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The l-basis of a planar rational curve—properties and

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The l -basis of a planar rationalcurve—properties and computationFalai Chen a,*and Wenping Wang baDepartment of Mathematics,University of Science and Technology of China,Hefei,Anhui 230026,PR China b Department of Computer Science and Information Systems,University of Hong Kong,Hong Kong,PR ChinaReceived 29October 2001received in revised form 19November 2002accepted 3December 2002AbstractA moving line L ðx ;y ;t Þ¼0is a family of lines with one parameter t in a plane.A moving line L ðx ;y ;t Þ¼0is said to follow a rational curve P ðt Þif the point P ðt 0Þis on the line L ðx ;y ;t 0Þ¼0for any parameter value t 0.A l -basis of a rational curve P ðt Þis a pair of lowest degree moving lines that constitute a basis of the module formed by all the moving lines fol-lowing P ðt Þ,which is the syzygy module of P ðt Þ.The study of moving lines,especially the l -basis,has recently led to an efficient method,called the moving line method ,for computing the implicit equation of a rational curve [3,6].In this paper,we present properties and equivalent definitions of a l -basis of a planar rational curve.Several of these properties and definitions are new,and they help to clarify an earlier definition of the l -basis [3].Furthermore,based on some of these newly established properties,an efficient algorithm is presented to compute a l -basis of a planar rational curve.This algorithm applies vector elimination to the moving line module of P ðt Þ,and has O ðn 2Þtime complexity,where n is the degree of P ðt Þ.We show that the new algorithm is more efficient than the fastest previous algorithm [7].Ó2003Elsevier Science (USA).All rights reserved.Keywords:Rational curve;Implicitization;Moving line;l -Basis;SyzygymoduleGraphical Models 64(2003)368–381/locate/gmod*Corresponding author.E-mail addresses:chenfl@ (F.Chen),wenping@csis.hku.hk (W.Wang).1524-0703/03/$-see front matter Ó2003Elsevier Science (USA).All rights reserved.doi:10.1016/S1524-0703(02)00017-5F.Chen,W.Wang/Graphical Models64(2003)368–3813691.IntroductionA moving line is a family of lines with one parameter t in the plane,and can thus be represented by Lðx;y;tÞ AðtÞxþBðtÞyþCðtÞ¼0,where AðtÞ;BðtÞ;CðtÞare polynomials.A moving line Lðx;y;tÞis said to follow a rational curve PðtÞ¼ðaðtÞ;bðtÞ;cðtÞÞif the point Pðt0Þis on the line Lðx;y;t0Þ¼0for any parameter value t0.As a convention,we call a moving line that follows the curve PðtÞa moving line of PðtÞ.A l-basis of a rational curve PðtÞis a pair of lowest degree moving lines that constitute a basis of the module formed by all the moving lines of PðtÞ.The l-basis of a planar rational curve has been proposed for computing the im-plicit equation of a planar rational curve[3].Applying a variant of BezoutÕs resultant to the l-basis,we can write the implicit equation of PðtÞas anðnÀlÞÂðnÀlÞde-terminant,where0<l6½n=2 ;in contrast,with other resultant based methods,the implicit equation must be written either as an nÂn determinant(using the Bezout resultant)or as a2nÂ2n determinant(using the Sylvester resultant).In the generic case where l¼b n=2c,the implicit equation can be written as an d n=2eÂd n=2e de-terminant.Thus,the l-basis provides a compact solution to the implicitization of a planar rational curve.Recent studies also show that the l-basis can be used to derive inversion formulas or study the singular points of a planar rational curve[2];besides,the rational para-metric equation of the curve can easily be obtained from the l-basis.Hence,the l-basis serves as a compact and useful representation of a planar rational curve—connecting its implicit equation and parametric equation,and facilitating the study of many properties of the curve.The definition and some properties of the l-basis of a planar rational curve have been studied in[3].However,the definition of the l-basis has not been given consis-tently,and the properties,especially those characterizing properties,of the l-basis have not been presented completely or systematically in the literature.In light of this, we present in Section2a list of properties and equivalent definitions of the l-basis of a planar rational curve.Several of these properties and definitions are new and pro-vide insight into efficient computation of the l-basis.These results have recently been used in studying the reparameterization of rational ruled surfaces[1].As an applica-tion,in Sections3we describe a new algorithm for computing the l-basis of a planar rational curve,and show that,besides being conceptually simpler,this algorithm is more efficient than the previous known fastest algorithm for computing the l-basis [7].We conclude the paper in Section4with some open problems.2.Definition and properties of l-basisWe begin with some preliminary knowledge about modules.Let R½t and R½x;y;t be the polynomial rings over thefield of real numbers,R,and R½t d denote the set of d-dimensional row vectors with entries in R½t .A set of vector polynomials M&R½t d is called a module over R½t if h1f1þh2f22M for any f1;f22M and h1;h22R½t .Suppose that M&R½t d is a module.If there exists afinite set of elements f i2M, i¼1;...;m,such that any f2M can be expressed byf¼h1f1þÁÁÁþh m f m;ð1Þwith h i2R½t ,i¼1;...;m,then M is said to befinitely generated,f f1;...;f m g is called a generating set of M,and we write M¼h f1;...;f m i.If expression(1)is un-ique for any f2M,then the generating set f f1;...;f m g is called a basis of the module M.Afinitely generated module over R½t always has a basis and is called a free module[4].Vector polynomials gi ðtÞin R½t 3,i¼1;2;...;m,are said to be R½t -linearly inde-pendent if P mi¼1h iðtÞg iðtÞ¼0,with h iðtÞ2R½t ,implies h iðtÞ¼0for i¼1;2;...;m.Let a planar rational curve be given byPðtÞ¼ðaðtÞ;bðtÞ;cðtÞÞð2Þin homogeneous form,where aðtÞ,bðtÞ,cðtÞ2R½t are relatively prime.The degree of PðtÞis defined to be n max f degðaðtÞÞ;degðbðtÞÞ;degðcðtÞÞg.For brevity,we will often denote aðtÞ;bðtÞ,and cðtÞby a,b,and c,respectively,where there is no danger of confusion.A moving line is a family of lines with one parameter t,defined by[5]Lðx;y;tÞ:¼AðtÞxþBðtÞyþCðtÞ¼0;ð3Þwhere AðtÞ;BðtÞ;CðtÞ2R½t .The degree of Lðx;y;tÞis defined to bedegðLðx;y;tÞÞ max f degðAðtÞÞ;degðBðtÞÞ;degðCðtÞÞg:For convenience,we also denote a moving line by LðtÞ ðAðtÞ;BðtÞ;CðtÞÞ.The moving line(3)follows the rational curve PðtÞ¼ðaðtÞ;bðtÞ;cðtÞÞif and only if LðtÞÁPðtÞ¼AðtÞaðtÞþBðtÞbðtÞþCðtÞcðtÞ 0:ð4ÞA moving line Lðx;y;tÞ¼0follows the curve PðtÞif the point Pðt0Þis on the line Lðx;y;t0Þ¼0for any parameter value t0.So one may say that an arbitrary point Pðt0Þof the curve PðtÞis at the intersection of two different moving lines of PðtÞwith the same parameter value t0.The moving line ideal of a rational curve PðtÞis defined to beI P¼f h1ðcxÀaÞþh2ðcyÀbÞj h1;h22R½x;y;t g&R½x;y;t :ð5ÞLet J1denote the set of all polynomials of degree one in x and y but somefinite degree in t.Then M P I P\J1consists of all moving lines in I P and is a module over R½t [3].It is proved in[3]that a moving line Lðx;y;tÞis a moving line of PðtÞif and only if Lðx;y;tÞ2M P.Thus M P is called the moving line module of PðtÞ.The moving line module M P of the rational curve PðtÞis isomorphic to the moduleM P¼fðAðtÞ;BðtÞ;CðtÞÞj AðtÞaðtÞþBðtÞbðtÞþCðtÞcðtÞ 0g&R½t 3under the isomorphism AðtÞxþBðtÞyþCðtÞ!ðAðtÞ;BðtÞ;CðtÞÞ.So we will use M P and M P interchangeably to represent the moving line module of PðtÞ.370 F.Chen,W.Wang/Graphical Models64(2003)368–381F.Chen,W.Wang/Graphical Models64(2003)368–381371Let a vector polynomial pðtÞ2R½t 3be written asX mp¼ðp1ðtÞ;p2ðtÞ;p3ðtÞÞ¼ðp i1;p i2;p i3Þt i;i¼0with the leading coefficient vectorðp m1;p m2;p m3Þ¼0.We denoteðp m1;p m2;p m3Þby LVðpÞand the degree of p by degðpÞ¼m.For example,forp¼ð2t2þ3tþ1;tþ4;t2þ2tþ3Þ;LVðpÞ¼ð2;0;1Þand degðpÞ¼2.Definition1.Two moving lines p¼p1ðtÞxþp2ðtÞyþp3ðtÞand q¼q1ðtÞxþq2ðtÞyþq3ðtÞare called a l-basis of a rational curve PðtÞ,or equivalently,a l-basis of M P,if 1.p and q are a basis of M P,i.e.,any moving line L2M P can be expressed byL¼h1pþh2q;ð6Þwith h1;h22R½t ;and2.p and q have the lowest degrees among all bases of M P;that is,assumingdegðpÞ6degðqÞ,then there does not exist another basis~p and~q of M P,with degð~pÞ6degð~qÞ,such that degð~pÞ<degðpÞor degð~qÞ<degðqÞ.The above definition of a l-basis appears to be different from a definition given in [3]for the l-basis for a rational curve in R d,d P2.Actually,two definitions for the l-basis are given in[3];thefirst one on page809is for a planar rational curve and the second one on page824for a rational curve in R d.Thefirst definition is one of the equivalent definitions for the l-basis to be given in the present paper;how-ever,the second condition,which covers the case of a planar rational curve,though equivalent to thefirst one,contains a redundant condition.The following is proved in[3]:Proposition1.Any planar rational curve PðtÞhas a l-basis.2.1.PropertiesThe existence of a l-basis of a planar rational curve,along with some of its prop-erties,are proved in[3].In the following we will present a more complete list of prop-erties of the l-basis,including some from[3],and then give several equivalent definitions of the l-basis.Theorem1.Let p¼p1ðtÞxþp2ðtÞyþp3ðtÞand q¼q1ðtÞxþq2ðtÞyþq3ðtÞbe a l-basis of a planar rational curve PðtÞ,where degðpÞ6degðqÞ.Denote p¼ðp1;p2;p3Þand q¼ðq1;q2;q3Þ.Then the following properties hold:1.p and q are R½t -linearly independent.2.pðt0Þ¼0and qðt0Þ¼0for any parameter value t0.3.pðt0Þand qðt0Þare linearly independent for any parameter value t0.4.LVðpÞand LVðqÞare linearly independent.372 F.Chen,W.Wang/Graphical Models64(2003)368–3815.Expression(6)is unique for any moving line L of PðtÞ.6.Any moving line L of PðtÞcan be expressed in(6)with degðh1pÞ6degðLÞanddegðh2qÞ6degðLÞ.7.pÂq¼k PðtÞfor some non-zero constant k.8.degðpÞþdegðqÞ¼n.Denote degðpÞ¼l and degðqÞ¼nÀl in t,where06l6b n=2c.Then l is unique for a given curve PðtÞ;furthermore,l is the lowest degree of any moving line of the curve PðtÞ.9.I P¼h p;q i.10.The implicit equation of PðtÞis given by Resðp;q;tÞ,i.e.,the resultant of p and qwith respect to t.Proof.(1)Consider the two moving lines of PðtÞ¼ðaðtÞ;bðtÞ;cðtÞÞ:u¼ðc;0;ÀaÞand v¼ð0;c;ÀbÞ,which are clearly R½t -linearly independent.Then u¼h11pþh12q and v¼h21pþh22q for some polynomials h ij2R½t ,i;j¼1;2.Hencecða;b;cÞ¼uÂv¼h pÂq;where h¼h11h22Àh12h21¼0.Thus pÂq¼0except for possiblyfinitely many values,and it follows that p and q are R½t -linearly independent.(2)Suppose pðt0Þ¼0for some t0,thenðtÀt0Þj pðtÞ.Let~pðtÞ¼pðtÞ=ðtÀt0Þ.Then ~pðtÞand qðtÞare also a basis of M P.But degð~pðtÞÞ<degðpðtÞÞ,a contradiction to that pðtÞand qðtÞare a l-basis.Similarly,one can show that qðt0Þ¼0.(3)Suppose that pðt0Þand qðt0Þare linearly dependent for some t0.Then there ex-ist constants a and b,both of them are nonzero,such that a pðt0Þþb qðt0Þ¼0.Let L¼a pþb q.Then L2M P,and L is not identically zero,since,by Property(1),p and q are R½t -linearly independent.Since Lðt0Þ¼0,ðtÀt0Þj L.Let~q¼L=ðtÀt0Þ. Then it is easy to verify that p and~q also form a basis of the module M P.But degð~qÞ<degðqÞ,contradicting that p and q are a l-basis.Hence,pðt0Þand qðt0Þare linearly independent for any t0.(4)Suppose that LVðpÞand LVðqÞare linearly dependent.Then a LVðpÞþb LVðqÞ¼0for some constants a and b,both are nonzero.Let~q¼a p t dþb q,where d¼degðqÞÀdegðpÞ.Then p and~q also form a basis of M P.However,since the lead-ing coefficient vectors of a p t d and b q cancel each other,degð~qÞ<degðqÞ,contradict-ing that q has the lowest degree among all the bases of PðtÞ.Hence,LVðpÞand LVðqÞare linearly independent.(5)For any moving line L of P,suppose L¼h1pþh2q and L¼g1pþg2q.Then ðh1Àg1Þpþðh2Àg2Þq¼0.It follows that h1¼g1and h2¼g2,since,by Property (1)above,p and q are R½t -linearly independent.Hence expression(6)is unique.(6)Suppose L¼h1pþh2q.If degðh1pÞ>degðLÞ,then LVðh1pÞþLVðh2qÞ¼0, thus LVðpÞand LVðqÞare linearly dependent,contradicting Property(4)above. Hence,degðh1pÞ6degðLÞ.Similarly,degðh2qÞ6degðLÞ.(7)Since pÁPðtÞ¼qÁPðtÞ¼0,and p and q are R½t -linearly independent,we have pÂq¼ðgðtÞ=hðtÞÞPðtÞfor gðtÞ;hðtÞ2R½t ,where g and h are relatively prime.Since aðtÞ,bðtÞ,and cðtÞare relatively prime,hðtÞmust be a constant.If gðtÞis not a con-stant,letting t0be a zero of gðtÞover the complexfield,then pðt0ÞÂqðt0Þ¼0,i.e.pðt0Þand qðt0Þare linearly dependent;but this contradicts Property(3)above.So gðtÞis also a constant,and plainly,g¼0.Hence,pÂq¼k PðtÞfor some nonzero con-stant k.(8)From Property(7),degðpÂqÞ¼degðPÞ¼n.Let p¼pl t lþp lÀ1t lÀ1þÁÁÁþp0and q¼qm t mþq mÀ1t mÀ1þÁÁÁþq0.Then pÂq¼ðp lÂq mÞt lþmþÁÁÁþp0Âq0.ByProperty(4),LVðpÞ¼pl and LVðqÞ¼qmare linearly independent,so plÂq m¼0.Itfollows thatdegðpÞþdegðqÞ¼lþm¼degðpÂqÞ¼degðPÞ¼n:The uniqueness of l is implied by the minimality of the degrees of p and q.On the other hand,let L2M P be any moving line of PðtÞ,then L¼h1pþh2q for some h1;h22R½t .If degðLÞ<l,then LVðh1pÞand LVðh2qÞcancel with each other. Hence LVðpÞand LVðqÞare linearly dependent,a contradiction to Property4above. Thus l is the lowest degree of any moving line of PðtÞ.(9)First recall that the moving line ideal I P is generated by cxÀa and cyÀb. Since cxÀa;cyÀb2M P,and p and q are a basis of M P,cxÀa;cyÀb can be ex-pressed by linear combinations of p and q over R½t .Hence,I P can be generated by p and q.(10)See[3].This completes of the proof of Theorem1.ÃThe relationship between different l-bases of a rational curve PðtÞis given by the following theorem.Theorem2.Let p;q,and~p;~q be two l-bases of a rational curve PðtÞ,with degðpÞ6degðqÞand degð~pÞ6degð~qÞ.Then degðpÞ¼degð~pÞand degðqÞ¼degð~qÞ. Furthermore,if degðpÞ¼degðqÞ,then~p¼a1pþb1q;~q¼a2pþb2qfor some constants a1,a2,b1,and b2with a1b2Àa2b1¼0;if degðpÞ<degðqÞ,then ~p¼a p;~q¼h pþb qfor some nonzero constants a and b,and h2R½t with degðhÞ6degðqÞÀdegðpÞ.Proof.By the definition of the l-basis,it is straightforward to see that degðpÞ¼degð~pÞand degðqÞ¼degð~qÞ.This is also Property(8)of Theorem1.Since~p;~q2M P,and p;q are a basis of M P,~p¼a1pþb1q;~q¼a2pþb2q;with a i;b i2R½t ,i¼1;2.If degðpÞ¼degðqÞ,then degð~pÞ¼degð~qÞ¼degðpÞ¼degðqÞ.It then follows from by Property(6)of Theorem1that a1,a2,b1,and b2are constants.We have a1b2Àa2b1¼0since~p and~q are R½t -linearly independent.If degðpÞ<degðqÞ,then degð~pÞ¼degðpÞ<degðqÞ.Again by Property(6)of Theorem1,b1¼0and a1is a nonzero constant.Similarly,from degð~qÞ¼degðqÞ>degðpÞ,one has b2is a nonzero constant and degða2Þ6degðqÞÀdegðpÞ.ÃF.Chen,W.Wang/Graphical Models64(2003)368–381373374 F.Chen,W.Wang/Graphical Models64(2003)368–3812.2.Equivalent definitionsNext we provide some equivalent definitions of a l-basis of a planar rational curve.Theorem3.Let p;q2M P be two moving lines of a planar rational curve PðtÞof degree n.Assume degðpÞ6degðqÞ.Then p and q form a l-basis of PðtÞif and only if one of the following conditions holds1.Any moving line L can be expressed in(6)with degðh1pÞ6degðLÞanddegðh2qÞ6degðLÞ.2.Any moving line L can be expressed in(6),and LVðpÞand LVðqÞare linearly inde-pendent.3.Any moving line L can be expressed in(6)and degðpÞþdegðqÞ¼n.4.pÂq¼k P for some non-zero constant k and degðpÞþdegðqÞ¼n.5.pÂq¼k P for some non-zero constant k,and LVðpÞand LVðqÞare linearly inde-pendent.6.degðpÞþdegðqÞ¼n,and p and q are R½t -linearly independent.7.degðpÞþdegðqÞ¼n,and LVðpÞand LVðqÞare linearly independent.8.degðpÞþdegðqÞ¼n and I P¼h p;q i.Here p¼pÁðx;y;1Þand q¼qÁðx;y;1Þ. Proof.The necessity of all the above conditions has been proved in Theorem1.So in the following we consider only their sufficiency.(1)We just need to show that p and q have the lowest possible degree among all the bases of M P.Let~p and~q be any basis of M P with degð~pÞ6degð~qÞ.Wefirst show degðpÞ6degð~pÞ.Clearly,~p¼h11pþh12q;~q¼h21pþh22qfor some h ij2R½t ,i;j¼1;2.If h11¼0,since degðh11pÞ6degð~pÞ,it follows that degðpÞ6degð~pÞ;if h11¼0and h12¼0,then degðpÞ6degðqÞ6degð~pÞ.So there is always degðpÞ6degð~pÞ.Next we show degðqÞ6degð~qÞ.If h22¼0,since degðh22qÞ6degð~qÞ,we have degðqÞ6degð~qÞ.If h22¼0,then h12¼0,for otherwise~p and~q cannot be a basis of P.In this case,since degðh12qÞ6degð~pÞ,we have degðqÞ6degð~pÞ6degð~qÞ.So there is always degðqÞ6degð~qÞ.We have shown that p and q are a basis of M P with the lowest degrees.Hence,p and q form a l-basis of M P.(2)Given a moving line L of P,suppose L¼h1pþh2q.Since LVðpÞand LVðqÞare linearly independent,LVðh1pÞand LVðh2qÞdo not cancel each other;hence, degðh1pÞ6degðLÞand degðh2qÞ6degðLÞ.Then,by condition(1),p and q form a l-basis of M P.(3)Again,we just need to show that p and q have the lowest degrees possible.First note that p and q form a basis of M P since,by assumption,any moving line in M P is expressible in(6).Let~p and~q be a l-basis of M P with degð~pÞ¼l and degð~qÞ¼nÀl,where0<l6b n=2c.Because of the minimality of the degrees ofF.Chen,W.Wang/Graphical Models64(2003)368–381375 the l-basis,we have degðpÞP l.If degðpÞ>l,then degðpÞ6degðqÞ<nÀl since degðpÞþdegðqÞ¼n.By Property(6)of the l-basis in Theorem1,p¼h1~p and q¼h2~p for some h1;h22R½t .Thus h2pÀh1q¼0,that is,p and q are R½t -linearly dependent.This contradicts that p and q form a basis of M P.Thus degðpÞ¼l and degðqÞ¼nÀl,and hence,p and q are a l-basis of M P.(4)Since pÂq¼k P for some nonzero constant k,for any moving line L of PðtÞ, LÁpÂq¼LÁP 0.Hence g L¼h1pþh2q for some g;h1;h22R½t ,where GCDðg;h1;h2Þ¼1.Further,if g is not a constant,letting t0be a zero of g over the complexfield,then h1ðt0Þpðt0Þþh2ðt0Þqðt0Þ¼0,but this contradicts Property (3)of Theorem1stating that pðt0Þand qðt0Þare linearly independent for any param-eter value t0.Hence g is a nonzero constant.So we may write L¼h1pþh2q for some h1;h22R½t .Then,by condition(3)above,p and q form a l-basis of M P.(5)Similar to the argument in the proof above for condition(4),it can be shown that any moving lines L of PðtÞcan be expressed by L¼h1pþh2q with h1;h22R½t . Then,by condition(2),the proof is completed.(6)Since p and q are R½t -linearly independent moving lines of P,gðpÂqÞ¼h P for g;h2R½t ,where g;h are relatively prime.Since the components a;b;c2R½t of P are relatively prime,g must be a nonzero constant.Thusn¼degðpÞþdegðqÞP degðpÂqÞ¼degðhÞþdegðPÞ¼degðhÞþn:It follows that degðhÞ¼0.Hence,we may write pÂq¼k P for some nonzero constant k.Now,by condition(4),p and q are a l-basis of M P.(7)Since LVðpÞand LVðqÞare linearly independent,LVðpÂqÞ¼LVðpÞÂLVðqÞ¼0.It follows that pÂq¼0;thus p and q are R½t -linearly independent. Now,by condition(6),p and q form a l-basis of M P.(8)Since cxÀa;cyÀb2I¼h p;q i,similar to the proof of Property(1)of Theo-rem1,one can show that p and q are R½t -linearly independent.By condition(6),p and q are a l-basis of PðtÞ.The proof of Theorem3is completed.ÃTwo definitions for the l-basis are given in[3].Thefirst definition is for planar rational curves and equivalent to condition(6)in Theorem3above.The second def-inition is for the general case of a rational curve in R d,d P2.This second definition, when d¼2,is equivalent to thefirst one but contains a redundant condition degðpÞþdegðqÞ¼n.putation of l-basisWe present in this section a new method for computing a l-basis of a planar ra-tional curve.The conditions(2)and(3)in Theorem3will play a key role in devel-oping this algorithm.We begin with the review of some existing methods.Thefirst method for computing a l-basis of a rational curve PðtÞusing undeter-mined coefficients is described by Sederberg et al.[6].This algorithm needs Oðn3Þarithmetic operations,where n is the degree of PðtÞ.376 F.Chen,W.Wang/Graphical Models64(2003)368–381A more efficient algorithm for computing the l-basis has recently been developed by Zheng and Sederberg[7];we will call it the ZS algorithm.Given a set of three gen-erating vector polynomials of the moving line module M P,the ZS algorithm sorts all the terms of the vector polynomials in the order of their degrees and component in-dices,and exploits the observation that the leading coefficients of at least two gener-ating vector polynomials have the same basis vector;this observation enables one to reduce the degree of one component of one vector polynomial in each reduction step, until a l-basis is reached.The ZS algorithm is similar to BuchbergerÕs algorithm for computing the Gr€o ebner basis of a module,and its efficiency lies in that no more than three generators of the module M P are maintained at any moment.The com-plexity of the ZS algorithm is Oðn2Þ[7].We will present,in this paper,an improved algorithm for computing the l-basis of a planar rational curve.Our algorithm is similar to the ZS algorithm in that it also operates in the moving line module.However,based on the linear dependency of the leading coefficient vectors of the generating vector polynomials of M P,we are able to eliminate simultaneously the highest degree terms of all components of a vector poly-nomial in each reduction step,thus reducing the degrees of the polynomials more quickly than the ZS algorithm does.Moreover,our approach is conceptually simpler since it skips the need of sorting all the terms of the generating vector polynomials. In addition,the effective vector elimination scheme used makes the new algorithm easily extendable to computing the l-basis of a rational curve in higher dimensions.Based on operation count,the new algorithm also has the time complexity Oðn2Þ, but with the proportional constant about twice as small as that of the ZS algorithm. In the following we present the algorithm and analyze its time complexity.Before going on,we provide two lemmas.3.1.LemmasLemma1.The moving line module M P of a planar rational curve PðtÞ¼ðaðtÞ;bðtÞ;cðtÞÞis generated by the three vector polynomials v1;v2,and v3,where v1¼ðÀb;a;0Þ,v2¼ðÀc;0;aÞ,and v3¼ð0;c;ÀbÞ.Proof.The proof will be given by adapting the proof of Lemma1in[3].See also[7]. For completeness,we sketch the proof in the following.Since a,b,and c are relatively prime,there exist u;v;w2R½t such that uaþvbþwc¼1.Suppose that AðtÞxþBðtÞyþCðtÞ¼0is a moving line in M P. For brevity,AðtÞ,BðtÞ,and CðtÞwill be denoted by A,B,and C,respectively,in the following.Since aAþbBþcC¼0,it follows thatA¼uaAþvbAþwcA¼uðÀbBÀcCÞþvbAþwcA¼ÀbðuBÀvAÞÀcðuCÀwAÞ:Similarly,B¼aðuBÀvAÞþcðÀvCþwBÞ。

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空间六面体三线性坐标唯一的充要条件

空间六面体三线性坐标唯一的充要条件

空间六面体三线性坐标唯一的充要条件木效文;白凯娟;邓重阳【摘要】利用三角形重心坐标的唯一性,重新证明了平面四边形的双线性坐标唯一的充要条件是四边形为梯形.通过类比平面四边形的双线性坐标唯一的充要条件,利用四面体重心坐标的唯一性,给出了空间六面体三线性坐标唯一的充要条件.【期刊名称】《杭州电子科技大学学报》【年(卷),期】2016(036)002【总页数】4页(P93-96)【关键词】重心坐标;双线性坐标;空间六面体;三线性坐标【作者】木效文;白凯娟;邓重阳【作者单位】杭州电子科技大学理学院,浙江杭州310018;杭州电子科技大学理学院,浙江杭州310018;杭州电子科技大学理学院,浙江杭州310018【正文语种】中文【中图分类】O182.2文献[1]给出了定义在平面凸多边形上一般化的重心坐标，其中“平面上的双线性坐标”的特例已经得到了深入的研究[2-4]，并在计算机图形学[5]和有限元方法[6]上得到了广泛的应用.然而对于3维空间中的三线性坐标，目前的文献中却很少提及.作为空间重心坐标的一种，它可以应用于3维图形建模和变形、计算机图形学.本文分两部分：首先证明了四边形双线性坐标唯一的充要条件，接着研究了空间六面体三线性坐标唯一的充要条件.设由平面上4点P00,P01,P10,P11组成的四边形κ如图1所示.对于平面上的点P,由κ的4个顶点P00,P01,P10,P11定义的双线性坐标为：P=(1-s)(1-t)P00+(1-s)tP01+s(1-t)P10+stP11=[1-s-t+(2-λ01-λ10)st]P00+(t-st+λ01st)P01+(s-st+λ10st)P10=(1-u01-u10)P00+u01P01+u10P10.其中，1-λ01-λ10,λ01,λ10，1-u01-u10,u01,u10分别为点P11，点P在三角形P00P01P10中的重心坐标.因为三角形的重心坐标是唯一的，可得：当时，分别考虑λ01=1且u01=t；λ01≠1等两种情况.解式(1)的第2个方程，令t2的系数等于0，解得λ01=1或者λ10=1.事实上，若λ01=1，则P11=(1-λ01-λ10)P00+λ01P01+λ10P10=P01+λ10(P10-P00)，所以边P11P01平行于边P10P00；若λ01=1，则边P11P10平行于边P00P01.所以四边形κ是一个梯形.反之，若四边形κ是一个梯形，则存在非零常数k1,k2，使得P11-P01=k1(P10-P00)，或者P11-P10=k2(P01-P00)，所以λ01=1或者λ10=1.故四边形双线性坐标唯一的充要条件为四边形是梯形.设空间六面体τ(侧面可能不是平面)如图2所示.空间点P的三线性坐标为：设：式中：aij(i,j=1,2,3,4)，bj(j=1,2,3,4)分别是顶点P110,P011,P101,P111及空间点P在由顶点P000,P001,P010,P100构成的四面体中的重心坐标[7].将式(3)代入式(2)，并与式(4)比较系数得：将式(5)中的s提出来，得：设B1,B2为2个平面，q1,q2,q3,q4为4条线段.记集合J={B1,B2,q1,q2,q3,q4}. 若J中元素同时满足条件：B1∥B2；q1,q2,q3,q4互相平行；q1,q2,q3,q4的并集包含于B1,B2的并集，则记这时的J为J-.若J中元素同时满足条件：B1∥B2；q1∥q2,q3∥q4；q1,q3异面；线段长度的比值满足；q1,q2,q3,q4的并集包含于B1,B2的并集，则记这时的J为J+.分别记平面P100P000P001P101，P110P010P011P111，P100P000P010P110，P101P001P011P111，P100P101P111P110，P000P001P011P010为A1，A2，A3，A4，A5，A6，记线段P000P001，P100P101，P110P111，P010P011为l1,l2,l3,l4，线段P000P010，P100P110，P101P111，P001P011为l5,l6,l7,l8，线段P000P100，P001P101，P011P111，P010P110为l9,l10,l11,l12.令S1={A1,A2,l1,l2,l4,l3}，S2={A1,A2,l9,l10,l12,l11}，S3={A3,A4,l5,l8,l6,l7}，S4={A3,A4,l9,l12,l10,l11}，S5={A5,A6,l1,l4,l2,l3}，S6={A5,A6,l5,l6,l8,l7}.S={S1,S2,S3,S4,S5,S6}，则有：定理空间六面体三线性坐标唯一的充要条件为：集合S中存在元素为J-或者具有J+.证明分4种情况求解式(6)并证明定理.1)n1，n2不恒等于0.从式(6)的前2个、后2个方程中分别消去s，得：令式(7)中高次项的系数a1=0,a2=0,b1′=0,b2′=0,c2=0,d1=0,d2=0,e1=0，解得a14=a34=a44=1,a24=0，对式(7)余下方程组考虑t或者u唯一，解得：式(8)等价于六面体满足∥∥∥，平面P000P001P010P011平行于平面P100P101P110P111.事实上，式(8)等价于：式(12)等价于a44=1,a24=0等价于∥∥∥，平面P110P100P111P101平行于平面P010P000P011P001.其余情况的证明与式(8)的证明类似.2)n1≡0，n2不恒等于0.因为n1≡0，所以a12=0,a32=1,a22=a42.这时，m1=0.在下，考虑式(7)的第2个方程，令t4,t3,t2的系数恒等于0；在下，分别就a22=1且u=b2，a22≠1考虑式(7)的第2个方程，并分别令t2以及u4,u3,u2的系数恒等于0，解得：3)n1不恒等于0，n2≡0.因为n2≡0，所以a33=0,a13=1,a23=a43.这时m2=0.在下，考虑式(7)的第1个方程，令u4,u3,u2的系数恒等于0；在下，分别就a23=1且t=b3，a23≠1考虑式(7)的第1个方程，并分别令u2和t4,t3,t2的系数恒等于0，解得：4)n1≡0且n2≡0.因该条件，故有a12=0,a32=1,a22=a42,a33=0,a13=1,a23=a43.这时m1=0，且m2=0.解得：如果六面体的顶点按照图1排列，则将式(8)，式(9)，式(13)，式(16)，式(19)，式(20)结合起来，同时将式(10)，式(11)，式(14)，式(15)，式(17)，式(18)结合起来，便得到定理.证毕.本文给出了空间三线性坐标唯一的充要条件的几何描述，可以看出这时六面体包括四棱柱等基本空间几何体，不足之处是没有给出具体的简单代数公式或者等式.接下来，将对空间三线性坐标有理情形做进一步研究.参考文献[1]FLOATER M S,HORMANN K,KS G.A general construction of barycentric coordinates over convex polygons[J].Advances in Computational Mathematics,2006,24(1):311-332.[2]FLOATER M S. Generalized barycentric coordinates and applications[J]. Acta Numerica,2015,24:161-170.[3]FLOATER M S.The inverse of a rational bilinear mapping[J].Computer Aided Geometric Design,2015,33:46-50.[4]SEDERBERG T W,ZHENG J.Birational quadrilateral maps[J].Computer Aided Geometric Design,2015,32:1-4.[5]HORMANN K, FLOATER M S. Mean Value Coordinates for Arbitrary Planar Polygons[J]. ACM Transactions on Graphics, 2006,25(4): 1424-1441.[6]WACHSPRESS E L. A Rational Finite Element Basis[M]. Burlington, MA: Elsevier, 1975: 88-125.[7]梁俊.体积坐标系的坐标四面体重心定理[J].四川师范大学学报(自然科学版),1999,2(4):428-432.。

变速箱壳体机械加工工艺设计外文文献翻译、中英文翻译、外文翻译

变速箱壳体机械加工工艺设计外文文献翻译、中英文翻译、外文翻译

Gearbox shell machining process design《Manufacturing Engineering and Technology—Machining》Mechanical Industry Press In March 2004, version 1 p560—564（Serope kalpakjian）（Steven R.Schmid）AbstractGearbox shell is a more complex structure of spare parts box, its high precision, complex process, and the processing quality will affect the overall performance engine, so it has become the engine manufacturer's focus parts one.Machining process planning must guarantee the machining quality of parts, to meet the technical requirements stipulated in drawings, at the same time should also have high productivity and efficiency. Therefore, machining process planning design is an important work, requires designers must have a rich experience in production practice and wide range of mechanical manufacturing technology basic theory knowledge. In the specified procedure, should according to the production of parts and the existing equipment conditions, taking the processing quality into account, productivity and economy requirements, after repeated analysis and comparison, to determine the optimal or the best solution.1.Technical Characteristics of the gearbox shellThe gearbox shell process features are: the structure of complex shape; processing plane, more than holes; uneven wall thickness and stiffness is low; processing of high precision typical of box-type processing part. The main processing of the surface of cylinder block top surface, the main bearing side, cylinder bore, the main and camshaft bearing bore holes and so on, they will directly affect the machining accuracy of the engine assembly precision and performance, mainly rely on precision equipment, industrial fixtures reliability and processing technology to ensure the reasonableness.2.The gearbox shell process design principles and the basisDesign Technology program should be to ensure product quality at the same time, give full consideration to the production cycle, cost and environmental protection; based on the enterprises ability to actively adopt advanced process technology and equipment, and constantly enhance their level of technology. Gearbox shell machining process design should follow the following basic principles: 2.1 The selection of processing equipmentThe principle of selection adopted the principle of selection adopted the principle of combining rigid-flexible, processing each horizontal machining center is located mainly small operations with vertical machining center, the key process a crank hole, cylinder hole, balancer shaft hole High-speed processing of high-precision horizontal machining center, an upper and lower non-critical processes before and after the four-dimensional high-efficiency rough milling and have a certain adjustment range of special machine processing;2.2 Concentration process principleFocus on a key process in principle process the body cylinder bore, crankshaft hole, Balance Shaft hole surface finishing and the combination of precision millingcylinder head, using a process focused on a setup program to complete all processing elements in order to ensure product accuracy The key quality processes to meet the cylinder capacity and the relevant technical requirements;3.The gearbox shell machining process design the main contentGearbox shell complex structure, high precision, arge size, is thin-walled parts, there are a number of high precision plane and holes. Engine block machining process characteristics; mainly flat and the hole processing, processing of flat generally use planing, milling methods such as processing, processing of hole used mainly boring, processing and multi-purpose drilling holes. As the cylinder complex structure. so how to ensure that the mutual position of the surface processing precision is an important issue.3.1 The selection of blankGearbox shell on the materials used are generally gray cast iron HT150,HT200,HT250,there is also cast aluminum or steel plate, this engine block using high-strength alloy cast iron. Cylinder in the processing prior to aging treatment in order to eliminate stress and improve the rough casting mechanical properties. Improve the rough accuracy, reduction of machining allowance, is to improve the automated production line system productivity and processing quality of the important measures. As the foreign box-type parts of rough quality and high precision, and its production-line system has been implemented directly on the blank line, not only eliminating the need for blank check device also saves the rough quality problems due to waste of machining time, increase overall efficiency. Therefore, the refinement of rough is to improve the productivity of the most promising way out. For the engine block production line, can be rough in parts on-line pre-milling six face, removing most of the margin, to facilitate direct on-line parts.3.2 Machining process selection and processing of the benchmarkChoose the right processing technology base is directly related to the processing quality can ensure the parts. Generally speaking, process benchmarking can be divided into coarse and fine reference base.3.2.1 Coarse reference baseThe baseline for the on-line thick rough ,which is particularly important the choice of benchmark crude, if crude benchmark choice unreasonable, will the uneven distribution of machining allowance, processing and surface offsets, resulting in waste. In the cylinder production line, we have adopted for the coarse side of the base;3.2.2 Fine reference baseRefined the base of this box for the engine block parts, the general use of "side two sales "for a full range of uniform benchmarks, For the longer automated production positioning. In the gearbox shell of the process, we have adopted to the side, bottom and the spindle hole positioning, in the processing center on the process.3.3 Machining Processing Stages and processes of the arrangementsOften a part of many apparent need for processing, of course, the surface machining accuracy are different. Processing of high precision surface, often afterrepeated processing; As for the processing of the surface of low precision, only need to go through one or two on the list. Thus, when the development process in order to seize the "processing high precision surface, "this conflict, the reasonable arrangement processes and rational division stage of processing. Arrange the order of the principle of process is: after the first coarse refined, the first surface after the hole, the first benchmark other. In the engine block machining, the same should follow this principle.3.3.1 roughing stage engine block machining processThe arrangements for roughing process, to fully carry out rough rough, trim most of the margin in order to ensure production efficiency;3.3.2 semi-finishing phase of the engine block machiningIn order to ensure the accuracy of the middle of some important surface processing, and arrange some semi-finishing operations, will be required accuracy and surface roughness of the surface of the middle of some processing to complete, while demanding the surface of semi-finished, to prepare for future finishing;3.3.3 The finishing stageThe finishing stage of requiring high accuracy and surface roughness of the surface processing; 3.3.4 secondary processingSecondary processing such as small surface screw holes, you can finish of the major surface after the one hand, when the workpiece deformation process little impact at the same time also reduced the rejection rate;In addition,if the main surface of a waste,these small the surface will not have to be processed,thus avoiding a waste of man-hours.However,if the processing is very easy for a small surface bumps the main surface,it should be placed on a small surface finish prior to the main surface finishing;3.3.5 make proper arrangements for secondary processesMake proper arrangements for secondary processes such as product inspection process,in part roughing stage,the key process before and after processing,spare parts all the processing has been completed,should be appropriate arrangements. Stage of processing division,has the following advantages:First,it can take measures to eliminate the rough workpiece after the stress,to ensure accuracy; second,finishing on the back,and will not damage during transport the surface of the workpiece has been processed;again,first roughing the surface defects can be detected early and promptly deal with rough,do not waste working hours.But most small parts,do not sub very thin.3.4 The gearbox shell surface of the main processing and secondary processes3.4.1 Plane processingPlane processing at present, the milling of engine blocks is the primary means of planar processing,domestic milling feed rate is generally 300-400mm/min,and foreign 2000-4000mm/min milling feed rate compared to far cry,to be on increasing,therefore,improve the milling feed rate,reduce overhead time is to improve the productivity of the major means of finishing a number of plane engine block when the milling feed rate to reach 2399mm/min,greatly improved efficiency; Top surface of the cylinder milling is a key process in the process,the flatnessrequirements for 0.02/145mm,the surface roughness of Ra1.6um.Processing in the cylinder,the use of side and spindle bearing bore positioning,top,bottom and middle vagay only aperture while processing used in the processing line outside of the knife device can better meet the engine block machining accuracy;3.4.2 General holes MachiningGeneral holes Machining holes in general are still using the traditional processing of drilling,expansion,boring,reaming,tapping and other craft approach. Issues in the design process of specific programs,use of coated cutting tools,cutting tools and other advanced tools within the cooling,and using a large flow of cooling systems,greatly improving the cutting speed,improved productivity;3.4.3 Deep hole processingDeep hole processing of the traditional processing method is used to grade twist drill feed,low efficiency of their production,processing and quality is poor.The deep hole in the engine block processing,the use of gun drilling process;3.4.4 CleaningCleaning is divided into wet cleaning and dry cleaning.Machining cylinder automatic production line using a large flow of wet cleaning;3.4.5 DetectionDetect points outside the line detection and line detection of two kinds. Quality inspection in the engine block,according to the actual situation with lines outside the detection,the main use of coordinate measuring machine integrated measurements of the cylinder,each 200 samples 1-5 pieces,each class random one.变速箱壳体机械加工工艺设计摘自：《机械工程与技术（机加工）》（英文版）机械工业出版社2004年3月第1版美国卡卡尔帕基安·施密德摘要变速箱壳体是变速器机零件中结构较为复杂的箱体零件，其精度要求高，加工工艺复杂，并且加工加工质量的好坏直接影响发动机整个机构的性能，因此，它成为各个发动机生产厂家所关注的重点零件之一。

小学下册第十四次英语第1单元测验卷(有答案)

小学下册第十四次英语第1单元测验卷(有答案)

小学下册英语第1单元测验卷(有答案)考试时间：90分钟（总分:110）A卷一、综合题(共计100题共100分)1. 填空题：My brother loves to play __________. (足球)2. 听力题：We go _____ (fishing/camping) every summer.3. 填空题：The ________ (生态影响因素) shape ecosystems.4. 填空题：My toy _____ can fly high.5. 听力题：The chemical formula for cyclohexane is ______.6. 选择题：What is the name of the famous character who lives in a shoe?A. Old Mother HubbardB. The Old Woman Who Lived in a ShoeC. CinderellaD. Goldilocks7. 听力题：You can find _______ in a garden or park.8. 填空题：I enjoy making ______ (美食) for my friends during gatherings.9. 听力题：A _______ can measure the amount of energy consumed by an appliance over time.10. 填空题：The __________ (历史的传承角色) connect generations.What is the name of the famous tower in Paris?A. Eiffel TowerB. Leaning Tower of PisaC. Burj KhalifaD. Space Needle答案: A12. 填空题：The ________ was a major turning point in the history of Europe.13. 填空题：The __________ (历史的纪录片) offer visual insights into the past.14. 听力题：Plants absorb carbon dioxide through their ______.15. 选择题：What do we call the time when the sun rises?A. SunriseB. SunsetC. DuskD. Dawn答案: A16. 听力题：Carbon dioxide is produced when we _______.17. 听力题：I have ___ (four/five) friends at school.18. ts are adapted to ______ (高海拔) environments. 填空题：Some pla19. 选择题：What is 20 divided by 5?A. 2B. 4C. 5D. 6答案:B20. 填空题：A sunflower turns towards the _____.My grandma makes the best __________ (甜点).22. 填空题：I like to visit ______ during summer break.23. 选择题：What is the opposite of hot?A. WarmB. CoolC. ColdD. Mild答案:C24. 听力题：The ______ helps make important decisions.25. 选择题：What is the capital city of the United Kingdom?A. ManchesterB. LondonC. EdinburghD. Cardiff答案: B26. 选择题：What do you call a large, slow-moving animal with a shell?A. TortoiseB. TurtleC. SnailD. Armadillo答案:A27. 选择题：What is the name of the fairy tale character who had a magic mirror?A. CinderellaB. Snow WhiteC. RapunzelD. Belle28. 听力题：My sister is ______ to a party this weekend. (going)29. 填空题：I have a pet ______ (兔子) named Fluffy. It is very soft and loves to be ______ (抚摸).What do we call the process of a liquid turning into a gas?A. EvaporationB. CondensationC. FreezingD. Melting答案: A31. 选择题：What do we call the water cycle's process of water vapor turning into liquid?A. EvaporationB. CondensationC. PrecipitationD. Collection答案: B32. 选择题：What is the opposite of "fast"?A. QuickB. SlowC. RapidD. Speedy答案:B33. 填空题：The _______ (虎) is a powerful hunter.34. 选择题：What do we call the process of cooking food using steam?A. BoilingB. SteamingC. FryingD. Baking答案:B35. 选择题：Which planet is known as the "Blue Planet"?A. EarthB. MarsC. VenusD. Neptune36. 填空题：The parrot has bright _________. (羽毛)The _______ of a plant can be very long.38. 填空题：_____ (花卉销售) supports local economies.39. 选择题：What term describes the shape of the orbit of planets?A. CircularB. EllipticalC. LinearD. Irregular40. 填空题：My ________ (玩具名称) comes with a set of stickers.41. 选择题：What is the name of the famous Scottish lake said to be home to a monster?A. Loch NessB. Lake SuperiorC. Lake TahoeD. Lake Victoria答案：A42. 选择题：What do we call the area of land near the sea?A. CoastB. DesertC. ForestD. Mountain43. 听力题：The Earth's surface is constantly ______ due to natural forces.44. 听力题：We have a _____ (庆典) for the festival.45. 听力题：The sky is _____ (blue/green) today.46. 选择题：What is the name of the fairy tale character who lost her glass slipper?a. Snow Whiteb. Cinderellac. Sleeping Beautyd. Rapunzel答案:B47. 听力题：A __________ is a geological feature that can impact agricultural practices.48. 听力题：The color of phenolphthalein changes in acidic and basic solutions, indicating ______.49. 听力题：My sister loves to ________.50. 听力题：I like _____ (to run/to walk).51. 选择题：What is the name of the tree that produces acorns?A. PineB. OakC. MapleD. Birch答案:B52. 选择题：What do you call the process of creating a new plant from a cutting?A. GraftingB. CloningC. PropagationD. All of the above53. 选择题：What do we call a person who studies animals?A. BiologistB. ZoologistC. BotanistD. Ecologist54. 填空题：The first successful heart transplant was performed in ______ (20世纪).55. 填空题：I love to go ________ (露营) in the summer.56. 填空题：I like to listen to ______ (故事) before I go to sleep.The process of oxidation involves the ______ of electrons.58. 选择题：What is the name of the famous artist known for his works in the Renaissance?A. Leonardo da VinciB. Vincent van GoghC. Pablo PicassoD. Claude Monet答案: A59. 选择题：What do you call a baby kangaroo?A. JoeyB. CubC. CalfD. Kit答案:A60. 听力题：Chemical changes often involve the formation of _____ or new substances.61. 听力题：A __________ is a natural resource that can be recycled.62. 选择题：How many days are in February during a leap year?a. 28b. 29c. 30d. 31答案:B63. 听力题：The ________ (dog) is barking loudly.64. 填空题：He is a _____ (评论家) who reviews films.65. 填空题：I created a _________ (玩具动物园) with all my stuffed animals.66. 填空题：There are many _______ (昆虫) in the garden.A ______ (植物的保育措施) can protect vulnerable species.68. 选择题：Which animal is known as the "king of the jungle"?A. ElephantB. LionC. TigerD. Bear答案:B69. 填空题：I can enjoy playful activities with my ________ (玩具类型).70. 填空题：At recess, we play ________ (游戏) on the playground. I love to play ________ (足球) with my classmates.71. 填空题：The __________ is a major geographical feature in Europe. (阿尔卑斯山)72. 填空题：A _____ (城市绿化) initiative can improve living conditions.73. 听力题：They are ___ (singing/playing) together.74. 选择题：What is the name of the famous river in Egypt?A. AmazonB. MississippiC. NileD. Yangtze75. 填空题：Certain plants are known for their ______ (药用性质).76. 听力题：The _____ (hedgehog) is spiky.77. 听力题：The process of oxidation involves __________ losing electrons.78. 选择题：What do we call a person who studies the weather?A. ClimatologistB. MeteorologistC. EnvironmentalistD. Geologist79. 选择题：What is the capital of Mongolia?A. UlaanbaatarB. HohhotC. ErdenetD. Darkhan答案:A. Ulaanbaatar80. 填空题：The chef, ______ (厨师), teaches cooking classes.81. 听力题：A ______ is a cold-blooded animal that lays eggs.82. 听力题：A __________ is an area of land that is covered with trees.83. 填空题：The anteater's long snout is perfect for eating ______ (蚂蚁).84. 填空题：The ________ was a famous naval battle during World War II.85. 选择题：Which of these animals can swim?A. LionB. WhaleC. HorseD. Monkey86. 填空题：My sister loves to read ____.87. 听力题：The chemical properties of an element depend on its ______.88. 听力题：A __________ is a low area that collects water.89. 填空题：My favorite superhero _________ (玩偶) has a cool _________ (披风).90. 填空题：My __________ (玩具名) is made of __________ (材料).91. 听力题：My favorite sport is _____ (篮球).92. 选择题：What do you wear on your feet?A. HatB. GlovesC. ShoesD. Scarf93. 听力题：The snow is _______ (white).94. 听力题：The chemical symbol for cesium is _______.95. 选择题：What do we call a person who repairs cars?A. AccountantB. MechanicC. ArchitectD. Chef答案:B96. 听力题：A chemical that can cause a reaction to occur is called a ______.97. 听力题：It is ___ (raining/sunny) outside.98. 选择题：What do we call the study of the earth's physical structure?A. GeographyB. GeologyC. EcologyD. Astronomy答案: B99. 选择题：What do we call a story that teaches a moral lesson?A. FableB. MythC. LegendD. Folklore答案: A100. 填空题：I like to ride my ______ (自行车) through the neighborhood with my friends.。

数学专业英语词汇（N）

数学专业英语词汇（N）

数学专业英语词汇(N)n ary relation n元关系n ball n维球n cell n维胞腔n chromatic graph n色图n coboundary n上边缘n cocycle n上循环n connected space n连通空间n dimensional column vector n维列向量n dimensional euclidean space n维欧几里得空间n dimensional rectangular parallelepiped n维长方体n dimensional row vector n维行向量n dimensional simplex n单形n dimensional skeleton n维骨架n disk n维圆盘n element set n元集n fold extension n重扩张n gon n角n graph n图n homogeneous variety n齐次簇n person game n人对策n simplex n单形n sphere bundle n球丛n th member 第n项n th partial quotient 第n偏商n th power operation n次幂运算n th root n次根n th term 第n项n times continuously differentiable n次连续可微的n times continuously differentiable function n次连续可微函数n tuple n组n tuply connected domain n重连通域n universal bundle n通用丛nabla 倒三角算子nabla calculation 倒三角算子计算nabla operator 倒三角算子napier's logarithm 讷代对数natural boundary 自然边界natural boundary condition 自然边界条件natural coordinates 自然坐标natural equation 自然方程natural equivalence 自然等价natural exponential function 自然指数函数natural frequency 固有频率natural geometry 自然几何natural injection 自然单射natural isomorphism 自然等价natural language 自然语言natural logarithm 自然对数natural mapping 自然映射natural number 自然数natural oscillation 固有振荡natural sine 正弦真数natural transformation 自然变换naught 零necessary and sufficient conditions 必要充分的条件necessary and sufficient statistic 必要充分统计量necessary condition 必要条件necessity 必然性negation 否定negation sign 否定符号negation symbol 否定符号negative 负数negative angle 负角negative binomial distribution 负二项分布negative complex 负复形negative correlation 负相关negative definite form 负定型negative definite hermitian form 负定埃尔米特形式negative definite quadratic form 负定二次形式negative function 负函数negative number 负数negative operator 负算子negative parity 负电阻negative part 负部分negative particular proposition 否定特称命题negative proposition 否定命题negative rotation 反时针方向旋转negative semidefinite 半负定的negative semidefinite eigenvalue problem 半负定特盏问题negative semidefinite form 半负定型negative semidefinite quadratic form 半负定二次形式negative sign 负号negative skewness 负偏斜性negative variation 负变差negligible quantity 可除量neighborhood 邻域neighborhood base 邻域基neighborhood basis 邻域基neighborhood filter 邻域滤子neighborhood retract 邻域收缩核neighborhood space 邻域空间neighborhood system 邻域系neighborhood topology 邻域拓扑neighboring vertex 邻近项点nephroid 肾脏线nerve 神经nested intervals 区间套net 网net function 网格函数net of curves 曲线网net of lines 直线网network 网络network analysis 网络分析network flow problem 网络潦题network matrix 网络矩阵neumann boundary condition 诺伊曼边界条件neumann function 诺伊曼函数neumann problem 诺伊曼问题neumann series 诺伊曼级数neutral element 零元素neutral line 中线neutral plane 中性平面neutral point 中性点newton diagram 牛顿多边形newton formula 牛顿公式newton identities 牛顿恒等式newton interpolation polynomial 牛顿插值多项式newton method 牛顿法newton potential 牛顿位势newtonian mechanics 牛顿力学nice function 佳函数nil ideal 零理想nil radical 幂零根基nilalgebra 幂零代数nilpotency 幂零nilpotent 幂零nilpotent algebra 幂零代数nilpotent element 幂零元素nilpotent group 幂零群nilpotent ideal 幂零理想nilpotent matrix 幂零矩阵nilpotent radical 幂零根基nine point circle 九点圆nine point finite difference scheme 九点有限差分格式niveau line 等位线niveau surface 等位面nodal curve 结点曲线nodal line 交点线nodal point 节点node 节点node locus 结点轨迹node of a curve 曲线的结点noetherian category 诺特范畴noetherian object 诺特对象nomogram 算图nomographic 列线图的nomographic chart 算图nomography 图算法non additivity 非加性non archimedean geometry 非阿基米德几何non archimedean valuation 非阿基米德赋值non countable set 不可数集non critical point 非奇点non denumerable 不可数的non denumerable set 不可数集non developable ruled surface 非可展直纹曲面non enumerable set 不可数集non euclidean geometry 非欧几里得几何学non euclidean motion 非欧几里得运动non euclidean space 非欧几里得空间non euclidean translation 非欧几里得平移non euclidean trigonometry 非欧几里得三角学non homogeneity 非齐non homogeneous chain 非齐次马尔可夫链non homogeneous markov chain 非齐次马尔可夫链non isotropic plane 非迷向平面non linear 非线性的non negative semidefinite matrix 非负半正定阵non oriented graph 无向图non parametric test 无分布检验non pascalian geometry 非拍斯卡几何non ramified extension 非分歧扩张non rational function 无理分数non relativistic approximation 非相对性近似non reversibility 不可逆性non singular 非奇的non stationary random process 不平稳随机过程non steady state 不稳定状态non symmetric 非对称的non symmetry 非对称non zero sum game 非零和对策nonabsolutely convergent series 非绝对收敛级数nonagon 九边形nonassociate 非结合的nonassociative ring 非结合环nonbasic variable 非基本变量noncentral chi squre distribution 非中心分布noncentral f distribution 非中心f分布noncentral t distribution 非中心t分布noncentrality parameter 非中心参数nonclosed group 非闭群noncommutative group 非交换群noncommutative ring 非交换环noncommutative valuation 非交换赋值noncommuting operators 非交换算子noncomparable elements 非可比元素nondegeneracy 非退化nondegenerate collineation 非退化直射变换nondegenerate conic 非退化二次曲线nondegenerate critical point 非退化临界点nondegenerate distribution 非退化分布nondegenerate set 非退化集nondense set 疏集nondenumerability 不可数性nondeterministic automaton 不确定性自动机nondiagonal element 非对角元nondiscrete space 非离散空间nonexistence 不存在性nonfinite set 非有限集nonholonomic constraint 不完全约束nonhomogeneity 非齐性nonhomogeneous 非齐次的nonhomogeneous linear boundary value problem 非齐次线性边值问题nonhomogeneous linear differential equation 非齐次线性微分方程nonhomogeneous linear system of differential equations 非齐次线性微分方程组nonisotropic line 非迷向线nonlimiting ordinal 非极限序数nonlinear equation 非线性方程nonlinear functional analysis 非线性泛函分析nonlinear lattice dynamics 非线性点阵力学nonlinear operator 非线性算子nonlinear optimization 非线性最优化nonlinear oscillations 非线性振动nonlinear problem 非线性问题nonlinear programming 非线性最优化nonlinear restriction 非线性限制nonlinear system 非线性系统nonlinear trend 非线性瞧nonlinear vibration 非线性振动nonlinearity 非线性nonlogical axiom 非逻辑公理nonlogical constant 非逻辑常数nonmeager set 非贫集nonmeasurable set 不可测集nonnegative divisor 非负除数nonnegative number 非负数nonnumeric algorithm 非数值的算法nonorientable contour 不可定向周线nonorientable surface 不可定向的曲面nonorthogonal factor 非正交因子nonparametric confidence region 非参数置信区域nonparametric estimation 非参数估计nonparametric method 非参数法nonparametric test 非参数检定nonperfect set 非完备集nonperiodic 非周期的nonperiodical function 非周期函数nonplanar graph 非平面图形nonprincipal character 非重贞nonrandom sample 非随机样本nonrandomized test 非随机化检验nonrational function 非有理函数nonremovable discontinuity 非可去不连续点nonrepresentative sampling 非代表抽样nonresidue 非剩余nonsense correlation 产生错觉相关nonsingular bilinear form 非奇双线性型nonsingular curve 非奇曲线nonsingular linear transformation 非退化线性变换nonsingular matrix 非退化阵nonspecial group 非特殊群nonstable 不稳定的nonstable homotopy group 非稳定的同伦群nonstandard analysis 非标准分析nonstandard model 非标准模型nonstandard numbers 非标准数nonsymmetric relation 非对称关系nonsymmetry 非对称nontangential 不相切的nontrivial element 非平凡元素nontrivial solution 非平凡解nonuniform convergence 非一致收敛nonvoid proper subset 非空真子集nonvoid set 非空集nonzero vector 非零向量norm 范数norm axioms 范数公理norm form 范形式norm of a matrix 阵的范数norm of vector 向量的模norm preserving mapping 保范映射norm residue 范数剩余norm residue symbol 范数剩余符号norm topology 范拓朴normability 可模性normal 法线normal algorithm 正规算法normal basis theorem 正规基定理normal bundle 法丛normal chain 正规链normal cone 法锥面normal congruence 法汇normal coordinates 正规坐标normal correlation 正态相关normal curvature 法曲率normal curvature vector 法曲率向量normal curve 正规曲线normal density 正规密度normal derivative 法向导数normal dispersion 正常色散normal distribution 正态分布normal distribution function 正态分布函数normal equations 正规方程normal error model 正规误差模型normal extension 正规开拓normal family 正规族normal force 法向力normal form 标准型normal form problem 标准形问题normal form theorem 正规形式定理normal function 正规函数normal homomorphism 正规同态normal integral 正规积分normal linear operator 正规线性算子normal mapping 正规映射normal matrix 正规矩阵normal number 正规数normal operator 正规算子normal order 良序normal plane 法面normal polygon 正规多角形normal polynomial 正规多项式normal population 正态总体normal probability paper 正态概率纸normal process 高斯过程normal sequence 正规序列normal series 正规列normal set 良序集normal simplicial mapping 正规单形映射normal solvable operator 正规可解算子normal space 正规空间normal surface 法曲面normal tensor 正规张量normal to the surface 曲面的法线normal valuation 正规赋值normal variate 正常变量normal variety 正规簇normal vector 法向量normality 正规性normalization 标准化normalization theorem 正规化定理normalize 正规化normalized basis 正规化基normalized function 规范化函数normalized variate 正规化变量normalized vector 正规化向量normalizer 正规化子normalizing factor 正则化因数normed algebra 赋范代数normed linear space 赋范线性空间normed space 赋范线性空间northwest corner rule 北午角规则notation 记法notation free from bracket 无括号记号notation of backus 巴科斯记号notion 概念nought 零nowhere convergent sequence 无处收敛序列nowhere convergent series 无处收敛级数nowhere dense 无处稠密的nowhere dense set 无处稠密点集nowhere dense subset 无处稠密子集nuclear operator 核算子nuclear space 核空间nucleus of an integral equation 积分方程的核null 零null class 零类null divisor 零因子null ellipse 零椭圆null function 零函数null hypothesis 虚假设null line 零线null matrix 零矩阵null method 衡消法null plane 零面null point 零点null ray 零射线null relation 零关系null representation 零表示null sequence 零序列null set 空集null solution 零解null system 零系null transformation 零变换null vector 零向量nullity 退化阶数nullring 零环nullspace 零空间number 数number defined by cut 切断数number defined by the dedekind cut 切断数number field 数域number interval 数区间number line 数值轴number notation 数记法number of partitions 划分数number of repetitions 重复数number of replications 重复数number of sheets 叶数number sequence 数列number set 数集number system 数系number theory 数论number variable 数变量numeration 计算numerator 分子numeric representation of information 信息的数值表示numerical 数值的numerical algorithm 数值算法numerical axis 数值轴numerical calculation 数值计算numerical coding 数值编码numerical coefficient 数字系数numerical computation 数值计算numerical constant 数值常数numerical data 数值数据numerical determinant 数字行列式numerical differentiation 数值微分numerical equality 数值等式numerical equation 数字方程numerical error 数值误差numerical example 数值例numerical function 数值函数numerical inequality 数值不等式numerical integration 数值积分法numerical invariant 不变数numerical mathematics 数值数学numerical method 数值法numerical model 数值模型numerical operator 数字算子numerical quadrature 数值积分法numerical series 数值级数numerical solution 数值解numerical solution of linear equations 线性方程组的数值解法numerical stability 数值稳定性numerical table 数表numerical value 数值numerical value equation 数值方程nutation 章动。