuracy desired. Most students are introduced to polynomials at a very early stage in their studies mathematics, and would probably recall them in the form below P(t) = a n t n + a n-1 t n a 1 t + a 0 which represents a polynomial as a linear combination of certain elementary polynomials : { {1,t, t 2,.t n ) }. In general, any polynomial function that has degree less than or equal to n, can be written in this way, and the reasons are simply The set of polynomials of degree less than or equal to n forms a vector space: polynomials can be added together, can be multiplied by a scalar, and all the vector space properties hold. The set of functions { {1,t, t 2,.t n ) } form a basis for this vector space which is any polynomial of degree less than or equal to n can be uniquely written as a linear combinations of these functions. This basis, commonly called the power basis, is only one of an infinite number of bases for the space of polynomials. 9

10 Piecewise Polynomials Shortcomings of Curves Curves consisting of just one polynomial or rational segment are often inadequate. Their shortcomings are: A high degree is required in order to satisfy a large number of constraints; e.g., (n - I)-degree is needed to pass a polynomial Bezier curve through n data points. However, high degree curves are inefficient to process and are numerically unstable; A high degree is required to accurately fit some complex shapes; Single-segment curves (surfaces) are not well-suited to interactive shape design; although Bezier curves can be shaped by means of their control points (and weights), the control is not sufficiently local. The solution is to use curves (surfaces) which are piecewise polynomial, or piecewise rational. 10

11 Piecewise Polynomials The given Figure shows a curve, C(u), consisting of m (= 3) nth degree polynomial segments. C(u) is defined on u E [0,1]. The parameter values are called breakpoints. They map into the endpoints of the three polynomial segments. Each segment is denoted by C i (u), 1< i < m. These segments are constructed so that they can join with some level of continuity. This construction is not needed to be same at every breakpoint. Let C i ( j ) denote the j th derivative of C i. C(u) is said to be C k continuous at the breakpoint U i if C i ( j ) (u i ) = C i+1 j (u i ) for all 0 j k. 11

12 Piecewise Polynomials in Bezier Form The given figure shows with the three segments in cubic Bezier form where P j i denotes the ith control point of the jth segment. If the degree equals three and the breakpoints U = {uo, u1, u2, u3} remain fixed, and if we allow the twelve control points, P j i, to vary arbitrarily, we obtain the vector space, V, consisting of all piecewise cubic polynomial curves on U. V has dimension twelve, and a curve in V may be discontinuous at u1 or u2. Now suppose we specify that P 1 3 = P 2 0 and P 2 3 = P 3 0. This gives rise to V 0, the vector space of all piecewise cubic polynomial curves on U which are at least C 0 continuous everywhere. V 0 has dimension ten, and V 0 V. 12

13 Piecewise Polynomials in Bezier Form V has dimension twelve, and a curve in V may be discontinuous at u1 or u2. Now suppose we specify that P 1 3 = P 2 0 and P 2 3 = P 3 0. This gives rise to V 0, the vector space of all piecewise cubic polynomial curves on U which are at least C 0 continuous everywhere. V 0 has dimension ten, and V 0 V. Imposing C 1 continuity is a bit more involved. Let us consider u = u 1. Assume that P 1 3 = P 2 0. Let be local parameters on the intervals [uo, u1] and [u1 u2], respectively. Then 0 v, w 1. C 1 continuity at u 1 implies 13

14 Piecewise Polynomials in Bezier Form Using property P1. 7, it is easy to derive the general expression for the derivative of a Bezier curve The above equation says that P 1 3 and P 2 3 can be written in terms of P 1 2, P 2 1 and P 2 2, P 3 1, respectively. Hence, V 1, the vector space of all C 1 continuous piecewise cubic polynomial curves on U, has dimension eight, and V 1 V 0 V. We want a curve representation of the form where the P i are control points, and the { fi ( u), i=0,1,,.,n} are piecewise polynomial functions forming a basis for the vector space of all piecewise polynomial functions of the desired degree and continuity (for a fixed breakpoint sequence, U = {u i }, 0 i m). 14

15 Piecewise Polynomials in Bezier Form Property : local support : This means that each f i (u) is nonzero only on a limited number of subintervals, not the entire domain, [u 0, u m ]. Since P i is multiplied by f i (u) moving P i affects curve shape only on the subintervals where f i (u) is nonzero. Finally, given appropriate piecewise polynomial basis functions, we can construct piecewise rational curves Nonrational and rational tensor product surfaces 15

16 Spline In the computer science subfields of computer-aided design and computer graphics, the term spline more frequently refers to a piecewise polynomial parametric curve. Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design. The term spline comes from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes. It also is an acronym for "Smooth Polynomial Lines Interpolating Numerical Estimates". 16

17 B-splines Defined B-spline can be defined in many number of ways by divided differences of truncated power functions and by a recurrence formula. We use the recurrence formula, since it is the most useful for computer implementation. Let U= { u 0,u 1,..u m } be a nondecreasing sequence of real numbers, i.e., u i < u i+l, i = 0,..., m -1. The ui are called knots, and U is the knot vector. The ith B-spline basis function of p-degree (order p + 1), denoted by Ni, p ( U ), is defined as 17

18 B-splines Defined N i,o (u) is a step function, equal to zero everywhere except on the half-open interval u E [u i, u i+1 ); For p > 0, N i,p (u) is a linear combination of two (p - 1) degree basis functions Computation of a set of basis functions requires specification of a knot vector - U, and the degree - p The above equation can yield the quotient % (see examples later); we define this quotient to be zero; 18

19 The N i,p (u) are piecewise polynomials, defined on the entire real line; generally only the interval [u o, u m ] is of interest; The half-open interval, [U i, U i+1 ), is called the ith knot span; it can have zero length, since knots need not be distinct The computation of the pth-degree functions generates a truncated triangular table We used the term breakpoint and required u i < u i+1 for all I and now we use the term knot and assume u i u i+1. The breakpoints correspond to the set of distinct knot values, and the knot spans of nonzero length define the individual polynomial segments. Hence, we use the word knot with two different meanings: a distinct value (breakpoint) in the set U, and an element of the set U (there can exist additional knots in U having the same value). 19

20 Example: Degree Zero Fig: nonzero zeroth-degree basis functions U = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} 20

21 Example: Degree One Fig: nonzero first-degree basis functions U = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} 21

22 Example: Degree Two Fig: nonzero Second-degree basis functions U = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} 22

23 N 3,2 AS PIECEWISE POLYNOMIAL where N 3,0 =1, for 1<=u<2 N 4,0 =1, for 2<=u<3 N 5,0 =1, for 3<=u<4 =0, elsewhere =0, elsewhere =0, elsewhere

24 LOCAL SUPPORT N i,p (u)=0 if u is outside the interval[u i,u i+p+1 ). N 1,3 is a combination of N 1,0, N 2,0, N 3,0, N 4,0 and therefore N 1,3 is only non-zero for u that belongs to the range[u 1, u 5 ).

25 LOCAL IMPACT For any given knot span[u j, u j+1 ), there are atmost p+1 of N i,p are non-zero, namely the functions N j-p,p,,n j,p.. ON [u 3,u 4 ) the only non-zeroth degree function is N 3,0. Hence, the only cubic functions that are non-zero on [u 3, u 4 ) are N 0,3,.,N 3,3.

26 PROPERTIES OF B-SPLINES- NON-NEGATIVITY N i,p (u)>=0 for all i,p,u. Proof can be done by induction on p. Proof By Induction on P: Its clearly true for p=0 and assuming that it is true for p-1 and for all p>=0 for arbitrary i and u values. By Local support property, N i,p (u)=0,if u does not belong to [u i, u i+p ).

27 NON-NEGATIVITY (CONT D) But u belonging to the interval[u i,u i+p ) and also assuming the term N i,p-1 (u) is non-negative then the first term will be non-negative. The Same will hold true for the second term and therefore N i,p (u) is non-negative. Therefore for all i,p,u N i,p,u is always non-negative.

28 PARTION OF UNITY PROPERTY For an arbitrary knot span, [u i, u i+1 ), the summation of N j,p (u) for the range j= i-p to i equals to 1 for all u that belongs to [u i,u i+1 )

29 PARTION OF UNITY PROPERTY(CONT D) By changing the summation from i-p to i-p+1, and assuming that N i-p,p-1 (u)=n i+1,p-1 (u)=0,we get BY APPLYING RECURSIVELY WE GET,

30 DIFFERENTIABILITY AND MAXIMUM VALUE PROPERTY According to differentiability property, at a knot N i,p (u) is p-k time differentiable where k is the multiplicity of the knot and all derivatives of N i,p (u) exists in the interior of the knot span. Maximum value property states that N i,p (u) attains exactly one maximum value except in the case of p=0

31 DERIVATIVES OF B-SPLINES The derivative of B-splines basis function is PROOF: By Induction on p. Base case p=1 can be proved and assuming that the above equation is true for p-1,p>1

32 DERIVATIVES OF B-SPLINES(CONT D) By using Product rule on, yields,

33 DERIVATIVES OF B-SPLINES(CONT D) By using the values of N i,p-1 and N i+1,p-1 we get, By using Cox-de Boor formula the expression inside the parenthesis can be replaced by N i,p-1 and N i+1,p-1

34 DERIVATIVES OF B-SPLINES(CONT D)

35 HIGHER ORDER DERIVATIVES OF B-SPLINES K TH order derivative N i,p is obtained by repeated differentiation of the general derivative of B-splines function.

36 DERIVATIVES INTERMS OF B-SPLINE FUNCTION Where,

37 DERIVATIVES INTERMS OF B-SPLINE FUNCTION(CONT D) CONDITIONS WHILE DEFINING DERIVATIVES IN TERMS OF B-SPLINE ARE: 1. k should not exceed the p value 2. Denominators involving knot differences can become Zero in which case the quotients are defined as zero.

38 RECURSIVE For p>0, N i,p (u) is a linear combination of two (p-1) degree basis function

39 DERIVATIVES OF B-SPLINE WITH MULTIPLE KNOTS For m+1 knots there are n+1 basis functions where n=m-p-1 Example,N 0,p (a)=1 because N 0,0,..,N p-1,0 =0 which therefore implies N P,0 (a)=n 0,p (a)=1.by partition of unity property N i,p (a)=0 for i!=0 and N i,p (b)=0 for i!=n

40 Knots Pth degree B-spline curve Cox-de Boor recursive formula Where: Pi represents the control points represents the basis functions defined on the knot vector m+1

41 Knot Vectors Where values of u range We get a uniform B-spline if the knots are equidistant Amount of knots in U (m+1) is directly related to degree (k) and number of control points (p+1) M = k + p + 1 Example: A cubic B-spline using control points needs m = = 8 Hence, our knot vector

42 Knot Vector Notation Ū : U {u k } Û: U {u k + du}= U + {u k } U + = same as U except u k is replaced by u k + du N ī : basis function after inserting u k into U N î : basis functions after inserting u k + du into U Fig.1. Knot vector notation

43 B-spline derivatives Assume: The final point on the first Bézier curve has the same coordinates as the first point of the second Bézier curve The first derivative at the end of the first Bézier curve is the same as the first derivative at the start of the second Bézier curve The second derivative at the end of the first Bézier curve is the same as the second derivative at the start of the second Bézier curve

44 Derivatives with respect to a knot Given a nonrational B-spline curve of degree p = n+p+1 Defined over knot vector, m The derivative is of the form: where {u k is a knot of multiplicity less than or equal to p, and

45 Right derivative For the right derivative: Let du be such that Step 2 yields new control points 1) insert into 2) insert into Step 1 yields new control points The difference in the numerator:

46 Right and Left Derivative For the left derivative: Let du be such that SAME FORMULA! Finally, The right derivative of a non rational B- spline curve of degree p is another degree p curve: In right derivative k must satisfy: In left derivative k must satisfy:

47 Basis function derivatives Given the basis function knot vector of degree p defined over where m = n+p+1 The derivative is of form: To find the right derivative, let du be such that: Evaluating the definition of the right derivative Then rearranging and manipulating the numerator to be compatible yields the limit in fig.2. Fig.2. Definition of a basis function derivative with respect to u 10

48 Derivatives with respect to knots Note: Difference between left and right derivatives of curves is due to the fact that different sets of functions are nonvanishing for the left and right indexes. For knots of multiplicity of 1, the left and right derivative functions are the same For knots of higher multiplicity different sets are nonvanishing Overlap index range where derivative functions are the same Fig.3. Nonvanishing derivative basis functions with respect to and

49 Computational Algorithms Algorithm 1: An algorithm to compute the point on a B-spline curve and all derivatives up to and including the d-th, at a fixed u-value n: the number of control points is n + 1 p: the degree of the curve U: the knots P: the control points CK[]: output of k-th derivative nders[][]: local array used to store derivative of basis functions Algorithm.1.

50 Computational Algorithms Now, we want to formally differentiate the p-th degree B- spline curve defined on the knot vector, Now dropping the first and last knots of the knot vector U to let it become U. Since U has m - 1 knots then the function, computed on U, is equal to, computed on U Hence, C (u) is a (p-1)th degree curve

51 Computational Algorithms Example: Fig.4. C(u) Fig.5. C (u)

52 Computational Algorithms This algorithm is based on the previous algorithm with the exception of using (AllBasisFuns) instead of (BasisFuns) to return all nonzero basis functions of all degrees (from 0 up to p). Like the previous algorithm, this one computes the point on a B-spline curve and all derivatives up to and including the d-th, at a fixed u-value In particular, N[j][i] is the value of the i-th degree basis function, where 0 i p and 0 j i

53 Part II: Design examples

54 (by Alicia Cantón, Gabinete de Tele-Educación de la Universidad Politécnica de Madrid) Modeling a cup/glass using Kevin Burgess) Modeling a plane engine

55 Part III: GM lab

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61 References The NURBS Book (Monographs in Visual Communication) by Les A. Piegl, Wayne Tiller Piegl, Les A., and Wayne Tiller. Computing the Derivative of NURBS with Respect to a Knot. Computer Aided Geometric Design, vol. 15, no. 9, 1998, pp , doi: /s (98)