We describe here a properly recent application of the Gröbner Cover algorithm (GC) providing an algebraic support to Dynamic Geometry computations of geometrical loci. It provides a complete algebraic solution of locus computation as well as a suitable taxonomy allowing to distinguish the nature of the different components. We included a new algorithm Locus into the Singular grobcov.lib library for this purpose. A web prototype has been implemented using it in Geogebra

Abánades Astudillo, Miguel Ángel

Botana Ferreiro, Francisco Ramón

Montes, Antonio

Recio Muñiz, Tomás

English

Miguel A. Abánades

Francisco Botana

Antonio Montes

Tomás Recio

Crossref

Software Using the Gröbner Cover for Geometrical Loci Computation and Classification

UCrea

Software using the Gröbner Cover forgeometrical loci computation and classificationMiguel A. Abánades1, Francisco Botana2, Antonio Montes3, and Tomás Recio41 Universidad Complutense de Madrid, Spainabanades@ajz.ucm.es,2 Universidad de Vigo, Spainfbotana@uvigo.es,http://webs.uvigo.es/fbotana/3 Universitat Politècnica de Catalunya, Spainantonio.montes@upc.edu,http://www-ma2.upc.edu/montes/4 Universidad de Cantabria, Spaintomas.recio@unican.es,http://www.recio.tkAbstract. We describe here a properly recent application of the GröbnerCover algorithm (GC) providing an algebraic support to Dynamic Ge-ometry computations of geometrical loci. It provides a complete algebraicsolution of locus computation as well as a suitable taxonomy allowing todistinguish the nature of the different components. We included a newalgorithm Locus into the Singular grobcov.lib library for this purpose.A web prototype has been implemented using it in Geogebra.Keywords: Locus, Taxonomy, Dynamical Geometry, Groebner Cover.1 Introduction5 One of the defining characteristics of Dynamical Geometry (DG) is obtaininggeometrical loci problems. Neverthless, the existing DG software are not able togive a satisfactory answer. This is the case for the first standard DG systemsdeveloped in the late 80’s (such as Cabri and The Geometer’s Sketchpad), aswell as for more recent ones, such as GeoGebra or Java Geometry Expert. InDG systems, it is often the case that locus determination is purely graphical,producing an output that is not robust enough and not reusable by the givensoftware. Moreover, extraneous objects are frequently appended to the true locusas side products of the locus determination process.Using the Gröbner Cover we are able to give an exact algebraic result,allowing moreover to give a taxonomy of the different locus components. In5 Authors partially supported by the Spanish “Ministerio de Economı́a y Competitivi-dad” and by the European Regional Development Fund (ERDF), under the ProjectMTM2011–25816–C02–02. The third author was supported by projects Gen. Cat.DGR 2009SGR1040 and MICINN MTM2009-07242.2 M. Abánades - F. Botana - A.Montes - T. Reciosection 2 we give a summary of the Gröbner Cover. In section 3 we givethe locus taxonomy obtained using it, and describe the new Locus algorithm.Finally, in section 4 we show characteristic examples justifying the taxonomyand showing its functionality. A web prototype [3] has been implemented usingthe new algorithm in Geogebra.2 The Gröbner CoverThe Gröbner Cover algorithm for discussing parametric polynomial idealsgives a canonical description, classifying the solutions by their characteristics(number of solutions, dimension, etc.). This is used here for defining a taxon-omy of geometrical loci and to implement it in the new algorithm Locus. It isincluded in the Singular grobcov.lib library allowing its use by DG software.The Gröbner Cover provides the analog of the reduced Gröbner basisof an ideal for parametric ideals. Its existence was proved by Wibmer’s Theo-rem [6], and the method and algorithms were developed in [5]. Montes imple-mented in Singular the grobcov.lib library [7], whose actual version incorporatesKapur-Sun-Wang algorithm [2] for computing the initial Gröbner System used ingrobcov algorithm, as described in [4], and recently also the Locus algorithmdescribed here. A more detailed description can be seen in [1].Let y = y1, . . . , yn be the set of variables and u = u1, . . . , um the set ofparameters. Given a generating set F = {f1, · · · , fs} ⊂ Q[u][y] of the parametricideal I = 〈F 〉 and a monomial order y in the variables, the grobcov algorithmdetermines– the unique canonical partition of the parameter space Cm into locally closedsets (segments) with associated generalized reduced Gröbner basis:GC = {(S1, B1, lpp1), . . . , (Sr, Br, lppr)}.– The segments Si are disjoint locally closed subsets of Cm and ⊕iSi = Cm.– The basisBi of a segment Si has fixed set of leading power products (lpp), whoensures that the type of solutions is the same over all points of the segment,and is the generalized reduced Gröbner basis of 〈F 〉 over the segment Si.– The lpp’s are included in the output, even if they can be seen on the basis,to characterize the segments and facilitate the applications.– Moreover, if the ideal is homogeneous, the lpp’s are characteristic of thesegment, no other segment having the same lpp’s.The generalized reduced Gröbner basisBi of a segment Si is formed by a set ofmonic I-regular functions over Si. An I-regular function, representing an elementof the basis, allows a full-representation in terms of a set of polynomials thatspecialize for every point u0 of the segment, either to the corresponding elementof the reduced Gröbner basis of the specialized ideal Iu0 after normalization, orto zero. It also allows a generic representation given by a single polynomial thatspecializes well on an open subset of the segment and to zero on the remainingGröbner Cover for locus computation 3points of it. Usually it is sufficient with the generic representation, and we can,if needed, compute the full representation from it using the extend algorithm.The segments Si are given in canonical P-representation, given by a set ofprime ideals of the formPrep(S) = {{pi, {pij : 1 ≤ j ≤ ri}} : 1 ≤ i ≤ s}representing the set:S =s⋃i=1V(pi) \ ri⋃j=1V(pij) .Each V(pi) \⋃rij=1 V(pij) is a component of the segment, and its representative{pi, {pij : 1 ≤ j ≤ ri}}, by abuse of language, is also denoted a component whenthere is no ambiguity. pi is called the top of the component, and {pij : 1 ≤ j ≤ ri}the holes.3 Locus taxonomyA geometric locus is a set of points satisfying some conditions. Locus computa-tion is an important issue in Dynamic Geometry, where the term locus generallyrefers to loci of the following kind: determine the trajectory determined by thedifferent positions of a point P (tracer), corresponding to the different positionsof a second point M (mover) along the path where it is constrained by the con-struction. Nevertheless, the actual existing DG software do not provide correctalgebraic solutions.Using the Gröbner Cover we are able to give a precise algebraic answer.We shall consider more general locus problems in the plane with a tracer pointP (u1, u2), whose coordinates u = (u1, u2) are considered as parameters andthe remaining coordinates, distances, etc. y = y1, . . . , yn of the construction asvariables. The locus problem will give rise to a system F ⊂ Q[u][y]. The locusdetermination consists now in obtaining the conditions over the parameters ufor which there are solutions for the y.In the Gröbner Cover the values of the parameters and variables areconsidered over C. Thus we can provide only locus solutions over the complexC, whereas DG is interested in the real projection, who is not always obvious.Moreover, we consider only problems that can be be formulated exactly in termsof equations with coefficients in Q. We also restrict the study to plane lociproblems, even if it can be generalized to higher dimensional spaces. LetV(F ) = {(u,y) ⊂ C2+n : ∀f ∈ F, f(u,y) = 0}be the set of solutions of F . Denote π1 and π2 the projections onto the parameterand variable spaces:π1 : C2+n −→ C2 π2 : C2+n −→ Cn(u,y) 7→ u (u,y) 7→ y4 M. Abánades - F. Botana - A.Montes - T. RecioL← Locus(G)Input: G = {(Si, Bi, lppi) : i ≤ i ≤ s} the Gröbner cover of an ideal in Q[u][y],where Si = ∪jCij and Cij = {(pij , {pijk : 1 ≤ k ≤ rij}) : 1 ≤ j ≤ ri}.Output: L, the components of the P-representation of the locusL = {{qi, {qij : 1 ≤ j ≤ si}, typei} : 1 ≤ i ≤ s}beginC1 = Select the segments of G with dim(lppi) = 0 # normal-segmentsC1 = Specialize the basis on every component of C1 and mark the component‘Normal’ if the basis continues to depend on the u’s and ‘Special’ if notC2 = Select all the components of the segments of G with dim(lppi) > 0# non-normal segmentsL1 = LCUnion(C1), marking the components of L1 as‘Normal’ or ‘Special’ inheriting the character of the fullL2 = LCUnion(C2);Mark the components of L2 of dim(C) = 0 and dim(C) > 0 respectively as‘Accumulation’ and ‘Degenerate’ componentsL = L1 ∪ L2endTable 1. Locus algorithmThe taxonomy that we give is motivated by the interpretation of the solutionsin a lot of loci problems of different kind (see [1]):Generic locus L associated to the parametric polynomial system F (u,y) isthe set L = π1(V(F )) ⊂ C2,(i.e. the set of values of the coordinates of the tracerfor which there exist solutions).Taxonomy:- Normal locus: are the points u ∈ C2 of the locus L for whichdim(π2(V(F ) ∩ π−11 (u))) = 0 (i.e. the set of points in the parameter spacethat correspond to a single (or a finite number of) positions of the variables).- Normal components: A component Cs of the normal locus is normalif dim(Cs) = dim(π2(V(F ) ∩ π−11 (Cs))) (i.e. the components of the nor-mal locus whose different points are generated by different points of thevariables).- Special components: A component Cs of the normal locus is specialif dim(Cs) > 0 and dim(π2(V(F ) ∩ π−11 (Cs))) = 0 (i.e. the componentsof the normal locus of dimension 1 that are generated by a single (or afinite number of) points of the variables).- Non-normal locus: are the points u ∈ C2 of the locus L for whichdim(π2(V(F ) ∩ π−11 (u))) > 0 (i.e. the set of points in the parameter spacethat correspond to infinite positions of the variables).- Degenerate components: are the components Cd of the non-normallocus with dim(Cd) > 0 (i.e. the components of the non-normal locus ofdimension 1).Gröbner Cover for locus computation 5- Accumulation components (points): are the components Ca of thenon-normal locus with dim(Ca) = 0 (i.e. the zero dimensional compo-nents of the non-normal locus).Problems in section 4 are chosen to justify the taxonomy. The geometric re-levance of this algebraic classification of the different components of a locus isopen to interpretation by the user. Dynamic Geometry systems could present thecollection of different parts (with the corresponding typology) of the computedlocus, letting to the user the decision of which pieces are to be considered ordiscarded in a particular context.Based on our experience (see section 4), we tend to discard the degeneratecomponents as geometrically irrelevant, as they usually correspond to degenerateinstances of a construction, such as two coincident vertices in a triangle. However,we consider the accumulation points as forming part of the (geometric) locus,since they represent special points that are determined by infinitely many valuesof the variables. The special components are generally also discarded, as they aregenerated by a single position of the mover, but sometimes they can be useful.We designed the Locus algorithm (Table 1) that takes the output of thegrobcov and classifies the appropriated components of the segments followingthe defined taxonomy. We show now some examples justifying our taxonomyand its functionality.4 Applications and Functionality4.1 Pascal’s LimaçonOQcPP ′Fig. 1. Pascal limaçonThe following Problem is considered here to justify the definitions of “Nor-mal” and “Special” components of the “Normal Locus”. Consider the following6 M. Abánades - F. Botana - A.Montes - T. Reciolocus problem. Let O be a fixed point on a circle c of radius r, and l be a linepassing through O(0, r) and Q(y1, y2), a general point on c. Let P (u1, u2) be apoint on l such that distance(P,Q) = k, where k is a constant. The limaçon ofPascal is the locus set traced by P as Q moves along c, as shown on Figure 1.Setting r = 2 and k = 1, the ideal (set of equations) determining the locus is:F = 〈y21 + y22 − 4, (u1 − y1)2 + (u2 − y2)2 − 1, (2− y2)u1 + y1(u2 − 2)〉Computing the solution using grobcov and locus algorithms in Singular we dothe following:Input:LIB "grobcov.lib";ring R=(0,u1,u2),(y1,y2),lp;short=0;ideal F= y1^2+y2^2-4, (u1-y1)^2+(u2-y2)^2-1, (2-y2)*u1+y1*(u2-2);def L=locus(grobcov(F));"locus(grobcov(F))="; L;Output:locus(grobcov(F))=[1]:[1]:_[1]=(u1^4+2*u1^2*u2^2-9*u1^2+u2^4-9*u2^2+4*u2+12)[2]:[1]:_[1]=1[3]:Normal[2]:[1]:_[1]=(u1^2+u2^2-4*u2+3)[2]:[1]:_[1]=1[3]:SpecialLocus algorithm determines and characterizes two components: a “Normal”component that is the Pascal limaçon’s concoid, and a “Special” circle generatedby the single mover point O, for which the construction is degenerate. Usuallythese Special components are to be discarded by Dynamical Geometry software,and the algorithm returns it as “Special” letting to the user the decision aboutits consideration.4.2 Offset of a circleNow we want to justify the definition of “Accumulation” and “Degenerate” com-ponents of a locus considering the locus of the offset of a circle of radius 1 atGröbner Cover for locus computation 7distance 1. The given circle has as equation g : y21 + y22 − 1. The family of circleswho generate the envelope is F = (u1 − y1)2 + (u2 − y2)2 − 1, where (u1, u2) issome point of the envelope. To compute the envelope we have to add the equa-tion∂F∂y1∂g∂y2− ∂F∂y2∂g∂y1, ensuring that the envelope is tangent in each point toa curve of the family. We have to consider thus the following ideal:H = 〈y21 + y22 − 1, (u1 − y1)2 + (u2 − y2)2 − 1, y1u2 − y2u1〉.The standard method will eliminate (y1, y2) to obtain the envelope. But we canconsider (y1, y2) as the mover and take (u1, u2) as the tracer using the ringR=(0,u1,u2),(y1,y2),lp; and the command locus(grobcov(H)).Doing so we also obtain two components: The “Normal” component consist-ing of the circle u21 + u22 − 4, plus the “Accumulation” point component (u1, u2)at the center generated by all the circles. The accumulation points are to beconsidered as part of the offset by Dynamical Geometry software,.4.3 Improvements: Detecting Bad Mover PositionsLocus algorithm as described in Table 1 is incomplete, and must be improved.It assumes that the generic segment of the Gröbner Cover has basis {1}, aswe do not expect a locus that contents the whole plane except some curves.Nevertheless, in the next example the generic segment does not have basis {1}.A(3, 4)O(3, 1)T (x, y)M(y1, y2)B(3,−2)P (y3, y4)Fig. 2. Locus described by T (and P ) as M runs along its circle.The reason is that there is a point of the mover for which the construction is8 M. Abánades - F. Botana - A.Montes - T. Reciodegenerate, and would give rise to a bidimensional solution. It is necessary toeliminate this point of the mover to obtain the correct expected locus.We programmed our algorithm to be able to detect and eliminate such badmover point positions producing degenerate solutions in the locus.Consider the following locus construction (see Figure 2). The moverM(y1, y2)runs over the circle with center at O(3, 1) and radius OA, where A = (3, 4).We consider the line parallel to the line AM passing through O and the lineperpendicular to it passing through the point B = (3,−2). Both lines intersectat point P (y3, y3). Consider the line AP and the circle with center M and radiusMP . We consider this intersection as the tracer point(s): T (x, y) and T ′ = P .The polynomial system describing the problem is the ideal F given byF = 〈(y1 − 3)2 + (y2 − 1)2 − 9, (4− y2)(y3 − 3) + (y1 − 3)(y4 − 1),(y1 − 3)(x1 − y1)− (4− y2)(y4 − y2),(y4 − 4)x− (y3 − 3)y + 4y3 − 3y4,(x− y1)2 + (y − y2)2 − (y1 − y3)2 − (y2 − y4)2〉For the computation we use the ring ing R=(0,x,y),(x1,x2,y1,y2),lp;. Withthis improvement we obtain the proper two irreducible “Normal” componentsV(x2 − 6x+ y2 + y + 7),V(x4 − 12x3 + 2x2y2 − 13x2y + 236x2 − 12xy2 + 78xy − 1200x+ y4−13y3 + 60y2 − 85y + 1495).The output produced by the algorithm includes a message informing of theremoval of bad mover positions (point A(3, 4) in our case).Geometrically, the problem is that when the mover is on the point (y1 =3, y2 = 4), the line AM is not defined. The algorithm eliminates the segments ofthe Gröbner Cover containing this point that is detected on the generic segment.This locus is example 9 in our prototype [3]. By clicking the Find locusbutton, we obtain the locus description shown on Figure 2.References1. Miguel Abanades, Francisco Botana, Antonio Montes, Tomas Recio. “An AlgebraicTaxonomy for Locus Computation in Dynamic Geometry”. Computer Aided Design(2014), submitted.2. Deepak Kapur, Yao Sun, Dingkang Wang. “A new algorithm for computing com-prehensive Gröbner systems”. Proceedings of ISSAC’2010, (2010), 29–36. ACMPress, New York.3. F. Botana, Locus Prototype: http://webs.uvigo.es/fbotana/LocusGC/ (2012).4. Antonio Montes. “Using Kapur-Sun-Wang algorithm for the Gröbner Cover”.Proceedings of EACA 2012, (2012), 135–138. ed: R. Sendra, C. Villarino. pub:Universidad de Alcalá de Henares5. Antonio Montes, Michael Wibmer. “Gröbner bases for polynomial systems withparameters”. Jour. Symb. Comp. 45 (2010), 1391–1425.6. Michael Wibmer. “Gröbner bases for families of affine or projective schemes”. Jour.Symb. Comp. 42: (2007), 803–834.7. Singular. http://www.singular.uni-kl.de/ (Last accessed February 2014).

Software using the Gröbner Cover for geometrical loci computation and classification

We describe here a properly recent application of the Gröbner Cover algorithm (GC) providing an algebraic support to Dynamic Geometry computations of geometrical loci. It provides a complete algebraic solution of locus computation as well as a suitable taxonomy allowing to distinguish the nature of the different components. We included a new algorithm Locus into the Singular grobcov.lib library for this purpose. A web prototype has been implemented using it in Geogebra.Peer ReviewedPostprint (published version

Abanades, Miguel A.

Botan, Francisco

Montes Lozano, Antonio

UPCommons. Portal del coneixement obert de la UPC

Software using the Gröbner cover for geometrical loci computation and classification

We describe here a properly recent application of the Gröbner Cover algorithm (GC) providing an algebraic support to Dynamic Geometry computations of geometrical loci. It provides a complete algebraic solution of locus computation as well as a suitable taxonomy allowing to distinguish the nature of the different components. We included a new algorithm Locus into the Singular grobcov.lib library for this purpose. A web prototype has been implemented using it in Geogebra.Peer Reviewe

UPCommons

Software using the Gröbner Cover for geometrical loci computation and classification

Abstract

Similar works

Full text

Available Versions

Crossref

UCrea

UPCommons. Portal del coneixement obert de la UPC

UPCommons