n is symmetric if it does not depend on the order
|
|
- Tobias Phelps
- 7 years ago
- Views:
Transcription
1 Chapter 18 Polynoial Curves 18.1 Polar Fors and Control Points The purpose of this short chapter is to show how polynoial curves are handled in ters of control points. This is a very nice application of affine concepts discussed in previous chapters and provides a stepping stone for the study of rational curves. This chapter is just a brief introduction. A coprehensive treatent of polynoial curves can be found in Gallier [70]. The key to the treatent of polynoial curves in ters of control points is that polynoials can be ultilinearized. 1 To be ore precise, say that a ap f: R d R } {{ d R } n is ultiaffine if it is affine in each of its arguents, and that a ap f: R d R } {{ d R } n is syetric if it does not depend on the order of its arguents, i.e., fa π1,...,a π =fa 1,...,a for all a 1,...,a, and all perutations π. Then, for every polynoial F t of degree, there is a unique syetric and ultiaffine ap f: R R R such that F t =ft,...,t, for all t R. This is an old folk theore, probably already known to Newton. The proof is easy. By linearity, it is enough to consider a onoial of the for x k, where k. The unique syetric ultiaffine ap corresponding to x k is σ k t 1,...,t, k where σ k t 1,...,t isthekth eleentary syetric function in variables, i.e. σ k = t i. I {1,...,} I =k Given a polynoial curve F : R R n of degree i I x 1 t =F 1 t,...=... x n t =F n t, 1 The ter ultilinearized is technicaly incorrect, we should say ultiaffinized! 571
2 57 CHAPTER 18. POLYNOMIAL CURVES where F 1 t,...,f n t are polynoials of degree at ost, F : R R n arises fro a unique syetric ultiaffine ap f: R R n,thepolar for of F, such that F t =ft,...,t, for all t R see Rashaw [141], Farin [58, 57], Hoschek and Lasser [90], or Gallier [70]. For exaple, consider the plane cubic defined as follows: We get the polar fors F 1 t = 4 t t 9 4, F t = 4 t t 9 4 t. f 1 t 1,t,t = 1 4 t 1t + t 1 t + t t 1 t 1 + t + t 9 4 f t 1,t,t = 4 t 1t t 1 t 1t + t 1 t + t t 4 t 1 + t + t. Also, for r s, the ap f: R R n is deterined by the +1control points b 0,...,b, where since ft 1,...,t = k=0 I J={1,...,} I J=, cardj=k b i = fr,...,r,s,...,s, i i i I s ti j J tj r fr,...,r, s,...,s. k k For exaple, with respect to the affine frae r = 1, s =, the coordinates of the control points of the cubic defined earlier are: b 0 =0, 0 b 1 = 4, 4 b = 4, 1 b =0, 0. Conversely, for every sequence of + 1 points b 0,...,b, there is a unique syetric ultiaffine ap f such that b i = fr,...,r,s,...,s, i i naely ft 1,...,t = k=0 I J={1,...,} I J=, cardj=k i I s ti j J tj r b k. Thus, there is a bijection between the set of polynoial curves of degree and the set of sequences b 0,...,b of + 1 control points. The figure below shows four contol points b 0, b 1, b, b specifying a polynoial curve of degree, where b 0 = fr, r, r, b 1 = fr, r, s, b = fr, s, s, b = fs, s, s.
3 18.1. POLAR FORMS AND CONTROL POINTS 57 b b 1 b 0 b Figure 18.1: Control points and control polygon The upshot of all this is that for algorithic purposes, it is convenient to define polynoial curves in ters of polar fors. Recall that the canonical affine space associated with the field R is denoted as A, unless confusions arise. Definition A paraeterized polynoial curve in polar for of degree is an affine polynoial ap F : A Eof polar degree, defined by its -polar for, which is soe syetric -affine ap f: A E, where A is the real affine line, and E is any affine space of diension at least. Given any r, s A, with r<s, a paraeterized polynoial curve segent F [r, s] in polar for of degree is the restriction F :[r, s] Eof an affine polynoial curve F : A Ein polar for of degree. We define the trace of F as F A, and the the trace of F [r, s] asf [r, s]. Typically, the affine space E is the real affine space A of diension. Reark: When defining polynoial curves, it is convenient to denote the polynoial ap defining the curve by an upper-case letter, such as F : A E, and the polar for of F by the sae, but lower-case letter, f. It would then be confusing to denote the affine space which is the range of the aps F and f also as F, and thus, we denote it as E or at least, we use a letter different fro the letter used to denote the polynoial ap defining the curve. Also note that we defined a polynoial curve in polar for of degree at ost, rather than a polynoial curve in polar for of degree exactly, because an affine polynoial ap f of polar degree ay end up being degenerate, in the sense that it could be equivalent to a polynoial ap of lower polar degree. For convenience, we will allows ourselves the abuse of language where we abbreviate polynoial curve in polar for to polynoial curve. We suarize the relationship between control points and polynoial curves in the following lea. Lea Given any sequence of +1 points a 0,...,a in soe affine space E, there is a unique polynoial curve F : A E of degree, whose polar for f: A E satisfies the conditions fr,...,r, s,...,s=a k, k k
4 574 CHAPTER 18. POLYNOMIAL CURVES where r, s A, r s. Furtherore, the polar for f of F is given by the forula ft 1,...,t = and F t is given by the forula where the polynoials k=0 I J={1,...,} I J=, J =k F t = B k [r, s]t = k i I s ti Bk [r, s]t a k, k=0 are the Bernstein polynoials of degree over [r, s]. s t j J k k t r tj r a k, Note that since the polar for f of a polynoial curve F of degree is syetric, the order of the arguents is irrelevant. Often, when arguent are repeated, we also oit coas between arguent. For exaple, we abbreviate fr,...,r,s,...,sasfr i s j. i j In the next section, we will abbreviate ft,...,t,r,...,r,s,...,sasft j r i j s i. j i j i 18. The de Casteljau Algorith The definition of polynoial curves in ters of polar fors leads to a very nice algorith known as the de Casteljau algorith, to draw polynoial curves. Using the de Casteljau algorith, it is possible to deterine any point F t on the curve, by repeated affine interpolations see Farin [58, 57], Hoschek and Lasser [90], Risler [14], or Gallier [70]. The exaple below shows F 1/. b 1 b 1, 1 b b 0, b 1, F 1/ = b 0, b 0, 1 b, 1 b 0 b Figure 18.: A de Casteljau diagra for t =1/
5 18.. THE DE CASTELJAU ALGORITHM 575 In the general case where a curve F is specified by + 1 control points b 0,...,b w.r.t. to an interval [r, s], let us define the following points b i,j used during the coputation of F t where f is the polar for of F : { bi if j =0, 0 i, b i,j = ft j r i j s i if 1 j, 0 i j. Then, we have the following equations: b i,j = s t b i,j 1 + t r b i+1,j 1. The result is F t =b 0,. The coputation can be conveniently represented in the following triangular for: j 1 j... k... b 0,0 b 0,1. b.. 1,0 b 0,j 1. b 0,j b i,j b i,j b 0, k b i+1,j 1... b k j,j.. b0, b k j+1,j 1... bk, k b k 1,1... b k,0. b 1,0 b,0. b j+1,j 1 b 1,1 b j,j When r t s, each interpolation step coputes a convex cobination, and b i,j lies between b i,j 1 and b i+1,j 1. In this case, geoetrically, the algorith consists of a diagra consisting of the polylines b 0,0,b 1,0, b 1,0,b,0, b,0,b,0, b,0,b 4,0,...,b 1,0,b,0
6 576 CHAPTER 18. POLYNOMIAL CURVES b 0,1,b 1,1, b 1,1,b,1, b,1,b,1,...,b,1,b 1,1 b 0,,b 1,, b 1,,b,,...,b,,b,... b 0,,b 1,, b 1,,b, b 0, 1,b 1, 1 called shells, and with the point b 0,, they for the de Casteljau diagra. Note that the shells are nested nicely. The polyline b 0,b 1, b 1,b, b,b, b,b 4,...,b 1,b is also called a control polygon of the curve. When t is outside [r, s], we still obtain shells and a de Casteljau diagra, but the shells are not nicely nested. One of the best features of the de Casteljau algorith is that it lends itself very well to recursion. Indeed, going back to the case of a cubic curve, it is easy to show that the sequences of points b 0,b 0,1,b 0,,b 0, and b 0,,b 1,,b,1,b are also control polygons for the exact sae curve see Farin [58, 57], Hoschek and Lasser [90], Gallier [70]. Thus, we can copute the points corresponding to t = 1/ with respect to the control polygons b 0,b 0,1,b 0,,b 0, and b 0,,b 1,,b,1,b, and this yields a recursive ethod for approxiating the curve. This ethod called the subdivision ethod applies to polynoial curves of any degree and can be used to render efficiently a curve segent F over [r, s]. b 1 b 1, 1 b b 0, b 1, F 1/ = b 0, b 0, 1 b, 1 b 0 b Figure 18.: Approxiating a curve using subdivision For uch ore on polynoial curves, see Gallier [70].
7 Chapter 19 Polynoial Surfaces 19.1 Polar Fors The purpose of this short chapter is to show how polynoial surfaces are handled in ters of control points. As Chapter 18, this Chapter is just a brief introduction and a stepping stone for the study of rational surfaces. A coprehensive treatent of polynoial surfaces can be found in Gallier [70]. The deep reason why polynoial surfaces can be effectively handled in ters of control points is that ultivariate polynoials arise fro ultiaffine syetric aps see Rashaw [141], Farin [58, 57], Hoschek and Lasser [90], or Gallier [70]. Denoting the affine plane R as P, traditionally, a polynoial surface in R n is a function F : P R n, defined such that x 1 = F 1 u, v,...=... x n = F n u, v, for all u, v R, where F 1 U, V,...,F n U, V are polynoials in R[U, V ]. There are two natural ways to polarize the polynoials defining F. The first way is to polarize separately in u and v. Ifp is the highest degree in u and q is the highest degree in v, we get a unique ultiaffine ap f:r p R q R n of degree p + q which is syetric in its first p arguents and syetric in its last q arguents, such that F u, v =fu,...,u; v,...,v. p q We get what is traditionally called a tensor product surface, or as we prefer to call it, a bipolynoial surface of bidegree p, q or a rectangular surface patch. We also say that the ultiaffine aps arising in polarizing separately in u and v are p, q -syetric. The second way to polarize is to treat the variables u and v as a whole. This way, if F is a polynoial surface such that the axiu total degree of the onoials is, we get a unique syetric degree ultiaffine ap f:r R n, such that F u, v =fu, v,...,u, v. 577
8 578 CHAPTER 19. POLYNOMIAL SURFACES We get what is called a total degree surface or a triangular surface patch. Using linearity, it is clear that all we have to do is to polarize a onoial u h v k. It is easily verified that the unique p, q -syetric ultiaffine polar for of degree p + q of the onoial u h v k is given by The denoinator f p,q h,k u 1,...,u p ; v 1,...,v q = f p,q h,k u 1,...,u p ; v 1,...,v q p h 1 q k I {1,...,p}, I =h J {1,...,q}, J =k p q is the nuber of ters in the above su. h k u i v j. j J It is also easily verified that the unique syetric ultiaffine polar for of degree of the onoial u h v k is given by The denoinator f h,ku 1,v 1,...,u,v = h h = k f h,ku 1,v 1,...,u,v h 1 h k hk h k i I I J {1,...,} I =h, J =k,i J= As an exaple, consider the following surface known as Enneper s surface: F 1 U, V =U U + UV F U, V =V V + U V F U, V =U V. u i v j. j J i I is the nuber of ters in the above su. We get the polar fors f 1 U 1,V 1, U,V, U,V = U 1 + U + U f U 1,V 1, U,V, U,V = V 1 + V + V U 1U U + U 1V V + U V 1 V + U V 1 V V 1V V + U 1U V + U 1 U V + U U V 1 f U 1,V 1, U,V, U,V = U 1U + U 1 U + U U V 1V + V 1 V + V V.
9 19.. CONTROL POINTS FOR TRIANGULAR SURFACES Control Points For Triangular Surfaces Given an affine frae rst in the plane where r, s, t Pare affinely independent points, it turns out that any syetric ultiaffine ap f: P E is uniquely deterined by a faily of +1+ points where E is any affine space, say R n. Let = {i, j, k N i + j + k = }. The following lea is easily shown see Rashaw [141] or Gallier [70]. Lea Given a reference triangle rst in the affine plane P, given any faily b i, j, k i,j,k of +1+ points in E, there is a unique surface F : P Eof total degree, defined by a syetric -affine polar for f: P E, such that I J K={1,...,} I,J,K pairwise disjoint fr,...,r,s,...,s,t,...,t=b i, j, k, i j k for all i, j, k. Furtherore, f is given by the expression fa 1,...,a = λ i µ j ν k fr,...,r,s,...,s,t,...,t, i I j J k K I J K where a i = λ i r + µ i s + ν i t,withλ i + µ i + ν i =1,and1 i. A point F a on the surface F can be expressed in ters of the Bernstein polynoials Bi,j,k! U, V, T = i!j!k! U i V j T k,as F a =fa,...,a= B i,j,kλ, µ, ν fr,...,r,s,...,s,t,...,t, i, j, k i j k where a = λr + µs + νt, withλ + µ + ν =1. For exaple, with respect to the standard frae rst = 1, 0, 0, 0, 1, 0, 0, 0, 1, we obtain the following 10 control points for the Enneper surface: fr, r, t fr, r, r, 0, 1 fr, r, s ft, t, t 0, 0, 0 fr, t, t 1, 0, 0, 0, 1 fs, t, t 0, 1, 0 fr, s, t 1, 1, 0,, 1 fs, s, t 0,, 1 fr, s, s,, 1 fs, s, s 0,, 1 A faily N =b i, j, k i,j,k of +1+ points in E is called a triangular control net, or Bézier net. Note that the points in = {i, j, k N i + j + k = },
10 580 CHAPTER 19. POLYNOMIAL SURFACES can be thought of as a triangular grid of points in P. For exaple, when = 5, we have the following grid of 1 points: We intentionally let i be the row index, starting fro the left lower corner, and j be the colun index, also starting fro the left lower corner. The control net N =b i, j, k i,j,k can be viewed as an iage of the triangular grid in the affine space E. It follows fro lea that there is a bijection between polynoial surfaces of degree and control nets N =b i, j, k i,j,k. 19. Control Points For Rectangular Surfaces Given any two affine fraes r 1, s 1 and r, s for the affine line A, it turns out that a p, q -syetric ultiaffine ap f:a p A q E is copletely deterined by the faily of p + 1q + 1 points in E b i, j = fr 1,...,r 1, s 1,...,s 1 ; r,...,r, s,...,s, p i i q j j where 0 i p and 0 j q. The following lea is easily shown see Rashaw [141] or Gallier [70]. Lea Let r 1, s 1 and r, s be any two affine fraes for the affine line A, andlete be an affine space of finite diension n. For any natural nubers p, q, for any faily b i, j 0 i p, 0 j q of p + 1q +1 points in E, there is a unique bipolynoial surface F : A A E of degree p, q, with polar for the p + q-ultiaffine p, q -syetric ap f:a p A q E, such that fr 1,...,r 1, s 1,...,s 1 ; r,...,r, s,...,s =b i, j, p i i q j j for all i, 1 i p and all j, 1 j q. Furtherore, f is given by the expression fu 1,...,u p ; v 1,...,v q = 0 i p 0 j q I J= I J={1,...,p} K L= K L={1,...,q} s1 u i uj r 1 s v k vl r s 1 r 1 s 1 r 1 s r s r i I j J k K l L b J, L. ApointF u, v on the surface F can be expressed in ters of the Bernstein polynoials B p i [r 1,s 1 ]u and B q j [r,s ]v, as F u, v = B p i [r 1,s 1 ]u B q j [r,s ]v fr 1,...,r 1, s 1,...,s 1 ; r,...,r, s,...,s. p i i q j j
11 19.4. THE DE CASTELJAU ALGORITHM AND SUBDIVISION 581 A faily N =b i, j 0 i p, 0 j q of p + 1q + 1 points in E, is often called a rectangular control net, or Bézier net. Note that we can view the set of pairs p,q = {i, j N 0 i p, 0 j q}, as a rectangular grid of p + 1q + 1 points in A A. The control net N =b i, j i,j, can be viewed p,q as an iage of the rectangular grid p,q in the affine space E. The portion of the surface F corresponding to the points F u, v for which the paraeters u, v satisfy the inequalities r 1 u s 1 and r v s,is called a rectangular surface patch, or rectangular Bézier patch, andf [r 1, s 1 ], [r, s ] is the trace of the rectangular patch. As an exaple, the onkey saddle is the surface defined by the equation z = x xy. It is easily shown that the onkey saddle is specified by the following rectangular control net of degree, over [0, 1] [0, 1]: sqonknet1 = {{0, 0, 0}, {0, 1/, 0}, {0, 1, 0}, {1/, 0, 0}, {1/, 1/, 0}, {1/, 1, -1}, {/, 0, 0}, {/, 1/, 0}, {/, 1, -}, {1, 0, 1}, {1, 1/, 1}, {1, 1, -}} In the next section, we review a beautiful algorith to copute a point F a on a surface patch using affine interpolation steps, the de Casteljau algorith The de Casteljau Algorith and Subdivision In this section, we quickly review how the de Casteljau algorith can be used to subdivide a triangular patch into three subpatches. For ore details, see Farin [58, 57], Hoschek and Lasser [90], Risler [14], or Gallier [70]. There are also versions of the de Casteljau algorith for rectangular patches, but we will not go into this topic in order to keep the size of this book reasonable. Again, readers are invited to consult Farin [58, 57], Hoschek and Lasser [90], Risler [14], or Gallier [70]. Given an affine frae rst, given a triangular control net N =b i, j, k i,j,k, recall that in ters of the polar for f: P Eof the polynoial surface F : P Edefined by N, for every i, j, k,we have b i, j, k = fr,...,r,s,...,s,t,...,t. i j k Given a = λr + µs + νt in P, where λ + µ + ν = 1, in order to copute F a =fa,...,a, the coputation builds a sort of tetrahedron consisting of + 1 layers. The base layer consists of the original control points in N, which are also denoted as b 0 i, j, k i,j,k. The other layers are coputed in stages, where at stage l, 1 l, the points b l i, j, k i,j,k l are coputed such that b l i, j, k = λb l 1 i+1,j,k + µbl 1 i, j+1,k + νbl 1 i, j, k+1. During the last stage, the single point b 0, 0, 0 is coputed. An easy induction shows that where i, j, k l, and thus, F a =b 0, 0, 0. b l i, j, k = fa,...,a,r,...,r,s,...,s,t,...,t, l i j k
12 58 CHAPTER 19. POLYNOMIAL SURFACES Assuing that a is not on one of the edges of rst, the crux of the subdivision ethod is that the three other faces of the tetrahedron of polar values b l i, j, k besides the face corresponding to the original control net, yield three control nets N ast =b l 0,j,k l,j,k, corresponding to the base triangle ast, N rat =b l i, 0,k i,l,k, corresponding to the base triangle rat, and N rsa =b l i, j, 0 i,j,l, corresponding to the base triangle rsa. If a belongs to one of the edges, say rs, then the triangle rsa is flat, i.e. rsa is not an afine frae, and the net N rsa does not define the surface, but instead a curve. However, in such cases, the degenerate net N rsa is not needed anyway. Fro an ipleentation point of view, we found it convenient to assue that a triangular net N = b i, j, k i,j,k is represented as the list consisting of the concatenation of the +1rows b i, 0, i,b i, 1, i 1,..., b i, i, 0, i.e., fr,...,r, t,...,t, fr,...,r, s, t,...,t,..., fr,...,r, s,...,s,t, fr,...,r, s,...,s, i i i i 1 i i 1 i i where 0 i. As a triangle, the net N is listed fro top-down as ft,...,t ft,...,t,s... ft, s,...,s fs,...,s fr,...,r,t fr,...,r,s 1 1 fr,...,r The ain advantage of this representation is that we can view the net N as a two-diensional array net, such that net[i, j] =b i, j, k with i + j + k =. In fact, only a triangular portion of this array is filled. This way of representing control nets fits well with the convention that the affine frae rst is represented as follows:
13 19.4. THE DE CASTELJAU ALGORITHM AND SUBDIVISION 58 r a t s Figure 19.1: An affine frae Instead of siply coputing F a =b 0, 0, 0, the de Casteljau algorith can be easily adapted to output the three nets N ast, N rat, andn rsa. Using the above version of the de Casteljau algorith, it is possible to recursiveley subdivide a triangular patch. It would see natural to subdivide rst into the three subtriangles ars, ast, and art, where a =1/, 1/, 1/ is the center of gravity of the triangle rst, getting new control nets N ars, N ast and N art using the functions described earlier, and repeat this process recursively. However, this process does not yield a good triangulation of the surface patch, because no progress is ade on the edges rs, st, andtr, and thus, such a triangulation does not converge to the surface patch. Thus, in order to copute triangulations that converge to the surface patch, we need to subdivide the triangle rst in such a way that the edges of the affine frae are subdivided. There are any ways of perforing such subdivisions, and we propose a ethod which has the advantage of yielding a very regular triangulation and of being very efficient. The subdivision strategy that we propose is to divide the affine frae rst into four subtriangles abt, bac, crb, and sca, where a =0, 1/, 1/, b =1/, 0, 1/, and c =1/, 1/, 0, are the iddle points of the sides st, rt and rs respectively, as shown in the diagra below: s sca a c abt bac crb t b r Figure 19.: Subdividing an affine frae rst
14 584 CHAPTER 19. POLYNOMIAL SURFACES It turns out that the four subpatches can be coputed in four calls to the subdivison version of the de Casteljau algorith. Details of such an algorith can be found in Gallier [70], as well as subdivison algoriths for rectangular surfaces. The onkey saddle shown in figure 19. was obtained using a version of the de Casteljau algorith. x y z Figure 19.: A onkey saddle
Data Set Generation for Rectangular Placement Problems
Data Set Generation for Rectangular Placeent Probles Christine L. Valenzuela (Muford) Pearl Y. Wang School of Coputer Science & Inforatics Departent of Coputer Science MS 4A5 Cardiff University George
More information5.7 Chebyshev Multi-section Matching Transformer
/9/ 5_7 Chebyshev Multisection Matching Transforers / 5.7 Chebyshev Multi-section Matching Transforer Reading Assignent: pp. 5-55 We can also build a ultisection atching network such that Γ f is a Chebyshev
More informationarxiv:0805.1434v1 [math.pr] 9 May 2008
Degree-distribution stability of scale-free networs Zhenting Hou, Xiangxing Kong, Dinghua Shi,2, and Guanrong Chen 3 School of Matheatics, Central South University, Changsha 40083, China 2 Departent of
More informationON SELF-ROUTING IN CLOS CONNECTION NETWORKS. BARRY G. DOUGLASS Electrical Engineering Department Texas A&M University College Station, TX 77843-3128
ON SELF-ROUTING IN CLOS CONNECTION NETWORKS BARRY G. DOUGLASS Electrical Engineering Departent Texas A&M University College Station, TX 778-8 A. YAVUZ ORUÇ Electrical Engineering Departent and Institute
More informationBEZIER CURVES AND SURFACES
Department of Applied Mathematics and Computational Sciences University of Cantabria UC-CAGD Group COMPUTER-AIDED GEOMETRIC DESIGN AND COMPUTER GRAPHICS: BEZIER CURVES AND SURFACES Andrés Iglesias e-mail:
More informationComputer Graphics CS 543 Lecture 12 (Part 1) Curves. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)
Computer Graphics CS 54 Lecture 1 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines and flat surfaces Real world objects include
More informationMINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS
MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS JIE HAN AND YI ZHAO Abstract. We show that for sufficiently large n, every 3-unifor hypergraph on n vertices with iniu
More informationCorollary. (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality
Corollary For equidistant knots, i.e., u i = a + i (b-a)/n, we obtain with (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality 120202: ESM4A - Numerical Methods
More informationApplying Multiple Neural Networks on Large Scale Data
0 International Conference on Inforation and Electronics Engineering IPCSIT vol6 (0) (0) IACSIT Press, Singapore Applying Multiple Neural Networks on Large Scale Data Kritsanatt Boonkiatpong and Sukree
More informationPhysics 211: Lab Oscillations. Simple Harmonic Motion.
Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.
More informationComputer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example.
An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: Wire-Frame Representation Object is represented as as a set of points
More informationPlanar Curve Intersection
Chapter 7 Planar Curve Intersection Curve intersection involves finding the points at which two planar curves intersect. If the two curves are parametric, the solution also identifies the parameter values
More informationMulti-Class Deep Boosting
Multi-Class Deep Boosting Vitaly Kuznetsov Courant Institute 25 Mercer Street New York, NY 002 vitaly@cis.nyu.edu Mehryar Mohri Courant Institute & Google Research 25 Mercer Street New York, NY 002 ohri@cis.nyu.edu
More informationFactor Model. Arbitrage Pricing Theory. Systematic Versus Non-Systematic Risk. Intuitive Argument
Ross [1],[]) presents the aritrage pricing theory. The idea is that the structure of asset returns leads naturally to a odel of risk preia, for otherwise there would exist an opportunity for aritrage profit.
More information1 Review of Newton Polynomials
cs: introduction to numerical analysis 0/0/0 Lecture 8: Polynomial Interpolation: Using Newton Polynomials and Error Analysis Instructor: Professor Amos Ron Scribes: Giordano Fusco, Mark Cowlishaw, Nathanael
More informationThe Australian Journal of Mathematical Analysis and Applications
The Australian Journal of Mathematical Analysis and Applications Volume 7, Issue, Article 11, pp. 1-14, 011 SOME HOMOGENEOUS CYCLIC INEQUALITIES OF THREE VARIABLES OF DEGREE THREE AND FOUR TETSUYA ANDO
More informationAudio Engineering Society. Convention Paper. Presented at the 119th Convention 2005 October 7 10 New York, New York USA
Audio Engineering Society Convention Paper Presented at the 119th Convention 2005 October 7 10 New York, New York USA This convention paper has been reproduced fro the authors advance anuscript, without
More informationLecture L9 - Linear Impulse and Momentum. Collisions
J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9 - Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationMachine Learning Applications in Grid Computing
Machine Learning Applications in Grid Coputing George Cybenko, Guofei Jiang and Daniel Bilar Thayer School of Engineering Dartouth College Hanover, NH 03755, USA gvc@dartouth.edu, guofei.jiang@dartouth.edu
More informationAirline Yield Management with Overbooking, Cancellations, and No-Shows JANAKIRAM SUBRAMANIAN
Airline Yield Manageent with Overbooking, Cancellations, and No-Shows JANAKIRAM SUBRAMANIAN Integral Developent Corporation, 301 University Avenue, Suite 200, Palo Alto, California 94301 SHALER STIDHAM
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More informationINTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE SYSTEMS
Artificial Intelligence Methods and Techniques for Business and Engineering Applications 210 INTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE
More informationLecture L26-3D Rigid Body Dynamics: The Inertia Tensor
J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L6-3D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall
More informationConstruction Economics & Finance. Module 3 Lecture-1
Depreciation:- Construction Econoics & Finance Module 3 Lecture- It represents the reduction in arket value of an asset due to age, wear and tear and obsolescence. The physical deterioration of the asset
More informationBinary Embedding: Fundamental Limits and Fast Algorithm
Binary Ebedding: Fundaental Liits and Fast Algorith Xinyang Yi The University of Texas at Austin yixy@utexas.edu Eric Price The University of Texas at Austin ecprice@cs.utexas.edu Constantine Caraanis
More informationA quantum secret ballot. Abstract
A quantu secret ballot Shahar Dolev and Itaar Pitowsky The Edelstein Center, Levi Building, The Hebrerw University, Givat Ra, Jerusale, Israel Boaz Tair arxiv:quant-ph/060087v 8 Mar 006 Departent of Philosophy
More informationHalloween Costume Ideas for the Wii Game
Algorithica 2001) 30: 101 139 DOI: 101007/s00453-001-0003-0 Algorithica 2001 Springer-Verlag New York Inc Optial Search and One-Way Trading Online Algoriths R El-Yaniv, 1 A Fiat, 2 R M Karp, 3 and G Turpin
More informationImage restoration for a rectangular poor-pixels detector
Iage restoration for a rectangular poor-pixels detector Pengcheng Wen 1, Xiangjun Wang 1, Hong Wei 2 1 State Key Laboratory of Precision Measuring Technology and Instruents, Tianjin University, China 2
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationThe Mathematics of Pumping Water
The Matheatics of Puping Water AECOM Design Build Civil, Mechanical Engineering INTRODUCTION Please observe the conversion of units in calculations throughout this exeplar. In any puping syste, the role
More informationWork, Energy, Conservation of Energy
This test covers Work, echanical energy, kinetic energy, potential energy (gravitational and elastic), Hooke s Law, Conservation of Energy, heat energy, conservative and non-conservative forces, with soe
More informationProblem Set 2: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka. Problem 1 (Marginal Rate of Substitution)
Proble Set 2: Solutions ECON 30: Interediate Microeconoics Prof. Marek Weretka Proble (Marginal Rate of Substitution) (a) For the third colun, recall that by definition MRS(x, x 2 ) = ( ) U x ( U ). x
More informationAnswer, Key Homework 7 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Hoework 7 David McIntyre 453 Mar 5, 004 This print-out should have 4 questions. Multiple-choice questions ay continue on the next colun or page find all choices before aking your selection.
More informationHow To Get A Loan From A Bank For Free
Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent,
More informationOn Computing Nearest Neighbors with Applications to Decoding of Binary Linear Codes
On Coputing Nearest Neighbors with Applications to Decoding of Binary Linear Codes Alexander May and Ilya Ozerov Horst Görtz Institute for IT-Security Ruhr-University Bochu, Gerany Faculty of Matheatics
More informationA GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM
A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM LARRY ROLEN Abstract. We study a certain generalization of the classical Congruent Nuber Proble. Specifically, we study integer areas of rational triangles
More informationTopics in Computer Graphics Chap 14: Tensor Product Patches
Topics in Computer Graphics Chap 14: Tensor Product Patches fall, 2011 University of Seoul School of Computer Science Minho Kim Table of contents Bilinear Interpolation The Direct de Casteljau Algorithm
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationMore Unit Conversion Examples
The Matheatics 11 Copetency Test More Unit Conversion Exaples In this docuent, we present a few ore exaples of unit conversions, now involving units of easureent both in the SI others which are not in
More informationA Gas Law And Absolute Zero Lab 11
HB 04-06-05 A Gas Law And Absolute Zero Lab 11 1 A Gas Law And Absolute Zero Lab 11 Equipent safety goggles, SWS, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution
More informationHow To Use Design Mentor
DesignMentor: A Pedagogical Tool for Computer Graphics and Computer Aided Design John L. Lowther and Ching Kuang Shene Programmers: Yuan Zhao and Yan Zhou (ver 1) Budirijanto Purnomo (ver 2) Michigan Technological
More informationTrading Regret for Efficiency: Online Convex Optimization with Long Term Constraints
Journal of Machine Learning Research 13 2012) 2503-2528 Subitted 8/11; Revised 3/12; Published 9/12 rading Regret for Efficiency: Online Convex Optiization with Long er Constraints Mehrdad Mahdavi Rong
More informationAlgebra (Expansion and Factorisation)
Chapter10 Algebra (Expansion and Factorisation) Contents: A B C D E F The distributive law Siplifying algebraic expressions Brackets with negative coefficients The product (a + b)(c + d) Geoetric applications
More informationSPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM
SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM ALEX WRIGHT 1. Intoduction A fixed point of a function f from a set X into itself is a point x 0 satisfying f(x 0 ) = x 0. Theorems which establish the
More information2. FINDING A SOLUTION
The 7 th Balan Conference on Operational Research BACOR 5 Constanta, May 5, Roania OPTIMAL TIME AND SPACE COMPLEXITY ALGORITHM FOR CONSTRUCTION OF ALL BINARY TREES FROM PRE-ORDER AND POST-ORDER TRAVERSALS
More informationThe Research of Measuring Approach and Energy Efficiency for Hadoop Periodic Jobs
Send Orders for Reprints to reprints@benthascience.ae 206 The Open Fuels & Energy Science Journal, 2015, 8, 206-210 Open Access The Research of Measuring Approach and Energy Efficiency for Hadoop Periodic
More informationCHAPTER 1 Splines and B-splines an Introduction
CHAPTER 1 Splines and B-splines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the
More informationA CHAOS MODEL OF SUBHARMONIC OSCILLATIONS IN CURRENT MODE PWM BOOST CONVERTERS
A CHAOS MODEL OF SUBHARMONIC OSCILLATIONS IN CURRENT MODE PWM BOOST CONVERTERS Isaac Zafrany and Sa BenYaakov Departent of Electrical and Coputer Engineering BenGurion University of the Negev P. O. Box
More informationCRM FACTORS ASSESSMENT USING ANALYTIC HIERARCHY PROCESS
641 CRM FACTORS ASSESSMENT USING ANALYTIC HIERARCHY PROCESS Marketa Zajarosova 1* *Ph.D. VSB - Technical University of Ostrava, THE CZECH REPUBLIC arketa.zajarosova@vsb.cz Abstract Custoer relationship
More informationModified Latin Hypercube Sampling Monte Carlo (MLHSMC) Estimation for Average Quality Index
Analog Integrated Circuits and Signal Processing, vol. 9, no., April 999. Abstract Modified Latin Hypercube Sapling Monte Carlo (MLHSMC) Estiation for Average Quality Index Mansour Keraat and Richard Kielbasa
More information1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
More informationMedia Adaptation Framework in Biofeedback System for Stroke Patient Rehabilitation
Media Adaptation Fraework in Biofeedback Syste for Stroke Patient Rehabilitation Yinpeng Chen, Weiwei Xu, Hari Sundara, Thanassis Rikakis, Sheng-Min Liu Arts, Media and Engineering Progra Arizona State
More informationDegree Reduction of Interval SB Curves
International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:13 No:04 1 Degree Reduction of Interval SB Curves O. Ismail, Senior Member, IEEE Abstract Ball basis was introduced
More informationA Gas Law And Absolute Zero
A Gas Law And Absolute Zero Equipent safety goggles, DataStudio, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution This experient deals with aterials that are very
More informationA magnetic Rotor to convert vacuum-energy into mechanical energy
A agnetic Rotor to convert vacuu-energy into echanical energy Claus W. Turtur, University of Applied Sciences Braunschweig-Wolfenbüttel Abstract Wolfenbüttel, Mai 21 2008 In previous work it was deonstrated,
More informationTransmission Eigenvalues in One Dimension
Transission Eigenvalues in One Diension John Sylvester Abstract We show how to locate all the transission eigenvalues for a one diensional constant index of refraction on an interval. 1 Introduction The
More informationAn Innovate Dynamic Load Balancing Algorithm Based on Task
An Innovate Dynaic Load Balancing Algorith Based on Task Classification Hong-bin Wang,,a, Zhi-yi Fang, b, Guan-nan Qu,*,c, Xiao-dan Ren,d College of Coputer Science and Technology, Jilin University, Changchun
More informationHW 2. Q v. kt Step 1: Calculate N using one of two equivalent methods. Problem 4.2. a. To Find:
HW 2 Proble 4.2 a. To Find: Nuber of vacancies per cubic eter at a given teperature. b. Given: T 850 degrees C 1123 K Q v 1.08 ev/ato Density of Fe ( ρ ) 7.65 g/cc Fe toic weight of iron ( c. ssuptions:
More informationREQUIREMENTS FOR A COMPUTER SCIENCE CURRICULUM EMPHASIZING INFORMATION TECHNOLOGY SUBJECT AREA: CURRICULUM ISSUES
REQUIREMENTS FOR A COMPUTER SCIENCE CURRICULUM EMPHASIZING INFORMATION TECHNOLOGY SUBJECT AREA: CURRICULUM ISSUES Charles Reynolds Christopher Fox reynolds @cs.ju.edu fox@cs.ju.edu Departent of Coputer
More informationCalculating the Return on Investment (ROI) for DMSMS Management. The Problem with Cost Avoidance
Calculating the Return on nvestent () for DMSMS Manageent Peter Sandborn CALCE, Departent of Mechanical Engineering (31) 45-3167 sandborn@calce.ud.edu www.ene.ud.edu/escml/obsolescence.ht October 28, 21
More informationRECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION. Henrik Kure
RECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION Henrik Kure Dina, Danish Inforatics Network In the Agricultural Sciences Royal Veterinary and Agricultural University
More informationRegions in a circle. 7 points 57 regions
Regions in a circle 1 point 1 region points regions 3 points 4 regions 4 points 8 regions 5 points 16 regions The question is, what is the next picture? How many regions will 6 points give? There's an
More informationAUC Optimization vs. Error Rate Minimization
AUC Optiization vs. Error Rate Miniization Corinna Cortes and Mehryar Mohri AT&T Labs Research 180 Park Avenue, Florha Park, NJ 0793, USA {corinna, ohri}@research.att.co Abstract The area under an ROC
More informationABSTRACT KEYWORDS. Comonotonicity, dependence, correlation, concordance, copula, multivariate. 1. INTRODUCTION
MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS BY INGE KOCH AND ANN DE SCHEPPER ABSTRACT In this contribution, a new easure of coonotonicity for -diensional vectors is introduced, with values between
More informationReconnect 04 Solving Integer Programs with Branch and Bound (and Branch and Cut)
Sandia is a ultiprogra laboratory operated by Sandia Corporation, a Lockheed Martin Copany, Reconnect 04 Solving Integer Progras with Branch and Bound (and Branch and Cut) Cynthia Phillips (Sandia National
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More informationDERIVATIVES AS MATRICES; CHAIN RULE
DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we
More informationIEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1. Secure Wireless Multicast for Delay-Sensitive Data via Network Coding
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1 Secure Wireless Multicast for Delay-Sensitive Data via Network Coding Tuan T. Tran, Meber, IEEE, Hongxiang Li, Senior Meber, IEEE,
More informationGrade 6 Mathematics Performance Level Descriptors
Limited Grade 6 Mathematics Performance Level Descriptors A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Grade 6 Mathematics. A student at this
More informationComputational Geometry. Lecture 1: Introduction and Convex Hulls
Lecture 1: Introduction and convex hulls 1 Geometry: points, lines,... Plane (two-dimensional), R 2 Space (three-dimensional), R 3 Space (higher-dimensional), R d A point in the plane, 3-dimensional space,
More information6. Time (or Space) Series Analysis
ATM 55 otes: Tie Series Analysis - Section 6a Page 8 6. Tie (or Space) Series Analysis In this chapter we will consider soe coon aspects of tie series analysis including autocorrelation, statistical prediction,
More informationGeometrico-static Analysis of Under-constrained Cable-driven Parallel Robots
Geoetrico-static Analysis of Under-constrained Cable-driven Parallel Robots M. Carricato and J.-P. Merlet 1 DIEM - Dept. of Mechanical Engineering, University of Bologna, Italy, e-ail: arco.carricato@ail.ing.unibo.it
More informationGeneral tolerances for Iinearand angular dimensions and geometrical=.tolerances
oa / - UDC 21 753 1 : 21 7 : 21 9 : 744 43 DEUTSCHE NORM April 1991 General tolerances for Iinearand angular diensions and geoetrical= tolerances (not to be used for new destgns) '-j' ;,, DIN 718 Allgeeintoleranzen
More informationLesson 44: Acceleration, Velocity, and Period in SHM
Lesson 44: Acceleration, Velocity, and Period in SHM Since there is a restoring force acting on objects in SHM it akes sense that the object will accelerate. In Physics 20 you are only required to explain
More informationOnline Bagging and Boosting
Abstract Bagging and boosting are two of the ost well-known enseble learning ethods due to their theoretical perforance guarantees and strong experiental results. However, these algoriths have been used
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional
More informationAnalyzing Spatiotemporal Characteristics of Education Network Traffic with Flexible Multiscale Entropy
Vol. 9, No. 5 (2016), pp.303-312 http://dx.doi.org/10.14257/ijgdc.2016.9.5.26 Analyzing Spatioteporal Characteristics of Education Network Traffic with Flexible Multiscale Entropy Chen Yang, Renjie Zhou
More informationBudget-optimal Crowdsourcing using Low-rank Matrix Approximations
Budget-optial Crowdsourcing using Low-rank Matrix Approxiations David R. Karger, Sewoong Oh, and Devavrat Shah Departent of EECS, Massachusetts Institute of Technology Eail: {karger, swoh, devavrat}@it.edu
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationPricing Asian Options using Monte Carlo Methods
U.U.D.M. Project Report 9:7 Pricing Asian Options using Monte Carlo Methods Hongbin Zhang Exaensarbete i ateatik, 3 hp Handledare och exainator: Johan Tysk Juni 9 Departent of Matheatics Uppsala University
More informationThe Velocities of Gas Molecules
he Velocities of Gas Molecules by Flick Colean Departent of Cheistry Wellesley College Wellesley MA 8 Copyright Flick Colean 996 All rights reserved You are welcoe to use this docuent in your own classes
More informationNURBS Drawing Week 5, Lecture 10
CS 430/536 Computer Graphics I NURBS Drawing Week 5, Lecture 10 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University
More informationPresentation Safety Legislation and Standards
levels in different discrete levels corresponding for each one to a probability of dangerous failure per hour: > > The table below gives the relationship between the perforance level (PL) and the Safety
More informationEfficient Key Management for Secure Group Communications with Bursty Behavior
Efficient Key Manageent for Secure Group Counications with Bursty Behavior Xukai Zou, Byrav Raaurthy Departent of Coputer Science and Engineering University of Nebraska-Lincoln Lincoln, NE68588, USA Eail:
More information1 Analysis of heat transfer in a single-phase transformer
Assignent -7 Analysis of heat transr in a single-phase transforer The goal of the first assignent is to study the ipleentation of equivalent circuit ethod (ECM) and finite eleent ethod (FEM) for an electroagnetic
More informationALLOWABLE STRESS DESIGN OF CONCRETE MASONRY BASED ON THE 2012 IBC & 2011 MSJC. TEK 14-7C Structural (2013)
An inforation series fro the national authority on concrete asonry technology ALLOWABLE STRESS DESIGN OF CONCRETE MASONRY BASED ON THE 2012 IBC & 2011 MSJC TEK 14-7C Structural (2013) INTRODUCTION Concrete
More informationThe Virtual Spring Mass System
The Virtual Spring Mass Syste J. S. Freudenberg EECS 6 Ebedded Control Systes Huan Coputer Interaction A force feedbac syste, such as the haptic heel used in the EECS 6 lab, is capable of exhibiting a
More informationEvaluating Inventory Management Performance: a Preliminary Desk-Simulation Study Based on IOC Model
Evaluating Inventory Manageent Perforance: a Preliinary Desk-Siulation Study Based on IOC Model Flora Bernardel, Roberto Panizzolo, and Davide Martinazzo Abstract The focus of this study is on preliinary
More informationChapter 5. Principles of Unsteady - State Heat Transfer
Suppleental Material for ransport Process and Separation Process Principles hapter 5 Principles of Unsteady - State Heat ransfer In this chapter, we will study cheical processes where heat transfer is
More informationSAMPLING METHODS LEARNING OBJECTIVES
6 SAMPLING METHODS 6 Using Statistics 6-6 2 Nonprobability Sapling and Bias 6-6 Stratified Rando Sapling 6-2 6 4 Cluster Sapling 6-4 6 5 Systeatic Sapling 6-9 6 6 Nonresponse 6-2 6 7 Suary and Review of
More informationCPU Animation. Introduction. CPU skinning. CPUSkin Scalar:
CPU Aniation Introduction The iportance of real-tie character aniation has greatly increased in odern gaes. Aniating eshes ia 'skinning' can be perfored on both a general purpose CPU and a ore specialized
More informationModeling Curves and Surfaces
Modeling Curves and Surfaces Graphics I Modeling for Computer Graphics!? 1 How can we generate this kind of objects? Umm!? Mathematical Modeling! S Do not worry too much about your difficulties in mathematics,
More informationPREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW
PREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW ABSTRACT: by Douglas J. Reineann, Ph.D. Assistant Professor of Agricultural Engineering and Graee A. Mein, Ph.D. Visiting Professor
More informationNURBS Drawing Week 5, Lecture 10
CS 430/536 Computer Graphics I NURBS Drawing Week 5, Lecture 10 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationReading 13 : Finite State Automata and Regular Expressions
CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model
More informationIntroduction to Unit Conversion: the SI
The Matheatics 11 Copetency Test Introduction to Unit Conversion: the SI In this the next docuent in this series is presented illustrated an effective reliable approach to carryin out unit conversions
More informationExperimental and Theoretical Modeling of Moving Coil Meter
Experiental and Theoretical Modeling of Moving Coil Meter Prof. R.G. Longoria Updated Suer 010 Syste: Moving Coil Meter FRONT VIEW Electrical circuit odel Mechanical odel Meter oveent REAR VIEW needle
More informationNovember 16, 2015. Interpolation, Extrapolation & Polynomial Approximation
Interpolation, Extrapolation & Polynomial Approximation November 16, 2015 Introduction In many cases we know the values of a function f (x) at a set of points x 1, x 2,..., x N, but we don t have the analytic
More information