On Degeneracy of Lower Envelopes of Algebraic Surfaces

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1 CCCG 010, Wiipeg MB, August 9 11, 010 O Degeeracy of Lower Evelopes of Algebraic Surfaces Kimikazu Kato Abstract We aalyze degeeracy of lower evelopes of algebraic surfaces. We focus o the cases omitted i the existig complexity aalysis of lower evelopes [Halperi ad Sharir 1993], ad re-defie the degeeracy from the viewpoit of the adjacecy structure ad the umber of coected compoets. We also defie badess of degeeracy from such viewpoit ad show how bad the degeeracy ca be. This research is iteded to cotribute to a robust geometrical computatio i the tolerat model. 1 Itroductio The importace of o-euclidea Vorooi diagrams is arisig. For example, a power diagram weighted Vorooi diagram) is used to aalyze the structure of the protei [7]. The agular Vorooi diagram [] ad the aspect ratio Vorooi diagram [1] are proposed as a tool for aalysis of algorithms cocerig computer visio. Amog o-euclidea Vorooi diagrams, the algebraic Vorooi diagram is oe of the most importat classes. Here algebraic meas that the boudaries are expressed as polyomials. Actually, the examples eumerated above are all algebraic. Cocerig lower evelopes of algebraic surfaces, Halperi ad Sharir [3] showed a upper boud of computatioal complexity. By aalyzig the combiatorial structure, they showed the upper boud is O +ε ) for ay ε. However it is uder the assumptio that the surfaces are i a geeral positio. It meas that the discussio about degeerate cases is itetioally avoided. From the viewpoit of real world applicatios, aalysis of the degeeracy leads to robustess of computatio. I such systems as computer-aided desig CAD) ad geography iformatio system GIS), which deal with algebraic curves ad surfaces, the tolerat model is employed iside. I the tolerat model, i order to hadle the umerical error of computatio, two poits are assumed as a same poit if their distace is withi a give fixed small umber. For example, the crossig poit of two geeral splie curves caot be expressed rigidly with double precisio coordiate values. I the tolerat model, eve if three curves are coceptually assumed to meet at oe poit, there might be a small Niho Uisys, Ltd. kimikazu.kato@uisys.co.jp regio surrouded by the curves i reality. That kid of uexpected topological situatio is proe to errors, ad thus, for a robust computatio, it is importat to eumerate all the topological situatios i advace. I this paper, we defie the degeeracy of a lower evelope of algebraic surfaces ad aalyze the worst case of degeeracy. As for a Euclidea Vorooi diagram, Sugihara ad Iri [6] iveted a robust algorithm eve for a degeerate case. Although a similar robust algorithm ca be a ultimate goal, a algorithm is ot discussed i the preset paper. We will just classify possible topological situatio aroud degeerate poit. That will directly be helpful for a eumeratio of test cases for software which uses algebraic objects. As a start-up of this lie of research, Muta ad the author [5] showed classificatio of sigularities of the agular Vorooi diagram. It was cosidered that sigularities were key poit to aalyze the degeeracy of the algebraic Vorooi diagram ad classificatio of sigularities was a first step to discuss the degeeracy. However, i a geeral cotext, the curret paper shows a evil degeeracy may occur eve without a sigularity. The rest of paper is orgaized as follows. I Sect., we give some defiitio of terms ad defie degeeracy. We aalyze the worst cases of degeeracy i Sect. 3. The we coclude i Sect. 4. Prelimiaries I the paper by Halperi ad Sharir [3], they cosidered patches of algebraic surfaces to defie a lower evelope ad the algebraic surfaces used there ca be geeral; i.e. a surface is defied by fx, y, z) = 0 where f i polyomial i x, y, z. However, i this paper, we restrict the domai. We oly cosider surfaces expressed as z = fx, y) ad assume that a surface is ot a patch; i.e. the surface cotais o boudary coditio ad is expressed by z = fx, y) aloe. Defiitio 1 Lower evelope of surfaces) For a give set of surfaces i the xyz-space, the projectio of a set of lowest parts of S i z-directio, ito xy-plae is called a lower evelope of S. I other words, whe a set of surfaces S is give as S = S i Si = x, y, z) z = fi x, y) }}, 1)

2 d Caadia Coferece o Computatioal Geometry, 010 the the lower evelope D of S is defied as D = D i D i = x, y) f i x, y) < f j x, y) for j } }, D i. ) I this paper, we oly focus o the case whe f i s are all polyomials with coefficiets i R. I such a case, we simply say D is a diagram of f i }, or a algebraic diagram if ot metioig f i s. To itroduce a computatioal error model to a lower evelope, we cosider a parameterized algebraic diagram defied as follows. Defiitio A parameterized algebraic diagram D ε is a lower evelope of a set of algebraic surfaces S ε = S εi Sεi = x, y, z) z = f i x, y, ε) }}, 3) where f i is polyomial i x, y, ε. I other words, D ε is defied as D ε = D εi D εi = x, y) f i x, y, ε) < f j x, y, ε) for j } }, D εi, for a set of polyomials f i x, y, ε) 4) For a small ε, D ε emulates a behavior of a diagram with a umerical error. Whe we say give a perturbatio to D ε, it meas cosider D ε for a sufficietly small ε. I short, a diagram is called degeerate if its perturbatio gives a chage of its topological structure. The, what is a chage of topological structure? I the Euclidea Vorooi diagram, it oly meas the adjacecy structure of regios. However, i the algebraic diagram, there is aother case to cosider: the umber of coected compoets of a regio might chage. Before defiig the degeeracy, we defie a adjacet graph of a algebraic diagram as follows. Type II The case whe the umber of coected compoets of a regio chages with a perturbatio. The Type I degeeracy is a geeralizatio of degeeracy of Euclidea Vorooi diagram. Type II oly appears i a algebraic diagram. Note that, sice we do ot cosider patches of surfaces, other degeerate cases metioed i [3] the case whe two boudary meet, sigular poit of a surface lies o a boudary of aother patch, etc. are omitted itetioally. The followigs are examples of degeeracy. Example 1 Whe polyomials are give as f 1 = x 1 ε) + y, f = x ε) + y, f 3 = x + y 1 + ε), f 4 = x + y + 1 ε), 5) the D 0 is show as Fig. 1a, which is a degeerate case of a Euclidea Vorooi diagram. By givig ε a perturbatio, its topological structure chages as i Fig. 1b ε > 0) or Fig. 1c ε < 0). This is a example of Type I degeeracy. Actually, the adjacecy graph is chaged as ε perturbs. a) ε = 0 b) ε > 0 c) ε < 0 Figure 1: Example of Type I degeeracy Example Suppose that polyomials are give as f 1 = y x )y + x ε), f = 0. 6) The, whe ε = 0, the regio D 1 = x, y) f 1 x, y) < f x, y)) } has two coected compoets Fig. a). By givig ε a perturbatio, the umber of its coected compoets becomes three for ε > 0 Fig. b), ad becomes oe for ε < 0 Fig. c). Defiitio 3 Suppose that a diagram D is defied by polyomials f 1,..., f. The, a adjacet graph G = V, E) of a diagram D is a udirected graph whose vertex set is V = 1,..., } ad whose edge set is E = i, j) D i is adjacet to D j via a curve }. Now we defie two types of degeeracy as follows. a) ε = 0 b) ε > 0 c) ε < 0 Type I The case whe the adjacecy graph chages with a perturbatio. Figure : Example of Type II degeeracy

3 CCCG 010, Wiipeg MB, August 9 11, Aalysis of degeeracy For the Euclidea Vorooi diagram, the adjacecy graph of the regios ca oly be a plaar graph, but for the algebraic diagram, it ca become arbitrary. Sice adjacecy graph ca be arbitrary for the algebraic diagram, the thikable worst case is that a cyclic graph becomes a perfect graph. The followig shows it really happes. Theorem 1 Cosider a lower-evelope defied by f i x, y, ε) = x cos iπ y si iπ ) + ε x si iπ + y cos iπ ) 4 x si iπ + y cos iπ ). 7) The its adjacecy graph chages from a cyclic graph to a perfect graph as ε chages from ε = 0 to ε > 0 Fig. 3). Thus f i < f j for j 9) cos θ iπ ) > cos θ jπ ) for j 10) i 1)π i + 1)π < θ < i 1)π i + 1)π or π + < θ < π +. 11) Defie the regio domiated by f i as, the the iequality above meas is adjacet to +1 for i = 0,...,, ad R 1 is adjacet to R 0. Thus the adjacecy graph for f i s is cyclic. As for ε > 0, we will oly cosider the area sufficietly ear to the lie y = 0. By expadig the iequality f i x, 0) > f j x, 0), we obtai cos iπ + εx si 4 iπ > cos jπ + εx si 4 jπ, 1) Regard ad R i as the same regio, ad deote it by i.e. R i = R i ). The by the formula above, the diagram aroud y = 0 is equivalet to the lower evelope of g i x) = cos iπ +εx si 4 iπ. Sice the graph of y = g i x) appears as Fig.4, is proved to be adjacet to +1 for i = 0,..., /. y. y = g3x) y = gx) y = g1x) y = g0x) a) ε = 0 x Figure 4: The graph of g i s The, we will aalyze the compoet of R i. Because f i x, y) f i x, y) = 8xy cos iπ si iπ ε x cos iπ + y si iπ ) ) 1, b) ε > 0 Figure 3: Example of a worst case of Type I Proof. Whe ε = 0, by substitutig r cos θ, r si θ) for x, y), f i ca be expressed as f i r cos θ, r si θ, 0) = r cos θ iπ ), 8) 13) i the area x > 0, the domiace i depeds o the sig of y ad ε x cos iπ + y si ) iπ 1. Cosider the equatio ε ) x cos iπ 1 which is give by substitutig 0 for y i the latter formula, ad deote its solutio by α. The, whe defiig β i > 0 as the solutio of the equatio cos iπ +εx si 4 iπ i + 1)π = cos +εx 4 i + 1)π si, 14)

4 d Caadia Coferece o Computatioal Geometry, 010 x = α R 1 y = 0 R 1 Figure 5: The compoet of R i a) ε = 0 b) ε > 0 R 1 R 1 R R R i+1 R i Figure 8: Example of a worst case of Type II; whe d mod 4 = R 1 R 1 R R 1 R i Figure 6: Diagram aroud y = 0 R [ i+j+1] R [ i+j] 1 R [ i+j] a) ε = 0 b) ε > 0 Figure 7: Diagram aroud y = 0 after a rotatio With some simple calculatio, it ca be proved that β i 1 < α < β i. Sice β i is the boudary betwee ad +1, the boudary of ad R i appears i both y > 0 part ad y < 0 part i. Thus the compoet of appears as Fig. 5. Overall colorig aroud y = 0 i the area x > 0 is show as Fig. 6, ad it shows is adjacet to R i+1 ad R i. By trasferrig x, y) by x y ) = cos jπ si jπ ) x si jπ cos jπ y ) j = 0,, 1), ad applyig the same discussio as above, the diagram becomes Fig. 7 where [m] = m mod ), ad thus is adjacet to R [ 1+j] ad R [ i+j+1]. This meas all the possible combiatio of ad R j appears as adjacet regios. Now we will cosider the worst case for Type II. The more largely the umber of crossig poits of boudaries chages, the worst we ca say it is. How much the umber of crossig poits ca chage? Here, because of the difficulty of aalysis o geeral cases, we show some examples oly uder the assumptio that the lower-evelope is expressed by two polyomials, ad f 1 is degree d ad f = 0. Uder that coditio, Bézout s theorem explaied bellow tells that the thikable worst case is whe f 1 ca be factorized ito two polyomials whose degrees are Figure 9: Example of a worst case of Type II; whe d mod 4 = 0 1. d ad d for eve d; ad. d 1 ad d+1 for odd d. Theorem Bézout, see [4] for example) I the two dimesioal projective space P K for a algebraically closed field K, suppose that two polyomials f ad g whose degree are m ad l respectively are give. The the umber of crossig poit of f = 0 ad g = 0 is ml icludig the multiplicity. The followig example shows it really achieves the thikable maximum umber of self crossig poits. Example 3 Cosider the lower-evelope give by f 1 = where [ ε l1 y l 1 i=0 ][ x iε) ε l x l i=0 ] y iε), f = 0, l1 = l = d d is eve) l 1 = d 1, l = d+1 d is odd) 15) 16) The, the umber of crossig poits of boudaries chages from 1 to d 4 for eve d) or d 1 4 for odd d) as ε chages from 0 to ε > 0. Typical picture of this example for eve d is show i Fig. 8 ad Fig. 9. Note that for eve d/ ad ε = 0, the regio domiated by f is empty ad a boudary appears as y = 0 see Fig. 9a).

5 CCCG 010, Wiipeg MB, August 9 11, Coclusio We have defied degeeracy of a lower evelop of algebraic surfaces from the viewpoit of a perturbatio of topological structures, ad have show some examples which are cosidered as the worst cases. The examples show might seem to be too artificial because it is uder a very geeral settig ad especially the error model employed here is too arbitrary from the practical poit of view, but we showed possibility of eviless i computatio of the algebraic diagram. It is a importat step toward a robust implemetatio of a applicatio which utilizes geometric objects. Furthermore, we shed a light o a combiatorial aspect of a perturbatio of a set of algebraic surfaces. Ackowledgemets This research is supported by Promotio program for Reducig global Evirometal load through ICT iovatio PREDICT), Miistry of Iteral Affairs ad Commuicatios of Japa. Refereces [1] T. Asao. Aspect-ratio vorooi diagram ad its complexity bouds. Iformatio Processig Letters, ):6 31, 007. [] T. Asao, N. Katoh, N. Tamaki, ad T. Tokuyama. Agular Vorooi diagram with applicatios. I Proceedigs of 3rd Iteratioal Symposium o Vorooi Diagram i Sciece ad Egieerig, pages 3 39, Baff, Caada, 006. [3] D. Halperi ad M. Sharir. New bouds for lower evelopes i three dimesios, with applicatios to visibility i terrais. I SCG 93: Proceedigs of the Nith Aual Symposium o Computatioal Geometry, pages 11 18, New York, NY, USA, ACM. [4] R. Hartshor. Algebraic Geometry. Spriger Verlag, [5] H. Muta ad K. Kato. Degeeracy of agular Vorooi diagram. I Proceedigs of 4th Iteratioal Symposium o Vorooi Diagram i Sciece ad Egieerig, pages 88 93, Wales, UK, 007. [6] K. Sugihara ad M. Iri. A robust topology-orieted icremetal algorithm for Vorooi diagrams. I Iteratioal Joural of Computatioal Geometry ad Applicatios, volume 4, pages 179 8, [7] A. Varshey, F. Brooks, ad D. Richardso. Defiig computig ad visualizig molecular iterfaces. IEEE Visualizatio, 95:36 43, 1995.