Tangents to fractal curves and surfaces

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1 Tangents to fractal cures and surfaces Dmitry Sokolo, Christian Gentil, Hicham Bensoudane To cite this ersion: Dmitry Sokolo, Christian Gentil, Hicham Bensoudane. Tangents to fractal cures and surfaces. Cures and Surfaces, 22, 692, pp <hal-7989> HAL Id: hal Submitted on Mar 23 HAL is a multi-disciplinary open access archie for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or priate research centers. L archie ouerte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de nieau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou priés.

2 Tangents to fractal cures and surfaces Dmitry Sokolo, Christian Gentil, and Hicham Bensoudane LORIA - Alice, Campus Scientifique, BP Vandoeure-les-Nancy Cedex, France christian.gentil@u-bourgogne.fr Laboratoire LE2I UMR CNRS 558 Faculté des Sciences Mirande, Aile de l Ingénieur, 278 DIJON Cedex Hicham.Bensoudane@u-bourgogne.fr Laboratoire LE2I UMR CNRS 558 Faculté des Sciences Mirande, Aile de l Ingénieur, 278 DIJON Cedex Abstract. The aim of our work is to specify and deelop a geometric modeler, based on the formalism of iterated function systems with the following objecties: access to a new unierse of original, arious, aesthetic shapes, modeling of conentional shapes smooth surfaces, solids and unconentional shapes rough surfaces, porous solids by defining and controlling the relief surface state and lacunarity size and distribution of holes. In this context we intend to deelop differential calculus tools for fractal cures and surfaces defined by IFS. Using local fractional deriaties, we show that, een if most fractal cures are nowhere differentiable, they admit a left and right half-tangents, what gies us an additional parameter to characterize shapes. Keywords: fractal cure, fractal surface, local fractional deriatie, iterated function systems Introduction Our long-term goal is to deelop a geometric modeler based on iteratie process and fractal geometry to allow designers to access a new unierse of shapes. Special properties of fractal structures hae led us to new concepts inexistent on classical geometric objects. Fractal cures and surfaces, for example, can hae ery different aspects and ery different kinds of roughness. This ast ariety of shapes is not accessible with polynomial cures and surfaces exactly because of their differentiability. ACKNOWLEDGMENT: This work has been supported by French National Research Agency ANR through COSINUS program project MODITERE n ANR- 9-COSI-4.

3 2 Tangents to fractal cures and surfaces To control these aspects we are led to use the concept of geometric texture [2]. This geometric texture is ery tightly coupled with differentiability of cures and surfaces. We study and attempt to characterize differential behaior from a geometric point of iew by means of local fractional deriatie [3]. Of course, other works were performed to study differential properties of fractal cures and surfaces. Kolwankar and Gangal [2, ] applied the fractional calculus to study real-alued functions with few examples of fractal cures. Using their results the authors are able to describe roughness of a cure with the Hölder exponent. Cochran has proposed a method of calculating normals to a fractal surface [6]. Scealy [6] identifies C fractal cures. Howeer, these works are applicable in the cases where deriaties exist and do not permit to characterize rough shapes. In this paper we present a general approach to study differential properties of fractal cures and surfaces. We are using BC-IFS Boundary Controled Iterated Function System [9] to construct fractal structures. Then we are using this representation to study necessary and sufficient conditions of differentiability. In order to simplify the presentation we show in detail how it can be done for the family of local corner cutting cures. Differential behaiour of these cures was studied before by De Boor [5] and Gregory [8], the cited authors hae found necessary conditions of differentiability. Howeer, BC-IFS approach allows to describe a larger family of cures, and therefore while we find the same necessary conditions for the set of cures gien by De Boor, we also study other regions of the conergence domain. A cartography of domains is presented in late sections of the paper. Finally, we show that necessary conditions are also sufficient ones. The rest of the paper is organized as follows: Section 2 proides necessary background needed to introduce BC-IFS in section 3. Section 4 studies necessary conditions of differentiability and presents cartography of the conergence domain Section 5 shows that necessary conditions are also sufficient ones Section 6 introduces a new descriptor of roughness of a fractal shape. 2 Background 2. IFS Gien a complete metric space E, d, where d is the associated metric, an IFS Iterated Function System is a finite set of contractie operators T = {T i } N i= acting on points of E. Each T i : E E induces T i : HE HE, i.e. operators acting in the space HE of non-empty compact subsets of E. Thus it is possible to define so-called Hutchinson operator T : HE HE as a union of operators T i. The Hutchinson operator maps a non-empty compact K E onto N T i K. The operators T i are contractie in the space E,d, i= therefore the induced operators are contracting in the space HE,d HE,

4 Tangents to fractal cures and surfaces 3 where d HE is the Hausdorff metric []. Of course, the Hutchinson operator is also contractie in HE,d HE. The contraction theorem [9] states that there is a unique compact A such as TA = A, namely the fixed point, noted AT. Moreoer, the fixed point A may be found as a limit A = lim n Tn K, where the limit does not depend on the choice of the seed compact K. The top line of figure 2 proides an illustration. The underlying IFS is composed of four transformations: ] ] ] [ x [ ] [ [ x.85.4 x. T = + y.4.85 y.6 [ ] [ ] [ ] x.2.26 x. T 2 = + y y.6 [ x.5.28 T = y y [ ] [ ] x.. x T 3 = y..6 y ] + [ ]..44 We hae chosen a square as the seed K. Remember that the final shape is independent of the choice. Thus, at the first iteration we apply each {T i } n i= to the square to get four deformed quadrilaterals in place of two branches, the stem and the top of the fern. Then we take a union of the quadrilaterals and restart the process. In few iterations only, quadrilaterals anish being almost imperceptible, but their union being plenty engender the shape of the fern. 2.2 CIFS In regular IFS we start from a seed, then apply a set of rules transformations, and repeat as required. In CIFS Controlled, or graph-directed IFS not all rules need to be applied at each step, a directed graph controls directs rules [3, 5]. We associate work spaces to the nodes of the graph and the arcs represent the transformations to be applied at the current state. In such a way it is possible to blend attractors of different nature. The left image of figure represents the control graph it can be seen as an automaton for the regular IFS generating the Barnsley fern, the iteration process is shown in the top line of figure 2. But what happens if we modify the automaton? Let us add three more transformations to the IFS the corresponding automaton is shown on the right of figure : ] ][ x [ ] [ x /2 x T 5 = y /2 y [ ] [ x /2 x T 7 = y /2 y ] + [ ] /2 T 6 x y = [ /2 /2 y ] + [ ] /2 These three transformations on itself generate the Sierpiński s triangle. Note also that the destination of the transformation T 2 is changed. Thus, once the transformation T 2 was applied, the subdiision is made according to the rules of the Sierpiński s triangle. While the Barnsley s fern consists of infinite number of shrunk copies of itself, the attractor shown in the right bottom image of figure 2 is a fern that consists of infinite number of shrunk Sierpiński s triangles. In fact, the arrows of the automaton depict the data flow, or the order of application of transformations. Howeer actual transformations act in the other direction. That

5 4 Tangents to fractal cures and surfaces T T T5 T ÖÒ T2 T ÖÒ Ë ÖÔ T2 T6 T3 T3 T7 Fig.. Two automatons generating rules order of application of transformations. Fig. 2. Top line: the Barnsley fern; bottom line : C-IFS allows to mix up attractors of different nature. 2.3 Projected IFS The notion of projected IFS was introduced by Zaïr and Tosan [2]. If one separates the iteration space from the modelling space, it is possible to create free-form fractal shapes. The work was inspired by spline cures, which are created by a projection of basis functions defined in a barycentric space. In the same way, it is possible to construct an IFS attractor in a barycentric space whose dimension is equal to the number of control points and to project it into the modelling space. In other words, if we hae an attractor A BI n = {λ R n n i= λ i = }, where n is the number of control points, the projection can be made just by a matrix multiplication PA = { n i= P iλ i λ i A}. Here the matrix P = [P P P n ] is composed of control points. This construction imposes that transformations in IFS must act in a barycentric space. For linear operators expressed in matrix form it means that all columns sum up to. 3 Boundary Controlled IFS Boundary Controlled IFS BC-IFS is a graph-controlled IFS with a B-rep structure introduced by Tosan et al [9]. This is a conenient method to express faceedge-ertex hierarchies implicitly existing in many fractal attractors [7]. The notions of B-rep here are a bit more general that in the classical case. Here a topological cell may be bordered by a fractal object and not only with an edge ertex. For example, a face may be the Sierpiński s triangle, an edge the Cantor set. The adantage of this method is its power to express incidence and adjacency constraints for subdiision processes for a gien topology, what results into constraints in the subdiision matrices. To define free-form shapes with BC-IFS it is necessary to distinguish different work spaces: the modeling space is where the final shape lies, this is also the space where we place control points; is so, the right lowest branch of the new fern can be found as the following limit: T 2AT 5,6,7, where T 5,6,7K = T 5K T 6K T 7K. This implies that we hae to choose two seeding compacts for two different spaces, in the images we hae chosen a square and a triangle.

6 Tangents to fractal cures and surfaces 5 barycentric spaces where we construct attractors corresponding to different topological entities, and this is where IFS transformations act. Let us illustrate the approach by constructing a local corner cutting 2D or 3D cure. This type of cures demands at least 3 control points, and endpoints of a cure depend on two of them. For a cure the B-rep structure is simple: we will hae edges bounded by ertices, therefore, in general case we will hae four different spaces: R 2 or R 3 where the final cure is to be drawn, the control points are to be placed here a barycentric space of dimension 3 we construct the simplest case with three control points, this space is where the attractor of the B-rep edge lies a barycentric space of left endpoint of the edge, the dimension is 2 since it depends on two control points similary a two-dimensional barycentric space for the right endpoint. The edge and ertices are attractors in barycentric space to be projected to the modeling space, thus we need three IFS to build the attractors. Let us say that the edge is obtained with an IFS {T,T }, where T and T are 3 3 subdiision matrices for the edge. The ertices are obtained with IFS {T l } and {T r }, and the matrices are 2 2. Figure 4 shows the BC-IFS automaton. First of all we see four nodes corresponding to four spaces. The matrix P is the projection matrix composed of control points. The only thing we hae not yet defined are transformations b and b., 2, 2,,, b 2, 2, T ËÔ l BI 2 2, 2, b,, 2 2, ËÔ r BI 2 P e b b l r Tl Tr T,, ËÔ e BI 3,, Fig. 3. At the left and at the right : barycentric spaces l = r = BI 2 corresponding to the ertices left and right respectiely; in the middle: edge barycentric space e = BI 3. The operators b and b embed the spaces l and r into subspaces of e. Fig. 4. General edge-ertex B-rep BC-IFS automaton.

7 6 Tangents to fractal cures and surfaces In fact, up to this moment we hae not imposed any constraints on the IFS matrices. If we fill them with random coefficients, nothing guarantees any connectiity. Howeer, in B-rep, ertices are boundaries of the edge, so there must be some relationship between the matrices. To ensure this we need embedding operators, namely b and b. Figure 3 illustrates the approach. It shows the basis functions of a uniform B-spline quadratic cure drawn in the three-dimensional barycentric space e = BI 3. Basis functions for endpoints in fact, these are just points of the cure are drawn in corresponding two-dimensional spaces l = BI 2 and r = BI 2. Then b and b embed endpoint spaces into the edge space to impose that the edge has the ertices for its endpoints. Let us find shapes of b and b. The endpoints /2 hae coordinates in the spaces /2 l and r. At the same time in the space /2 e they are /2 and /2. Therefore, the mappings b = and /2 b = are indeed simple embeddings. In other words, b and b say on which control points depend corresponding endpoints. 3. Topology constraints Incidence and adjacency constraints may be easily obtained by expanding the control graph. Figure 5 is an unfolded ersion of figure 4. After the first iteration on the control graph we pass from the modeling space to the edge space e, where an edge is bounded by two ertices l and r. This situation is shown in the top line of the figure. After one more iteration bottom line we hae two smaller edges along with their own two endpoints. b b l e r T l T T T r l b e b l = b r e b r b T l = T b T b = T b b T r = T b Fig. 5. Unfolded ersion of the control graph. This subdiision must be constrained in order to get the desired topology here a cure. Incidence constraints are shown in red, while adjacency constraints are in blue.

8 Tangents to fractal cures and surfaces 7 Adjacency constraints Let us say that we want to get a just-touching cure. When an edge is split into two edges, the left subdiided edge must be connected to the right one in order to guarantee the topology of a cure. Therefore, we impose the right endpoint of the left edge to coincide with the left endpoint of the right edge and this implies that the left and right endpoints are of the same nature and actually lie in the same space haing common generating IFS. Otherwise, the connectiity will be broken at following stages of the subdiision process. So we hae T l = T r = T. When we say the right endpoint of the left edge this means that we can follow the path e T e b in the control graph. The same holds for the left endpoint of the right edge : e T e b. As mentioned aboe, the ertices coincide, thus we can write T b = T b. Let us fill T and T with some arbitrary coefficients: a b c a b c T = d e f T = d e f g h i g h i Then we rewrite the constraint: T b = T b a b c a b c d e f = d e f g h i g h i And it implies that last two columns of T are equal to two first columns of T : b c a b e f = d e h i g h Incidence constraints In the same manner, incidence constraints may be deduced from the fact that the subdiision of the endpoints must be in harmony with endpoints of subdiided edges. Thus, for the left endpoint let us follow the paths e b e T = e T e b, what results into the constraint b T = T b. The same holds for the right endpoint: b T = T b, the constraints are shown in red in figure 5. If the constraints are not fulfilled we will get a disconnected cure after two subdiisions. Let us sole the constraints on the matrices. Haing a b denoted T = we get: d e a a b b c = d d e e f g h i a b a b d e = d e g h

9 8 Tangents to fractal cures and surfaces and a b = a b c d d e e f g h i b c a b = e f d e h i Then it easy to see that: a s b s b s T = d s e s a s T = e s a s b s e s d s e s Let us add the fact that the matrices are stochastic all columns sum up to and we see that for a local corner cutting cure whose points depend on at most three control points and on two at least there are only two degrees of freedom: a b T = a b a a b T = b a b a b Conergence The conergence theorem [] states that in order to get the conergence of an IFS with linear transformations the operators must hae eigenalues strictly less than in absolute alue with one exception: all stochastic operators hae eigenalue that correspond to fixed points of the operators. T has eigenalues:,a, a b, while T has,b, a b. Thus in our case the conergence holds if and only if: < a < < b < < a b < Local corner cutting cures were studied earlier by Gregory, Qu, De Boor et al [8, 5, 4]. The notations we use here correspond exactly to their works, howeer there is a difference in the domain of definition. In fact, when the cited authors construct corner cutting cures, they suppose that all ertices of a polygon at iteration n belong to the polygon from the iteration n. Therefore, the studied domain is shown in gray in figure 6, it corresponds to the domain with positie eigenalues a,b and a b. Our construction does not use this assumption, so the domain we study here is all the region of conergence shown in red.

10 Tangents to fractal cures and surfaces 9 a b = b a b = a - - Fig. 6. In red: the domain of conergence, in gray: Gregory-Qu domain of definition. Parameterization and self-similarity Under latter constraints the attractor of the IFS {T,T } is a cure in three-dimensional barycentric space. It is easy to parameterize the cure with so-called natural parameterization t [,], where t = corresponds to the left endpoint of the cure, t = is the right endpoint and t = 2 corresponds to the junction point in the first leel of subdiision. Then if we denote the parameterized cure in the barycentric space as Ft, it is easy to get the parameterized cure in the modelling space Ct = PFt, where P = P P P 2 is the ector of control points. Let us rewrite the self-similarity property of the cure: F [, 2] = T F[,] and F [ 2,] = T F[,]. The parameterization is induced by the subdiision of the parameter space [,] for each iteration. 4 Differentiability Half-tangent ectors at endpoints are defined for large class of fractal cures [3]. Attractors are self-similar, so if a half-tangent exists for t = then it is easy to find a half-tangent ector for t = 2 : ector T is tangent to the cure F [ 2,] = T F[,]. Therefore, haing defined half-tangent ectors for endpoints of a fractal cure we automatically define it for a set dense in the parameter domain. In the same way if a fractal cure is not differentiable for an endpoint the singularity is copied by the self-similarity property. 4. Eigenectors and eigenalues T has real eigenalues λ =, λ = a b, λ 2 = a and T has λ =, λ = a b, λ 2 = b. Corresponding eigenectors are: b a+b = a, =, b 2 = 2a 2a + b a+b

11 Tangents to fractal cures and surfaces and = b a+b a a+b, =, a + 2b 2 = 2b a Eigenectors and corresponding to the eigenalue gie fixed points of T and T. Note that the fixed points are endpoints of the cure: = F and = F. It is easy to see that eigenectors corresponding to the sub-dominant eigenalues gie half-tangents to the endpoints. Note that sub-dominant eigenalues of T and T are non negatie. There are four cases:. a b is the sub-dominant eigenalue of T : a b a 2a b. In such a case the half-tangent to F is collinear with T and the half-tangent to C is collinear with P P. 2. a is the sub-dominant eigenalue of T : a > a b 2a b <. In this case the half-tangent to C is bp + 2aP + 2a + b P 2. This ector can hae different orientations depending on the alues of a and b. 3. a b is the sub-dominant eigenalue of T : a b b a 2b. The half-tangent to C is collinear with P P b is the sub-dominant eigenalue of T : b > a b a 2b <. The half-tangent to C has direction a + 2b P + 2bP ap 2. Again, the direction depends on a and b. The most interesting case is when a b is the sub-dominant eigenalue for both T and T. In such a case half-tangents are gien by two control points and therefore their directions do not depend on a and b. 4.2 Necessary conditions for differentiability Incidence and adjacency constraints of the BC-IFS guarantee C continuity for the limit cure. To hae C continuity half-tangents must be collinear at the junction point. Figure 7 shows an illustration. Let us suppose that the cure is differentiable, the half-tangents for the point t = 2 may be obtained by the self-similarity property from the half-tangents to endpoints. If t and t + are the directions of the half-tangents to endpoints, then T t+ must be equal to T t : C[,] = Q Q Q 2 F[,] Q Q 2 Q 3 F[,] = PT F[,] PT F[,] As we hae mentioned preiously, ectors t + and t depend on alues of a and b. To hae G continuity, it is obious that T t+ and T t must be at

12 Tangents to fractal cures and surfaces T t Q P + Q 2 + T t+ + Q 3 P 2 Q P Fig. 7. Half-tangent ectors at the joining point. least collinear for any configuration of control points Q Q Q 2 Q 3. The only possibility to fulfil the collinearity is when 2 : ector t belongs to the subspace corresponding to control points P and P 2, i.e. has zero first component. This is the case iff 2a b >. ector t + belongs to the subspace corresponding to control points P and P, i.e. has zero third component. This is the case iff a 2b >. Therefore if T t and T t+ are collinear, then a b is sub-dominant eigenalue for both T and T. The corresponding domain is shown by hatching in red and blue in figure 8. Howeer it is a necessary condition: it includes regions of differentiability zone as well as regions of cusp points zones 2 and 2. Therefore, collinearity is a rough tool and to distinguish the zones we hae to find direction of tangent ectors. 4.3 Cartography of differential behaiours To find direction of half-tangents and to identify differential behaiour of the other areas in the conergence domain, we use the following property etablished in [2]. Property. Let us find decomposition of FF in the eigenbases of T and T, respectiely: FF = α + α 2 2 = FF = β + β 2 2 = ba + b a 2a + b a + b + 2a + b a + b 2 aa + b b a + 2b a + b + a + 2b a + b 2 Let R,L {,2} such that L and R are the sub-dominant eigenectors of T and T, respectiely. Then if Ft has left and right half-tangents at 2 Except degenerated cases: if a = or b = or a+b = then the cure is degenerated piecewise linear and therefore differentiable except at ertices.

13 2 Tangents to fractal cures and surfaces b a b 2 4 zone of differentiability 6 a T t+ = α T t = β Q Q 2 Q Q 2 - Fig. 8. Cartography of regions according to differential properties. respectiely F and F, their directions t and t + are gien by: t = α L L t + = β R R Let us consider the subdiision of the cure : C[,] = Q Q Q 2 F[,] Q Q 2 Q 3 F[,] Now we focus on the point of junction of the two sub-cures C t = Q Q Q 2 Ft and C t = Q Q 2 Q 3 Ft. The directions of the half-tangents at the point are gien by Q Q 2 Q 3 t for C and Q Q Q 2 t+ for C. Depending on sub-dominant eigenalues of T and T we can hae three main different cases:. a b > a and a b > b: this case coers three regions in figure 8, namely regions, 2 and 2. Here we hae R = L = and Q Q Q 2 tr = β Q Q Q 2 = β Q Q 2 Q Q 2 Q 3 tl = α Q Q 2 Q 3 = α Q Q 2 with α = ba+b 2a+b a+b and β = aa+b a+2b a+b. As was explained in the preious section, the two half-tangent ectors at the joining point are collinear in this case. Howeer, the ectors hae the same direction if and only if α and β are of the same sign, and it is the case for the region of figure 8. For regions 2 and 2 half-tangent ectors hae opposite directions, resulting into cusp points.

14 Tangents to fractal cures and surfaces 3 2. a b > a or exclusie a b > b: Q Q 2 Q 3 t+ and Q Q Q 2 t are not collinear in general case. If one of the sub-dominant eigenalues of T or T is a b then the corresponding half-tangent ector is colinear with Q Q 2 regions 3 and 3 but the other half-tangent is not. 3. a b < a and a b < b: this case corresponds to regions 4 and 5 of figure 8. No half-tangent ector is collinear to Q Q 2 since the eigenectors hae three non-zero components and therefore the half-tangent ectors depend on three control points respectiely Q Q 2 Q 3 and Q Q Q 2. Regions 4 and 5 differ in the sign of the smallest in absolute alue eigenalue a b. For the region 5 the eigenalue is negatie, and it forces the cure to oscillate around the direction of the half-tangent, thus giing a fractal aspect to the cure. 5 Sufficient conditions Figure 9 shows the motiation for this section. If we use the natural parameterization, then een for differentiable cures, blending functions Ft are not differentiable in the sense of Lipschitz. Howeer, under a suitable parameterization the blending functions are differentiable. The image is obtained for alues a = /2 and b = /8.... Ft FT T = ft FT Ft F2T.2 F2t Fig. 9. Left image: The three components of blending function Ft in the natural parameterization. In the middle: non-singular parameterization as a functtion of natural parametrization. Right image: The three components blending function in the non-singular parameterization. This is ery similar to the situation with Stam s method [8] of exact ealuation of subdiision surfaces. Haing constructed the natural parameterization it is easy to find points of a cure surface, howeer the behaiour of deriaties is erratic and therefore many methods like [] may fail to work with this parameterization. There are seeral works that construct non-singular parameterizations, for example, we can cite [4] for Catmull-Clark subdiision surfaces. In this section we will show how to reparameterize any cure from Gregory region to garantee C blending functions. So instead of subdiiding the param-

15 4 Tangents to fractal cures and surfaces eter domain in equal hales as the natural parameterization does, we follow the same subdiision rules as for the control polygon. Figure illustrates the idea. We start with a control polygon P,P,P 2 ; in order to parameterize it we chose three real alues t,t,t 2 such that t < t < t 2. Then we say that the segments P,P and P,P 2 hae linear parameter domains t,t and t,t 2, respectiely. P P Q Q 2 Q 3 P 2 Q P t t t 2 u 3 u u u 2 u 3 Fig.. Left image: original control polygon P, P, P 2 and its parameter alues t, t, t 2. Right image: parameter domain is subdiided along with the control polygon. Then subdiided polygon Q,Q,Q 2,Q 3 is parameterized with three segments u,u, u,u 2 and u 2,u 3, where u i are obtained by the same rules of subdiision as Q i : Q Q Q 2 = P P P 2 T u u u 2 = t t t 2 T Q Q 2 Q 3 = P P P 2 T u u 2 u 3 = t t t 2 T The limit of the process gies us a well-parameterized cure. To erify the C continuity of the limit cure one may proceed as follows: construct a sequence of functions {f i } i= conerging pointwise to the limit function Ft. Here we start with a ector of blending functions f for the control polygon P,P,P 2. Then f is the ector of blending functions for the polygon Q,Q,Q 2,Q 3 etc. construct a sequence of deriaties {f i } i= and show that it conerges uniformly to a continuous function proe that the limit lim f i is indeed the deriatie of the cure Ft i In such a way we get sufficient conditions for differentiability, not only necessary ones. We do not want to oerload the presentation with technical questions of uniform conergence, all the proofs are detailed in a technical report [7]. The report proes that a cures is C continuous if and only if a and b are located in the Gregory region magenta zone in figure 8.

16 Tangents to fractal cures and surfaces 5 Moreoer, we hae proed that for any a and b in the conergence domain limit cures are differentiable almost eerywhere, i.e. eerywhere except on a set of measure zero [7]. As a matter of fact, this set consists of the junction point under all possible finite sequences of applications T and T. In other words, in the natural parameterization it is the point t = 2, t = 4, t = 3 4 etc all points of dyadic parameters. This set is denumerable. 6 Roughness of a cure There are few ways to describe roughness of a cure like Hölder exponent and fractal dimension. All the descriptors are good per se, but a cure may be fully described only by combining descriptors. Here we introduce a new descriptor, namely angles between half-tangent ectors T t and T t+. So we know that any cure from the Gregory-Qu domain is differentiable, but if we are not ery far from the domain; cures are not ery rough either. These almost smooth cures may be a good fit for computer graphics, where all geometry is discretized anyway. Or if one searches look for a really rough cure, where to look for it in the conergence domain? Angles between half-tangent ectors are ery easy to calculate, and therefore the search is ery efficient: αa,b = arccos < T t,t t+ > T t T t+ π,, Fig.. Left image: angle between two half-tangent ectors from red to π iolet; right image: thresholded ersion of the left one, the domain represents almost smooth cures the angle is less than 5 degrees. Left image of figure shows a graph of roughness s alues of a and b. The domain of Gregory-Qu is marked in red as half-tangent are collinear note that

17 6 Tangents to fractal cures and surfaces degenerate cases a = b, b = and = a + b are also in red. All cusp points are marked in iolet. Right image shows a thresholded ersion of the left one. Here we hae selected threshold of 5, so any cure generated with a and b from the black region is guaranteed to be almost smooth. 7 Conclusion Up to this moment we hae shown how a cure may be constructed. Constructing a surface may be done in exactly the same manner. For example, Doo-Sabin subdiision scheme may be described as a face-edge-ertex B-rep, where a patch topological face is bounded by four edges. Then each patch is subdiided into four smaller sub-patches and all them may be stitched together by implying adjacency of corresponding borders. Then it is immediate that for a regular Doo-Sabin patch there are three degrees of freedom. Either we set it to classic alues.5625,.875,.875 to get the Doo-Sabin subdiision surface, either we look for other shapes either smooth and differentiable or not. Figure 2 shows six different surfaces obtained by subdiiding a cube with different triples of weights. Fig. 2. Examples of different geometric textures obtained obtained by subdiiding a cube with different subdiision weights. In this paper we hae presented how to model cures and surfaces by means of iteratie process, namely linear BC-IFS Boundary Controled Iteratif Function System. This approach guarantees the required topology of the final shape by introducing incidence and adjacency constraints on a B-Rep model. For an linear BC-IFS it implies constraints on underlying matrices representing subdiision operators.

18 Tangents to fractal cures and surfaces 7 In this paper we hae explicitly constructed local corner cutting cures and studied the differential behaiour by analyzing eigenalues and eigenectors of the subdiision operators. While we find same necessary conditions as do Gregory and De Boor, we study a larger family of cures, since by using BC-IFS approach we are able to enrich the conergence domain, thus introducing new shapes. To characterize different families of shapes in the conergence domain we study eigenstructures of subdiision operators and propose a precise cartography of all the regions. We hae also proed that necessary conditions are also sufficient ones. Moreoer, we hae proed that stationary local corner cutting cures are differentiable almost eerywhere. Finally, we hae proposed a new roughness descriptor of fractal shapes. With this descriptor it is immediate to see where in the conergence domain we hae to look for rough or smooth cures. Indeed, een if a cure is not differentiable it may look ery smooth. References. Michael Barnsley. Fractals eerywhere. Academic Press Professional, Inc., San Diego, CA, USA, Hicham Bensoudane. Étude différentielle des formes fractales. PhD thesis, Uniersité de Bourgogne, Hicham Bensoudane, Christian Gentil, and Marc Neeu. Fractional half-tangent of a cure described by iterated function system. Journal Of Applied Functional Analysis, 42:3 326, April Ioana Boier-Martin and Denis Zorin. Differentiable parameterization of Catmull- Clark subdiision surfaces. In SGP 4: Proceedings of the 24 Eurographics/ACM SIGGRAPH symposium on Geometry processing, pages 55 64, New York, NY, USA, 24. ACM. 5. Carl De Boor. Local corner cutting and the smoothness of the limiting cure. Computer Aided Geometric Design, 75: , Wayne O. Cochran, Robert R. Lewis, and John C. Hart. The normal of a fractal surface. The Visual Computer, 74:29 28, June Christian Gentil. Les fractales en synthèse d image : le modèle IFS. PhD thesis, Uniersité LYON, John A. Gregory and Ruibin Qu. Nonuniform corner cutting. Computer Aided Geometric Design, 38: , John Hutchinson. Fractals and self-similarity. Indiana Uniersity Journal of Mathematics, 35:73 747, 98.. Andrei Khodakosky and Peter Schröder. Fine leel feature editing for subdiision surfaces. In SMA 99: Proceedings of the fifth ACM symposium on Solid modeling and applications, pages 23 2, New York, NY, USA, 999. ACM.. K.M. Kolwankar and A. D. Gangal. Hölder exponents of irregular signals and local fractional deriaties. Pramana, 48:49 68, January K.M. Kolwankar and A.D. Gangal. Local fractional deriaties and fractal functions of seeral ariables. in: Proc. of Fractals in Engineering, R. Daniel Mauldin and S. C. Williams. Hausdorff dimension in graph directed constructions. Transactions of the American Mathematical Society, 392:8 829, 988.

19 8 Tangents to fractal cures and surfaces 4. Marco Paluszny, Hartmut Prautzsch, and Martin Schäfer. A geometric look at corner cutting. Computer Aided Geometric Design, 45:42 447, Przemyslaw Prusinkiewicz and Mark S. Hammel. Language-restricted iterated function systems, koch constructions and l-systems. In New Directions for Fractal Modeling in Computer Graphics, ACM SIGGRAPH Course Notes. ACM Press, Robert Scealy. V -ariable fractals and interpolation. PhD thesis, Australian National Uniersity, Dmitry Sokolo and Christian Gentil. Sufficient conditions for differentiability of local corner cutting cures. Research report, LE2I, Uniersité de Bourgogne, Jos Stam. Exact ealuation of Catmull-Clark subdiision surfaces at arbitrary parameter alues. In Proceedings of SIGGRAPH, pages , E. Tosan, I. Bailly-Sallins, G. Gouaty, I. Stotz, P. Buser, and Y. Weinand. Une modélisation géométrique itératie basée sur les automates. In GTMG 26, Journées du Groupe de Traail en Modélisation Géométrique, Cachan, pages 55 69, Mars Chems Eddine Zair and Eric Tosan. Fractal modeling using free form techniques. Computer Graphics Forum, 53: , 996.