Adaptive and Smooth Surface Construction by Triangular A-Patches

Transcription

1 Adaptive and Smooth Surface Construction by Triangular A-Patches Guoliang Xu Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China Abstract We propose an approach to model a smooth surface from a surface triangulation by implicit surface patches which are subsets of zero contours of trivariate functions defined on a collection of irregular triangular prisms. Comparing with the earlier approaches of surface modeling by implicit surfaces, the proposed scheme uses much less patches, and can easily capture sharp features. Furthermore, the construction is adaptive in recovering the detail structures. Key words. Triangulation, implicit surface, interpolation, triangular patch. 1 Introduction Various approaches of using implicit surface representation the zero contour of trivariate function in modeling geometric objects or reconstructing the image to scattered data have been described in some papers (see for examples, [1], [4], [5], [6] and [8]). However, since the implicit surface can have multiple sheets, singularities and is not easy to evaluate, effective and easy used schemes are still under investigation. In this paper, starting from a triangulation (this is often the preprocessing stage of surface construction) of a unknown surface, we construct an implicit surface that interpolates the vertices of the triangulation. The constructed surface is G 0 at the edges that are labeled as sharp, and G 1 smooth (tangent plane continuous) elsewhere, and respects the topology of the triangulation. We assume that the surface triangulation is double sided so that we can label one side as positive and other as negative. A class of successful approaches of using implicit surface representation in interpolating a surface triangulation T with normals consists of the following two steps: (a) Build a surrounding simplicial hull (consisting of a series of tetrahedra) of the triangulation. (b) Construct a piecewise trivariate polynomial F on that hull, and use the zero contour of F to represent the surface. Dahmen [3] propose firstly an approach for constructing such a simplicial hull of T. In his approach, for each face, two tetrahedra are constructed. For each edge of T, two tetrahedra are formed that blending the neighboring face tetrahedra. For the construction of the surface over, Dahmen [3] uses six quadric patches for each face tetrahedron and four quadric patches for each edge tetrahedron. Guo[5] used a Clough-Tocher split and subdivided each face tetrahedron of the simplicial hull, hence utilizing six cubic patches per face of T and four cubic patches per edge. Dahmen and Support in part by National Innovation Fund , Chinese Academy of Sciences 1

2 Figure 1.1: Grouping the triangles by the sharp edges (thick lines) and assigning a normal for each group Thamm-Schaar [4] do not split the face tetrahedra, but the edge tetrahedra is split. All of these papers provided heuristics to overcome the multiple-sheeted and singularity problem of the implicit patches. In Bajaj Chen and Xu[1], their A-patches are guaranteed to be nonsingular and single sheeted within each tetrahedron. They use two surface patches for each face and four patches for each edge. Instead of using tetrahedra, we use, in this paper, prisms based on the idea of fat surface introduced by Barnhill, Opitz and Pottmann in [2]. The pipeline of the construction is as follows: 1. Compute the face normals and orientate them such that they point to the positive side of the surface triangulation. 2. For each edge, compute dihedral angle θ = π θ 1 for the two incident faces. If θ > α, then this edge is labeled as sharp edge. Here θ 1 is the angle between the two faces normals and α is a threshold value for controlling the sharp feature. 3. Estimate normal at each vertex of the triangulation. 4. Decimate the mesh if it is considered to be too dense. 5. For each triangle of T, construct an irregular triangular prism such that each of its edges passes through one vertex of the triangle and contain the given normal. 6. Define a trivariate function F on the union of the prisms, such that its zero contour in each prism passes through the corresponding triangle vertices and has the given normals. 7. Display the surface patches F = 0. Note that, each triangle of the triangulation corresponds to one triangular surface patch and there is no surface patch corresponding to edge and there is no splitting of the triangle. Hence the number of surface patches is significantly reduced comparing with the earlier implicit approaches. Another feature of the method is that evaluating one point of the surface is almost equivalent to solving a linear equation. Therefore, the surface can be easily and quickly displayed. Step 1 and 2 are straitforward. For the normal estimation at a vertex, we need to distinguish the cases of sharp vertex or non-sharp vertex. If there exist sharp edges incident to a vertex, then we say the vertex is sharp, otherwise, it is non-sharp. For non-sharp vertices, we use the limit normal of Loop s subdivision surface. For a sharp vertex, the triangles around the vertex are divided into some groups by the sharp edges (see Figure 1.1). For each group, we assign a single normal for 2

3 the vertex. This normal is computed by the weighted averaging Of face normals in this group. In the construction of surface patch for one triangle, there is only one normal used for one vertex of the triangle. This normal is vertex normal if the vertex non-sharp, otherwise the normal is group s normal to which the triangle belongs. When we mention the vertex normal of a triangle in the following, we always mean this normal. In the following we do not address the normal estimation problem. Instead of using geometric error based decimation scheme of a triangulation, we have developed a decimation scheme based on normal variation in order to capture the detail structures. This scheme will be reported elsewhere. Hence we omit it here. We mention this step here because it is in our implementation of the scheme and most of the examples given in the last section are produced from the decimated meshes. 2 Construction of the Triangular Surface Patches Suppose we are given a space surface triangulation T of the point set V = {v l } N l=1. For each triangle, say [v i v j v k ], we have three normals N (i) ijk, N (j) ijk, N (k) ijk for the vertices v i, v j, v k, respectively. Our aim is to construct a triangular surface patch for the triangle such that the surface patch passes through the three vertices of the triangle and has the given normal at the vertices and furthermore, the composite surface is G 1 smooth except at the sharp edges where it is G 0. The construction involves two steps: Construct a prism hull (step 5). Construction a trivariate function on the hull (step 6). 2.1 Construction of the Prism Hull For the given triangulation T, the prism hull, denoted as D, is a collection of prisms, that could be regarded as expansion of T in both its positive and negative directions. To describe how each of the triangles is expanded, we need to specify a direction at each vertex along which the triangles are extrude. At a non-sharp vertex, this direction is specified as the normal at the vertex. At a sharp vertex, say v i, there are several normals as described before. We choose the extrude direction, denoted as N i, as the average of all the face normals. Then the prism hull is build as follows. Let [v i v j v k ] be a triangle of T. Then we define a prism D ijk as D ijk := {p : p ijk (λ), λ I ijk } where ijk (λ) = {p IR 3 : p = p ijk (b 1, b 2, b 3, λ), b i 0} is a triangle for a fixed λ with p ijk (b 1, b 2, b 3, λ) = b 1 v i (λ) + b 2 v j (λ) + b 3 v k (λ), b 1 + b 2 + b 3 = 1 (2.1) and v l (λ) = v l +λn l, n l = N l / N l, l = i, j, k; and I ijk is a maximal open interval such that 0 I ijk and for any λ I ijk, the points v i (λ), v j (λ) and v k (λ) are not collinear and the three normals N i, N j and N k point to the same side of the plane P ijk (λ) = {p : p = p ijk (b 1, b 2, b 3, λ), b l IR}. We call D ijk as irregular triangular prism since the edges v l (λ), λ I ijk, l = i, j, k, are not parallel in general, and we call (b 1, b 2, b 3, λ) as D ijk coordinate of p if p = b 1 v i (λ) + b 2 v j (λ) + b 3 v k (λ). 2.2 Construction of the boundary curves, functions and gradients Let N i be the normal of vertex v i and N (i) ijk be the normals that is attached to the vertex v i of the triangle [v i v j v k ]. Then we assume the following: 3

4 Assumption 2.1. For each vertex v i, N T i N (i) ijk > 0 for any triangle [v iv j v k ] T. In order to define a function that is at least C 0 over D = i,j,k D ijk, we adjust the length of the normal N ijk, so that N T i N (i) ijk = N i 2. Let [v i v j ] be an edge of T and [v i v j v k ] and [v i v j v l ] be the adjacent triangles of the edge. Then we define N (i) ij = 1 ( N (i) ijk 2 + N (i) ) ijl, N (j) ij = 1 ( N (j) ijk 2 + N (j) ) ijl. In the following, we use the notations: F i = F Vi, F i = F Vi, v i V, F ij = F Hij, F ij = F Hij, [v i v j ] T, F ijk = F Dijk, F ijk = F Dijk, [v i v j v k ] T, where V i = {v i (λ) : λ (, )} D, H ij = {h ij (t, λ) : h ij (t, λ) = (1 t)v i (λ) + tv j (λ), t [0, 1], λ (, )} D and T denotes all the edges of T. For the construction of F ijk, there is no restriction on the interval I ijk. However, I ijk should be as large as possible, such that the surface F ijk = 0 is contained in D ijk. The problem of determining the largest I ijk is considered in section 5. In this section, we mainly construct the function F ijk. This will be done by firstly constructing function values and gradients on the edges and faces of D ijk and then using a transfinite triangle interpolation to extend the function to the interior of D ijk. On the edges V l, l = i, j, k, of D ijk, the function value is defined by (2.2) F (v l (λ)) = N l λ = n T l N (l) ijk λ, l = i, j, k, n l = N l / N l. (2.3) On the boundary face H lm of D ijk, the function value is defined by the cubic Hermite interpolation along the line segment [v l (λ)v m (λ)] = {h lm (t, λ) : t [0, 1]}. That is, the function interpolates the values F (v l (λ)), F (v m (λ)) and the derivatives [v m (λ) v l (λ)] T N (l) lm, [v m(λ) v l (λ)] T N (m) lm (2.4) at the end points v l (λ) and v m (λ), respectively, with an additional free quartic term. This leads to with F (h lm (t, λ)) = φ lm (t) + ψ lm (t)λ, t [0, 1], λ I ijk (2.5) φ lm (t) = (v m v l ) T N (l) lm t(1 t)2 + (v l v m ) T N (m) lm t2 (1 t) + θ lm t 2 (1 t) 2 (2.6) ψ lm (t) = n T l N (l) lm (1 t)3 + (n m + 2n l ) T N (l) lm t(1 t)2 + + (n l + 2n m ) T N (m) lm t2 (1 t) + n T mn (m) lm t3, (2.7) where θ lm is a free parameter that is used to control the shape of the curve: C lm (t, θ lm ) := v l + t(v m v l ) + λ(t)[n l + t(n m n l )] (2.8) 4

5 which will be the boundary of our constructed surface patch in D ijk, where λ(t) = φ lm(t) ψ lm (t) = λ 1(t)θ lm + λ 0 (t) satisfies the equation F (h lm (t, λ)) = 0. For any θ lm, curve C lm (t, θ lm ) passes through the vertices v l and v m and perpendicular to the normal n l and n m at the vertices. When θ lm increase (or decrease), the boundary curve C lm (t, θ lm ) goes away from the line segment [v l v m ] in the normal (or opposite normal) direction. The default choice of this parameter could be zero or to make some energy to be minimal. For example, 1 0 C lm(t, θ lm ) dt = min or 1 where k(t, θ lm ) is the curvature of C lm (t, θ lm ), or 0 k(t, θ lm )dt = min, max k(t, θ lm) = min. t [0,1] (2.9) To avoid producing bumpy surface, we prefer to use (2.9). In order to define the gradients on the boundary faces H lm, we take three different directions in IR 3 as follows: d 1 = v m (λ) v l (λ), d 2 = (1 t)n l + tn m, d 3 = d 1 d 2. (2.10) It is easy to see that D d1 F (h lm (t, λ)) = F (h lm(t, λ)) t D d2 F (h lm (t, λ)) = F (h lm(t, λ)) λ We artificially define = φ lm(t) + ψ lm(t)λ, (2.11) = ψ lm (t). (2.12) D d3 F (h lm (t, λ)) = (1 t)n (l) (m) ijk + tn ijk. From the equations we have Since d T i F lm = D di F lm, i = 1, 2, 3, F lm = { [d 1, d 2, d 3 ] T } 1 [Dd1 F lm, D d2 F lm, D d3 F lm ] T. [d 1, d 2, d 3 ] T [d 1 + αd 2, d 2 + βd 1, d 3 ] = d αd T 1 d d βd T 1 d d 3 2 where α = d T 1 d 2/ d 2 2, β = d T 1 d 2/ d 1 2, and d 1 2 d 2 2 (d T 1 d 2) 2 = d 3 2, we have {[d } 1, d 2, d 3 ] T 1 1 [ ] = d 1 2 d 2 2 (d T 1 d 2) 2 d 1 d 2 2 d 2 (d T 1 d 2 ), d 2 d 1 2 d 1 (d T 1 d 2 ), d 3. 5,

6 Hence F lm = [d 1 d 2 2 d 2 (d T 1 d 2)]D d1 F lm + [d 2 d 1 2 d 1 (d T 1 d 2)]D d2 F lm + d 3 D d3 F lm d 1 2 d 2 2 (d T 1 d 2) 2 (2.13) which could be written as F (h lm (t, λ)) = P 5(t) + Q 5 (t)λ + R 5 (t)λ 2 p 2 (t) + q 1 (t)λ + r 0 (t)λ 2, where P 5, Q 5, R 5 are polynomials of degree 5 and p 2, q 1, r 0 are polynomials of degree 2, 1, 0, respectively. 2.3 Construction of the function in the interior Having the function values and gradients on the boundary of D ijk, we can apply any transfinite triangular interpolant over the triangle ijk (λ) to construct the function F ijk. In this paper, we use the side-vertex scheme defined by Theorem 3.1 in [7] with some variations. The implementations show that the direct application of the scheme (3.9) in [7] leads to bad shaped surfaces. Hence we alter the equation (3.6) in [7] by introducing an additional term. The following is the modified scheme for a typical triangle [v 1 v 2 v 3 ]: F (p 123 (b 1, b 2, b 3, λ)) = 3i=1 3j=1,j i b 2 j D i(b 1, b 2, b 3, λ) 3i=1 3j=1,j i b 2 + E(b 1, b 2, b 3, λ), (2.14) j where D i (b 1, b 2, b 3, λ) are defined by interpolating function values and derivative at v i (λ) and b j 1 b i v j (λ) + b k 1 b i v k (λ): D i (b 1, b 2, b 3, λ) = (1 + 2b i )(1 b i ) 2 F (h jk (S i, λ)) b i (1 b i )[b j e T k (λ) + b ke T j (λ)] F (h jk(s i, λ)) + b 2 i (3 2b i)n T i n iλ + b 2 i [b je T k (λ) + b ke T j (λ)]n i + b 2 i (1 b i)(b j θ ij + b k θ ki ), (2.15) where (i, j, k) {(1, 2, 3), (2, 3, 1), (3, 1, 2)}, e k (λ) = v j (λ) v i (λ), e j (λ) = v k (λ) v i (λ), S i = E(b 1, b 2, b 3, λ) = b 2 1b 2 2b 2 3 l 1 +l 2 +l 3 =l (c l1 l 2 l 3 + λw l1 l 2 l 3 )B l l 1 l 2 l 3 (b 1, b 2, b 3 ) b k b j +b k. is free. That is, E(b 1, b 2, b 3, λ) has no influence on the function value and the first order partials of F at the boundary of D ijk since it has the factor b 2 1 b2 2 b2 3. This term can be used to fit data in the volume D ijk to get a better approximation. If there is no data available, E(b 1, b 2, b 3, λ) could be taken as zero. The function Bl l 1 l 2 l 3 (b 1, b 2, b 3 ) in E(b 1, b 2, b 3, λ) is the well-known Bernstein-Bézier polynomial on a triangle. Theorem 2.1. The function F defined by F Dijk = F ijk is C 1 over ijk D ijk everywhere except at the sharp edges where it is C 0 and it interpolates the function values N T n l λ at v l (λ), l = 1,, N. At non-sharp vertex v l (λ), it interpolates also the gradient N l. Proof. It is easy to see that the function F ijk is C 1 within D ijk. Hence we need only to consider the continuity of F on the face H lm. First we show that it is C 0. 6

7 On the edge V l the function value is uniquely defined by (2.3), hence the function is continuous there. On the face H lm, the function values defined by (2.5) use the values on the edge and derivatives defined by (2.4). These values are edge dependent. That is, For the two triangles that share the edge, the function values for the two function at the common face H lm are the same. Hence F is continuous. Now we show that the function is C 1 on the non-sharp edges. Since the constructions of boundary function F lm and the transfinite interpolation (2.14) have some variations from that of the referred papers [2] and [7] respectively, the validity of the theorem need to be verified. We assume function F and its gradient F are well defined on the union of D ijk (see Theorem 4.2 for the validity of this assumption). Let b 2 j b2 k W i = b 2 2 b2 3 + b2 1 b2 3 +, i = 1, 2, 3, i j k i. b2 1 b2 2 Then W i have the properties (see [7]): 3 W i = 1, W i bj =0 = θ ij, W i bj =0 = 0, i=1 where represents any first order differentiation. Consider a typical triangle [v 1 v 2 v 3 ]. We have F 123 bi =0,b j b k >0 = 3 [W l D l (b 1, b 2, b 3, λ)] bi =0,b j b k >0 l=1 = D i (b 1, b 2, b 3, λ) bi =0,b j b k >0 = F (h jk (b k, λ)), (2.16) where (i, j, k) {(1, 2, 3), (2, 3, 1), (3, 1, 2)}. That is, F is continuous on the face H jk. Now consider the gradient of F, F 123 bi =0,b j b k >0 = 3 [ W l D l (b 1, b 2, b 3, λ)] bi =0,b j b k >0 l=1 + W l D l (b 1, b 2, b 3, λ) bi =0,b j b k >0 = D i (b 1, b 2, b 3, λ) bi =0,b j b k >0. We now assume i = 1 and compute B i (b 1, b 2, b 3, λ) bi =0,b j b k >0. From (2.15) and using the relation (2.11) and (2.12), we can obtain the following relation [ D1, D 1, D ] T v 1 (λ) T v 3 (λ) T 1 b 1 b 2 λ = v 2 (λ) T v 3 (λ) T F (h 23 (b3, λ)), b1 =0,b 2 b 3 >0 b 2 n T j + b 3n T k where D 1 (b 1, b 2, λ) = D 1 (b 1, b 2, 1 b 1 b 2, λ). Then by (3.2) we have F 123 b1 =0,b 2 b 3 >0 = D 1 b1 =0,b 2 b 3 >0 = F (h 23 (b3, λ)). (2.17) Since two functions F 123 and F 23l defined on the two adjacent prisms D 123 and D 23l have the same gradients F 23 on the common face H 23, the two function is C 1 there. From the definition of F 123 and the properties of W i, the function F 123 is C 1 in the prism D 123, hence the function values (2.16) and gradients (2.17) could be extended to the boundaries V 2 and V 3 of the face H 23. Since for b 3 = 0 and b 3 = 1, the values of function (2.16) are N2 T n 2λ and N3 T n 3λ, and the gradients (2.17) are N 2 and N 3, respectively, the function F is C 1 there. 7

8 3 Evaluation of the Surface Patch Let [v i v j v k ] be any triangle of T. For each (b 1, b 2, b 3 ), b i 0, λ min (b 1, b 2, b 3 ) such that bi = 1, determine λ min = λ min = min{λ : F (p ijk (b 1, b 2, b 3, λ)) = 0}. (3.1) Then the surface point is defined by p = p ijk (b 1, b 2, b 3, λ min ). The main task here is to compute λ min for each (b 1, b 2, b 3 ) with b i 0. It follows from (2.15) that D i (b 1, b 2, b 3, λ) is a rational function of λ in the form F 0 + F 1 λ + N 0 + N 1 λ + N 2 λ 2 D 0 + D 1 λ + D 2 λ 2. Hence φ(λ) := F (p ijk (b 1, b 2, b 3, λ)) is a rational function in λ. The minimal zero in absolute value of φ(λ) is the required λ min. Although φ(λ) = 0 is a nonlinear equation, the computation shows that φ(λ) is nearly a linear function. Hence, starting from λ = 0, finding an interval with opposite signs of φ(λ) by searching, then using linear interpolation, an approximate minimal zero of φ(λ) is obtained by one or two Newton iterations. Let F ijk (b 1, b 2, λ) = F (p ijk (b 1, b 2, 1 b 1 b 2, λ)). Then the surface normal is computed by the following equation F ijk = {[v } [ 1 (λ) v 3 (λ), v 2 (λ) v 3 (λ), b 1 n i + b 2 n j + b 3 n k ] T 1 Fijk, b 1 F ijk b 2, F ] T ijk. (3.2) λ It should be noted that the 3 3 matrix in (3.2) is nonsingular if the points v 1 (λ), v 2 (λ) and v 3 (λ) are not collinear and the vector b 1 n i + b 2 n j + b 3 n k is not parallel to the plane P ijk (λ). 4 Error Computation After the decimation step of our algorithm (step 4). The points that are not the vertices of the decimated mesh are grouped into the volume D ijk for the triangle [v i v j v k ]. The surface patch for this triangle provide an approximation to these points. Now we consider the computation of the approximation error. Let p (l) ijk, l = 1, 2,, m ijk be the initial input points that are in the volume D ijk. The error bound is computed by the following steps: a. For each point p (l) ijk, compute the volume coordinate (b(l) 1, b(l) 2, b(l) 3, λ(l) ). b. For (b (l) 1, b(l) 2, b(l) 3 ) compute λ min(b (l) 1, b(l) 2, b(l) 3 ) as the (3.1) Then error of the point p (l) ijk to the surface is bounded by λ(l) λ min (b (l) 1, b(l) 2, b(l) 3 ), since p (l) ijk p ijk(b (l) 1, b(l) 2, b(l) 3, λ min(b (l) 1, b(l) +b (l) 2 v j(λ (l) ) v j (λ min (b (l) 1, b(l) 2, b(l) 3 2, b(l) 3 ) b(l) 1 v i(λ (l) ) v i (λ min (b (l) ) + b(l) 3 v k(λ (l) ) v k (λ min (b (l) λ (l) ) λ min (b (l) 1, b(l) 2, b(l) 3 ) (b(l) 1 n 1 + b (l) 2 n 2 + b (l) 3 n 3 = λ (l) λ min (b (l) 1, b(l) 2, b(l) 3 ). 1, b(l) 2, b(l) 1, b(l) 2, b(l) 3 ) 3 ) 8

9 5 The Condition of the Triangulation In this section, we will give conditions on the triangulation T under which the function F and its gradient F are well defined. Definition 4.1 (see definition 3.1 of [2]). A triangle ijk (λ) is called non-degenerate, if its vertices are not collinear, the normal vectors n i, n j, n k are not parallel to the plane P ijk (λ) and point to the same side of this plane. Theorem 4.1 (see Theorem 4 of [2]). Let [v i v j v k ] be a non-degenerate triangle of T with respect to the normal n i, n j, n k. Consider the real numbers λ 1,, λ s (s 6) that solve one of the following three quadratic equations: det(n l, v j (λ) v i (λ), v k (λ) v i (λ)) = 0, l = i, j, k and define a := max(, {λ i : λ i < 0}), b := min(, {λ i : λ i > 0}). Then I ijk = (a, b). Theorem 4.2. Let [v i v j v k ] be a non-degenerate triangle of T. Then both the function F ijk and its gradient F ijk are well defined on D ijk. Proof. Let λ I ijk. Since n i, n j, n k are not parallel to the plane P ijk (λ) and point to the same side of the plane det(n l, v j (λ) v i (λ), v k (λ) v i (λ)) 0, l = i, j, k and have the same sign. Hence for any b i 0, b j 0, b k 0, b i + b j + b k = 1 b l det (n l, v j (λ) v i (λ), v k (λ) v i (λ)) = det b l n l, v j (λ) v i (λ), v k (λ) v i (λ) 0. l=i,j,k l=i,j,k That is, the vector l=i,j,k b ln l is not parallel to the plane P ijk (λ). Hence the 3 3 matrix in (3.2) F ijk b 2, F ijk λ is nonsingular. This implies that the gradient F ijk is well defined if F ijk b 1, are well defined. This is true if F lm, (l, m) {(i, j), (j, k), (k, i)} are well defined. Hence we need to verify that the denominator of F lm is positive. Since the vector l=i,j,k b ln l is not parallel to the boundary of the triangle ijk (λ). Hence, the non-zero vectors d 1 and d 2 defined in (2.6) and (2.7) are not parallel. Therefore, the denominator of F lm, which is d 1 2 d 2 2 (d T 1 d 2) 2, is positive. 6 Conclusions and Examples In the study of using implicit surface patches for modeling a surface triangulation, we have been constantly seeking for the approaches of using one triangular patch for each face of the triangulation. The technique presented in this paper has ours wish fulfilled. Comparing with the earlier scheme of A-patch in [1, 3, 4, 5], the present scheme uses much less number of patches. For example, from Euler formula v + f e = 2 for a triangulation of a closed surface with genus zero, we have f = 2v 4, e = 3v 6, where v, e and f represent the the numbers of vertices, edges and faces, respectively. Since the early approaches use two or four pieces of surface patch for each edge, and one or two pieces of patch for each face. Hence the sum of the patch number is at least 8v 16 9

10 and as large as 2(8v 16). Therefore, the ratio of patch numbers is in the range [4 : 1, 8 : 1]. Furthermore, the proposed approach has the following features: 1. It is adaptive. 2. Could model sharp feature. 3. Error is easy to compute. The implementation of the paper shows another attractive feature of the approach. That is, the evaluation of the surface is quit easy. It is approximately to solve a linear equation for evaluating one surface point. More importantly, the constructed surfaces have better shape than previous A-patches. Examples to show the proposed scheme could model smooth surfaces with adaptive nature, sharp feature and detail structures are given in Fig 6.1, 6.2 and 6.3, respectively, where the left columns are the input decimated surface triangulation, the right columns are the constructed surface models. References [1] C. Bajaj, J. Chen, and G. Xu. Modeling with Cubic A-Patches. ACM Transactions on Graphics, 14(2): , [2] R. E. Barnhill, K. Opitz, and H. Pottmann. Fat surfaces: a trivariate approach to triangle-based interpolation on surfaces. Computer Aided Geometric Design, 9: , [3] W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages Academic Press, Boston, [4] W. Dahmen and T-M. Thamm-Schaar. Cubicoids: modeling and visualization. Computer Aided Geometric Design, 10:89 108, [5] B. Guo. Modeling Arbitrary Smooth Objects with Algebraic Surfaces. PhD thesis, Computer Science, Cornell University, [6] S. Lodha. Surface Approximation with Low Degree Patches with Multiple Representations. PhD thesis, Computer Science, Rice University, [7] G. M. Nielson. The side-vertex method for interpolation in triangles. J. Approx. Theory, 25: , [8] T.W. Sederberg. Piecewise algebraic surface patches. Computer Aided Geometric Design, 2:53 59,

11 Figure 6.1: Adaptive feature of the reconstruction: The flat parts use less patches than the curved parts 11

12 Figure 6.2: Capture sharp features of the objects 12

13 Figure 6.3: Capture the detail structures 13