Visualisation of implicit algebraic Lionel Alberti, Bernard Mourrain

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Lionel Alberti, Bernard Mourrain. Visualisation of implicit algebraic curves. Pacific Conference on Computer Graphics and Applications 2007, Ron Goldman, Oct 2007, Lahaina, Maui, Hawaii, United States. pp.303-312. ￿inria-00175062￿

HAL Id: inria-00175062 https://hal.inria.fr/inria-00175062 Submitted on 26 Sep 2007

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The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. inria-00175062, version 1 - 26 Sep 2007 opoueapeeielna prxmto wti a (within approximation distance Hausdorff linear given piecewise a produce to oooial qiaet(e oemrhc to homeomorphic) (ie. equivalent topologically f D e cuaeya mlctpaa planar implicit an accurately der hierarchi- computation the of precision cally. allow the which im- adapt approaches is to multilevel it scales, develop different to on portant coexists may models plicit ere( degree iglrrgos h te ein r iie into divided are regions curve other these or of the The boundary of the regions. topology on singular points the the that from deduced check be can to these order singu- around in degree the topological points isolates the compute which points, curves, lar algebraic implicit of scutn h ubro stp ye falgebraic of degree types given such isotopy a of questions param- of number curves many the the in Mathematics, counting curves In as implicit surfaces of domain. parametric analysis eter two the of to curve lead comput- intersection as the such ing operations, Geometric Fundamental of applications Modeling. many in appears curve plicit these on safely to zoom to coefficients, and regions. large interest of with defined regions polynomials curves identify render degree to high able is by exam- algorithm on this shown that is ples it basis, Bernstein polynomial the the in represented of techniques boundary. enveloping the with on Combined information from determined also are 0 ( 0 = nti ae,w ecieanwagrtmt ren- to algorithm new a describe we paper, this In edsrb e loih o h visualisation the for algorithm new a describe We h rbe faayigadvsaiaino im- of visualisation and analysing of problem The y [ = eua ein,i hc h rnhso h curve the of branches the which in regions, regular f ,b a, d ∈ ≥ ] Q × ) ic h iglrt tutrso im- of structures singularity the Since 8). 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[4, on surfaces rely to 4D extended 3D, and in 2] even 11, and 12, ap- 2D, 16, been in [13, has curves approach to The successfully plied exact tests. adjacency sweep- condition the and genericity involve the and tests They points to critical of singularities. critical computation and the as planes detects such ing that events, sweeping axis topological conceptual some a to projec- on use perpendicular based They techniques algorithm. tion [5] Decomposition braic these meet that algorithm requirements. of two types tech- main objects. two geometric algebra are of There representations computer exact applying on appear, niques curves when Such instance, degree. effi- for with large and polynomials or certified by coefficients the defined large is curves for of ends. treatment our look cient to we unsuitable feature [26] other in The described methods one accurately. based the them physically like on the zoom makes to requirement and con- This point which ”pixels” singular detect a to tain able be should algorithm a.J .Dieudonn´e A. J. Lab. oee,te sueeatipteutosand equations input exact assume they However, Alge- Cylindrical the by inspired is type first The ⋆ And non-degeneracy conditions have to be checked different steps in the method. (which can be difficult by itself) to ensure the correct- Here are some notations that we will need hereafter. ness of the algorithm. The problem is that the choice The implicit curve is defined by a squarefree polyno- of projection is not at all related to the geometry of mial f ∈ Q[x, y]. We denote by Z(f) = {(x, y) ∈ the curve. R2|f(x, y)=0} or C the locus of zeroes of f. The This is why the CAD methods are hardly efficient in domain in which we want to analyse the curve C is 2 practice, and are facing complexity problems in higher D0 := [a,b] × [c, d] ⊂ R . dimension. They are also intrinsically delicate to apply The set of singular points of C is denoted S := 2 using approximate computation. {(x, y) ∈ R |f(x, y)= ∂xf(x, y)= ∂yf(x, y)=0}. The other type of methods relies on subdivision The set of critical points or extremal points of f is 2 techniques of the original domain. This process is most denoted Ze(f) := {(x, y) ∈ R |∂xf(x, y)= ∂yf(x, y)= commonly used to get approximations of the curve in 0}. terms of Hausdorff distance. The most famous family For a subset S ⊂ R2, we denote by S◦. its interior, of algorithm using this approach is the marching cube by S its closure, and by ∂S its boundary. We call algorithms family [20]. It doesn’t not give any garantee domain any closed set D such that D◦ = D and D is on the topological correctness of its output, but it in- simply connected. spired some algorithms that do certified that their out- We call branch (relative to a domain D), any smooth put has the same topology as the curve (usually in the closed segment (ie. C∞ diffeomorphic to [0, 1]) whose smooth case). They have already been used for solv- endpoints are on ∂D. ing several complicated equations. See [29, 8] and the We call half branch at a point p ∈ D◦ or half branch recent improvements proposed in [21], exploiting pre- originating from p ∈ D◦, any smooth closed segment conditioning techniques. Extensions of this approach which has one endpoint on ∂D and whose other end- to higher dimensional objects have also been consid- point is p. ered [27, 16, 14, 28, 18, 1]. These subdivision methods Our objective is to determine the topology of C usually fail when singular points exist in the domain. inside D0. To do this, we find a partition of D0 If a threshold on the minimal size for boxes is not set, into what we call simple domains Di for which we the algorithm would run forever. Indeed at singulari- can compute the topology. For each kind of simple ties, no matter the scale of approximation, the shape domains, we have a so-called connection algorithms and topology of