The Delaunay Triangulation for Multidimensional ... - Semantic Scholar
c Allerton Press, Inc., 2012. ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2012, Vol. 56, No. 1, pp. 27–34. c V.A. Klyachin and A.A. Shirokii, 2012, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2012, No. 1, pp. 31–39. Original Russian Text
The Delaunay Triangulation for Multidimensional Surfaces and its Approximative Properties V. A. Klyachin* and A. A. Shirokii** Volgograd State University, Universitetskii pr. 100, Volgograd, 400062 Russia Received December 23, 2010
Abstract—We define the Delaunay triangulation for surfaces and prove an analog of the G. Voronoi empty sphere theorem. We also prove a convergence theorem for gradients of piecewise linear approximations constructed on the Delaunay triangulation for functions differentiable on smooth surfaces. DOI: 10.3103/S1066369X12010045 Keywords and phrases: simplex, triangulation, approximation of gradient.
1. INTRODUCTION The use of triangulation in computation and approximation problems is rather wide. See papers and references therein for some results and applications. The Delaunay triangulation is used most often, because the obtained triangles often tend to equilateral ones [5]. This property can be illustrated by the classical Schwarz example ([7], P. 191) of a piecewise linear approximation of the lateral surface of a straight circular cylinder. In this example the areas of the mentioned approximations tend to that of the cylinder, if the location of chosen points on the cylinder satisfies a certain condition. However, in our opinion, the key role here is played by the way of triangulation of the lateral surface of the cylinder, rather than by the mentioned condition. Namely, according to results obtained in this paper, the desired convergence of areas takes place, if the triangulation is neither arbitrary nor specified in the mentioned example, but the Delaunay one. Moreover, we have succeeded in obtaining qualitative characteristics, namely, approximation errors. Let us formulate the obtained results more precisely. Let D ⊂ Rn be some domain. For a function f from the class C 1 (D) we denote by ∇f (x) the gradient vector at the point x ∈ D, and for any vector ξ we do by ∂f ∂ξ = ∇f (x), ξ the derivative of f at x in the direction of the vector ξ. For a collection of points p0 , p1 , . . . , pk , 1 ≤ k ≤ n, in Rn we denote by S = S(p0 , . . . , pk ) the k-dimensional simplex with vertices at these points. We assume that vectors p1 −p0 , p2 −p0 , . . . , pk −p0 are linearly independent. Deleting the point pi , i = 0, . . . , k, from this collection, we can construct the (k − 1)-dimensional simplex; we denote it by Si . This simplex is a (k − 1)-dimensional face of the simplex S(p0 , . . . , pk ). Denote by ηi the unit vector of the outer normal to the face Si that lies in the plane of the simplex S(p0 , . . . , pk ). With respect to the simplex S an arbitrary vector X ∈ Rn can be represented as the sum X = N X + X T , where X T is the orthogonal projection of the vector X onto the plane of the simplex S, and X N is its orthogonal supplement. We also introduce the projection mapping onto the plane of the face of the simplex by setting πi (X) = X T − X, ηi ηi . Consider a simplex S ⊂ D. For f ∈ C 1 (D) we construct an affine function Lf (x) = A, x + a0 such that f (pi ) = Lf (pi ), i = 0, . . . , k, and AN = 0. One can easily prove the following assertion. * **
E-mail: [email protected]. E-mail: [email protected].
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Lemma 1. The affine function indicated above exists and is defined uniquely. For f ∈ C 1 (D) we introduce the following denotations: ωij = ∇f (pi ) − ∇f (pj ), βij =
|f (pi ) − f (pj ) − ∇f (pj ), pi − pj | , |pi − pj |
γij = f (pi ) − f (pj ) − ∇f (pj ), pi − pj ,
i, j = 0, . . . , k.
Lemma 2. For an arbitrary vector Y ∈ Rn and a pair of indices i = m, i, m = 0, . . . , k, the following inequality is valid: |Y, pi − pm | |pi − pm | + |πi (Y )| . (1) |Y, ηi | ≤ |ηi , pi − pm | |pi − pm | Proof. Since Y = Y N + Y T , taking into account that Y T = πi (Y ) + Y, ηi ηi , we get Y, pi − pm = Y, ηi ηi , pi − pm + πi (Y ), pi − pm . Therefore, correlation (1) takes place. Theorem 1. Let a simplex S(p0 , p1 , . . . , pk ) belong to D, k > 1, and let a function f (x) be differentiable at vertices of this simplex. Then for any number i = 0, . . . , k and any pair of numbers j, m = 0, . . . , k different from i the following inequality is valid: ∂f (pm ) ∂Lf (pm ) 1 ≤ βim + |πi (ωjm )| + |πi (∇f (pj ) − ∇Lf (pj ))| ; ∂ηi − ∂ηi sin θim here θim is the angle between the vector pi − pm and the plane of the face Si . If k = 1, then ∂f (p0 ) ∂Lf (p0 ) ≤ β10 , ∂η − ∂η where η = (p1 − p0 )/|p1 − p0 |. Proof. In Lemma 2 we set Y = ∇f (pm ) − ∇Lf (pm ). Then ∂f (pm ) ∂Lf (pm ) ∂ηi − ∂ηi |∇Lf (pm ) − ∇f (pm ), pi − pm | |pi − pm | + |πi (∇f (pm ) − ∇Lf (pm ))| . ≤ |pi − pm , ηi | |pi − pm | Since 1 |pi − pm | = , |pi − pm , ηi | sin θim
|∇Lf (pm ) − ∇f (pm ), pi − pm | = βim , |pi − pm |
and |πi (∇f (pm ) − ∇Lf (pm ))| ≤ |πi (∇f (pm ) − ∇f (pj ))| + |πi (∇f (pj ) − ∇Lf (pj ))| = |πi (ωjm )| + |πi (∇f (pj ) − ∇Lf (pj ))|, we obtain the desired assertion. In the case k = 1 the desired inequality immediately follows from the definition of β10 and the construction of the function Lf (x). Lemma 3. Assume that Π is some k-dimensional plane that forms with the plane of the simplex the angle 0 ≤ α < π. Let the symbol (v)Π stand for the orthogonal projection of the vector v onto this plane, and let (v)⊥ do for the corresponding orthogonal supplement. Then |(∇f (pm ) − ∇Lf (pm ))Π | ≤ |(∇f (pm ) − ∇Lf (pm ))T | + |∇f (pm )| sin α. RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 56 No. 1 2012
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Proof. Since the angle between planes equals the angle between their orthogonal supplements, any vector Y satisfies the inequality |(Y N )⊥ | ≥ cos α|Y N |, whence |(Y N )Π | ≤ sin α|Y N | ≤ sin α|Y |. Taking into account that |Y Π | ≤ |(Y N )Π | + |(Y T )Π |, we obtain |Y Π | ≤ |Y T | + sin α|Y |. Setting Y = ∇f (pm ) − ∇Lf (pm ) and using the fact that by construction [∇Lf (pm )]N = 0, we obtain the desired assertion. Remark 1. If Π is a plane tangent to some smooth surface at the point pm , then the proved inequality and Theorem 1 allow us to estimate the error of the approximation of the gradient of the function in the metric of this surface. Lemma 4. Assume that a function f belongs to C 1 (D) and a function ω(t) is the modulus of continuity of its gradient, i.e., any pair of points x , x ∈ D satisfies the inequality |∇f (x ) − ∇f (x )| ≤ ω(|x − x |). Then |ωij | ≤ ω(|pi − pj |), |pi −pj | 1 ω(t)dt. βij ≤ |pi − pj | 0 Proof. The first inequality is evident. We can obtain the second one by integrating the equality ∇f (x) = ∇f (pj ) + H(x, pj ), where by condition of the lemma |H(x, pj )| ≤ ω(|x − pj |), along the segment connecting points pi and pj . In the terminology of Theorem 1 we have ∂f (pm ) ∂Lf (pm ) T + |πi (∇f (pm ) − ∇Lf (pm ))|. − | ∇f (pm ) − ∇Lf (pm ) | ≤ ∂ηi ∂ηi We get the bound |pi −pm | 1 1 ω(t)dt |(∇f (pm ) − ∇Lf (pm )) | ≤ sin θim |pi − pm | 0 1 (1 + sin θim )(ω(|pm − pj |) + |πi (∇f (pj ) − ∇Lf (pj ))|) . (2) + sin θim T
The following corollaries of Theorem 1 are obtained immediately from bound (2) with the help of Lemmas 3 and 4. Corollary 1. Let k = 2. If m = 0, i = 1, and j = 2, while ϕS and dS denote values of the angle at the vertex p0 and the length of the maximal side of the triangle S, respectively, then dS 1 3 T ω(t)dt + 2ω(dS ) . |(∇f (p0 ) − ∇Lf (p0 )) | ≤ sin ϕS dS 0 Corollary 2. Let k = 2 and let vertices of the triangle S lie on a two-dimensional smooth surface F ⊂ D ⊂ Rn . Assume that the angle between the plane of this triangle and the tangent plane to the surface at the point p0 equals α, 0 ≤ α ≤ π/2. If a function f belongs to C 1 (D), then dS 1 3 ω(t)dt + 2ω(dS ) + sin α sup |∇f (x)|. |∇f (p0 ) − ∇Lf (p0 )| ≤ sin ϕS dS 0 x∈D Here the symbol ∇ denotes the gradient of a differentiable function in the metric of the surface F . Corollary 3. Let k = 3 and m = 0, i = 2, j = 1. Let the angle between the vector p2 − p0 and the two-dimensional face opposite to the vertex p2 equal θS , and let the angle at the vertex p1 of the triangle p0 p1 p3 equal ϕS . Denote by dS the length of the maximal edge of the simplex S. Then the following bound is valid: dS T 8 1 ω(t)dt + ω(dS ) . | ∇f (p0 ) − ∇Lf (p0 ) | ≤ sin ϕS sin θS dS 0 RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 56 No. 1 2012
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Corollary 4. Let k = 3. Assume that vertices of the tetrahedron S lie on a three-dimensional smooth surface F ⊂ D ⊂ Rn . Assume that the angle between the plane of the tetrahedron and the plane tangent to the surface at the point p0 equals α, 0 ≤ α ≤ π/2. If a function f belongs to C 1 (D), then dS 8 1 ω(t)dt + ω(dS ) + sin α sup |∇f (x)|. |∇f (p0 ) − ∇Lf (p0 )| ≤ sin ϕS sin θS dS 0 x∈D Here, as above, the symbol ∇ denotes the gradient of a differentiable function in the metric of the surface F . Note that analogous methods were applied in [8] for formulating conditions of the C 1 -convergence of the piecewise linear approximations of level surfaces of smooth functions. 2. THE DELAUNAY TRIANGULATION OF THE GRAPH OF A SMOOTH FUNCTION Consider an n-dimensional surface F in the space Rn+1 representing the graph of a function xn+1 = f (x1 , . . . , xn ), x ∈ D ⊂ Rn . Define the projection mapping π : F → D, π(x1 , . . . , xn+1 ) = (x1 , . . . , xn ). Let Pi , i = 1, . . . , N , be some set of points Pi ∈ Rn+1 that lie on the surface F such that at vertices of this set any n-dimensional simplex is nondegenerate. Denote by Pi = π(Pi ) the projection of the point Pi onto the domain D. We consider the triangulation of the surface F by n-dimensional simplexes whose vertices belong to the set Pi . We understand a triangulation of the surface as a set T of ndimensional simplexes such that 1) each point Pi of the given set is a vertex of one of simplexes S ∈ T ; 2) each vertex of any simplex S ∈ T is one of points Pi , i = 1, . . . , N ; 3) the interior of the intersection of any two simplexes is empty; 4) the projection of the system of simplexes Sj ∈ T , j = 1, . . . , m, is the usual triangulation of the set of points Pi in Rn . We say that a triangulation is acute-angle, if for each simplex its angles between any two faces are acute. Definition 1. Let S be some n-dimensional simplex in Rn+1 . We understand a circumscribed ball B(S) for S as an (n + 1)-dimensional ball of the least radius that contains vertices of the simplex. Definition 2. We say that a triangulation of a surface F satisfies the Delaunay condition, if for any simplex of the triangulation the interior of the circumscribed ball contains no triangulation vertices. Definition 3. We say that for two n-dimensional simplexes with a common (n − 1)-dimensional face the ball emptiness condition is fulfilled, if the interior of the circumscribed ball of some simplex contains no vertices of another one. From the point of view of the construction of optimal algorithms for triangulating surfaces, as well as the approximation of the gradient in the metric of a surface, the Delaunay triangulation has one important property. Namely, if the Delaunay condition is fulfilled locally, then it is also fulfilled globally [9–11]. The next theorem generalizes this property for the case of triangulation of a smooth surface representing the graph of a function. But as distinct from the plane case, it is necessary that an additional condition should be fulfilled. Below we illustrate this fact with an example. Condition α. Consider an arbitrary (n − 1)-dimensional simplex S of a triangulation. Construct the intersection US of all halfspaces that contain S which are defined by hyperplanes of (n − 1)-dimensional spheres that represent intersections of circumscribed spheres of the simplex S and the neighboring one. The number of such halfspaces does not exceed n for each simplex S. We assume that no vertex of the triangulation belongs to the interior of US for any S. Note that in the plane case, when triangulation vertices lie in one hyperplane, condition α, evidently, is fulfilled. RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 56 No. 1 2012
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Theorem 2. Let a surface F be defined over a convex domain D ⊂ Rn . A triangulation of this surface, for which condition α is fulfilled and any pair of simplexes with a common (n − 1)dimensional face satisfies the ball emptiness condition, is the Delaunay triangulation. Proof. Consider some triangulation of a set of points Pi with the mentioned in the theorem properties. For two neighboring simplexes S and S of this triangulation we construct a hyperplane Π going through the intersection of hyperspheres ∂B(S ) ∩ ∂B(S ). Consider an arbitrary simplex S1 of the given triangulation and some vertex A of an arbitrary simplex of this triangulation different from S1 . Consider the ray A1 A, where A1 ∈ S1 is some interior point of the simplex S1 . Assume that the theorem is not true, i.e., A ∈ B(S). In view of condition α we have A ∈ / US1 , therefore the ray A1 A first intersects one of planes Π constructed above that goes through some face of the simplex S1 . Denote this plane by + Π1 . It also contains one of faces of the neighboring simplex S2 . Denote by Π− 1 and Π1 halfspaces defined − + by the hyperplane Π1 , let A1 ∈ Π1 . Then, evidently, A ∈ Π1 . Note that in view of properties of balls one of the following inclusions is valid: + B(S1 ) ∩ Π+ 1 ⊂ B(S2 ) ∩ Π1
or + B(S2 ) ∩ Π+ 1 ⊂ B(S1 ) ∩ Π1 .
Assume that a vertex C of the simplex S2 does not belong to the plane Π1 . Since by assumption / B(S1 ), but, on the other hand, for the pair of simplexes S1 and S2 the ball is empty, we have C ∈ + + C ∈ B(S2 ) ∩ Π1 . Therefore, the inclusion B(S1 ) ∩ Π1 ⊂ B(S2 ) ∩ Π+ 1 is valid, because otherwise we + ⊂ B(S ) ∩ Π ⊂ B(S ). get a contradiction, namely, C ∈ B(S2 ) ∩ Π+ 1 1 Due to the above inclusion we 1 1 have A ∈ B(S2 ). Let us now construct a sequence of simplexes S1 , S2 , . . . , Sk such that A is a vertex of the simplex Ak . Then simplexes Si and Si+1 are neighboring. Let Πi be a hyperplane containing the (n − 1)-dimensional sphere ∂B(Si ) ∩ ∂B(Si+1 ). To this end we construct a two-dimensional vertical plane containing the ray A1 A. As a result we obtain a polyline connecting points A1 and A, whose segment lies in the corresponding simplexes S1 , S2 , . . . , Sk . The segment of this polyline that lies in S2 intersects the (n − 1)-dimensional face, which is common for simplexes S1 and S2 , and some other face S2 , which is common with the simplex S3 . In view of condition α this segment contains a point A2 , for which the ray A2 A intersects the plane Π2 . Repeating this reasoning, we construct sequences of simplexes Si , hyperplanes Πi , and points Ai such that + Ai ∈ Si , Si ⊂ Π− i , Si+1 ⊂ Πi , A ∈ B(Si ).
By construction, A ∈ B(Sk−1 ), which contradicts the ball emptiness condition for the pair of simplexes / B(S1 ). Since S1 and A are chosen arbitrarily, the theorem is proved. Sk−1 and Sk . Therefore, A ∈ Example. Note that condition α is necessary. The following example was proposed by V. V. Popov. In R3 there exist points A(104, 0, 0), B(134, 40, 120), C(0, 78, 0), D(0, 0, 0), and E(52, −20, 48) with the following properties: a) the ball circumscribed around the triangle ABC does not contain the point D, but contains the point E; b) the ball circumscribed around the triangle ACD does not contain points B and E; c) the ball circumscribed around the triangle ADE does not contain points C and B. Let F be the surface composed of the mentioned triangles. These triangles form a triangulation of F such that any two neighboring simplexes (i.e., those with a common edge) satisfy the ball emptiness condition, and the Delaunay condition is violated. On the other hand, note that the plane containing the intersection of spheres of balls circumscribed around triangles ABC and ACD obeys the equation 3x + 4y + 12z = 312. Then the set US for S = ABC represents the halfspace 3x + 4y + 12z > 312. One can immediately make sure that E ∈ US , therefore, condition α is not fulfilled.
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3. APPROXIMATION PROPERTIES OF THE DELAUNAY TRIANGULATION Let D ⊂ Rn , n > 1, be a domain containing a fixed sequence {Pm } of finite sets of points. For each of these sets we consider its triangulation Tm . We say that a triangulation is acute-angle, if for each simplex its angles between any two faces are acute. A triangulation of a set of points is called the Delaunay triangulation, if for any simplex of this triangulation the interior of the circumscribed ball contains no point from this set. For a simplex S ∈ Tm we denote by dS the length of its maximal edge. We set dm = max dS . S∈Tm
Consider sets of points Pm and their triangulations Tm such that dm → 0 as m → ∞, ∀ε > 0, ∀x ∈ D ∃m0 ∈ N : ∀m > m0 ∃a ∈ Pm : |a − x| < ε.
(3) (4)
The second condition means that Pm is the ε-mesh with all sufficiently great m. Consider some function f (x), x ∈ D, from the class C 1 (D). For any natural m we construct a piecewise affine function fm (x) such that fm (a) = f (a) for any point a ∈ Pm . One can easily prove that under conditions (3) and (4) the sequence fm (x) uniformly converges to the function f (x) on each compact subset K ⊂ D. But these conditions do not guarantee the convergence of derivatives of functions fm (x) to those of functions f (x) (even almost everywhere or in Sobolev spaces 1,p (D)). This fact is proved by the well-known Schwarz example ([7], P. 191). Wloc Below we assume that a function f : D → R is C 2 -smooth in a domain D ⊂ Rn , and all its second derivatives are bounded in D. Then there exists a constant M > 0 such that the modulus of continuity of the gradient ω(t) ≤ M · t. Our goal is to prove that the Delaunay triangulation and those close to it have (in a sense) the mentioned above approximation property. Let us first consider a two-dimensional case. For an arbitrary triangle S with vertices p0 , p1 , and p2 we introduce the value μS = sindSϕS , where ϕS is the value of the maximal acute angle of the triangle. Lemma 5. Let RS denote the radius of the circumference circumscribed around the triangle S. Then μS ≤ 4RS . Proof. Consider two cases. 1) The triangle is acute-angled. Then ϕS is the angle opposite to the longest side of length dS . According to the sine theorem, sindSϕS = 2RS . 2) The triangle is obtuse. Denote by ϕd the angle opposite to the longest side. Evidently, ϕd ≥ π/2 and ϕd + 2ϕS ≥ π, whence | sin ϕd | ≤ | sin 2ϕS |. Consequently, 2dS cos ϕS 2dS dS = ≤ = 4RS . sin ϕS sin 2ϕS sin ϕd
Theorem 3. Consider a finite set of points representing an ε-mesh in a plane domain D ⊂ R2 and assume that the Delaunay triangulation T for this set is given. Assume also that a point p is a vertex of the triangulation T such that for some triangle from T with a vertex at this point the center of the circumscribed circle lies in the domain D. Then |∇f (p) − ∇Lf (p)| ≤ 14M ε.
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Proof. Note that if conditions of the theorem are fulfilled, then the radius of the circle circumscribed around the considered triangle does not exceed ε. Really, assume the contrary, i.e., the radius of the circumscribed circle exceeds ε. Then in view of (4) there exists a vertex p of the triangulation (different from vertices of the mentioned triangle) such that the distance from it to the center of the circle circumscribed around the triangle does not exceed ε. But this contradicts the definition of the Delaunay triangulation. Therefore, taking into account the inequality in Corollary 1, where ω(t) ≤ M · t, we obtain the desired assertion. Remark 2. Note that in [6] one has proved that the error bound expressed in terms of the radius of the circumscribed circle is valid for at least one C 2 -smooth function, i.e., is a lower bound. Therefore, Theorem 3 implies that this bound also is an upper error bound, hence, it is asymptotically exact. The existence of this bound allows one to apply the Delaunay triangulation for approximating derivatives. Note that in the paper [12] one proposes a method for approximating the discrete curvature of surfaces based on the use of locally polar polyhedral surfaces. For studying cases of the triangulation of domains D ⊂ R3 we need the following lemma. Lemma 6. Let numbers 0 < λi ≤ π/2 + δ, i = 1, 2, 3, be given, and 0≤ δ ≤ π/2. If λ1 + λ2 + λ3 ≥ π, then at least two of these numbers are not less than λ0 = 12 π2 − δ . Proof. Assume the contrary, i.e., some of these two numbers are less than λ0 . Then for their sum we have 3
λi < 2λ0 +
i=1
π + δ = π, 2
which contradicts the condition of the lemma. Setting δ = 0 and taking into account that the sum of angles of plane and spherical triangles is not less than π, we obtain the following assertion. Corollary 5. Any acute-angled tetrahedron has the following properties: 1) at any vertex there exist two dihedral angles not less than π/4; 2) in any face there are two plane angles not less than π/4. Consider some acute-angled tetrahedron with vertices at points p0 , p1 , p2 , p3 ∈ R3 . According to the corollary, there exists a vertex, for example, p0 , with a plane angle not less than π/4. Evidently, the sum of two other plane angles at this vertex is not less than this angle. Therefore one of them is not less than π/8. Denote the obtained angles by ψ1 , ψ2 ≥ π/8. In addition, note that there also exist two dihedral angles at the same vertex, whose values are not less than π/4. Denote them by δ1 , δ2 ≥ π/4. These considerations imply that one of faces 0 p1 p2 or p0 p1 p3 forms with the face p0 p2 p3 an angle that equals δ1 or δ2 ; moreover, it also has a plane angle that equals ψ1 or ψ2 . Then, if θ is the angle between the vector p1 − p0 and the plane of the face p0 p2 p3 , then elementary geometric considerations give the equality sin θ = sin δi sin ψj ≥ sin π4 sin π8 with certain i, j = 0, 1. In the face p0 p2 p3 there exists one of plane angles ϕ not less than π/4. Therefore, in Corollary 3 we can estimate the multiplier as follows: 16 8 ≤ . sin ϕS sin θS sin π8 Thus, the following assertion is valid. Theorem 4. Consider a finite set of points representing an ε-mesh in a domain D ⊂ R3 . Assume that for this set an acute-angled triangulation T is given, and a point p is a vertex of the triangulation T . Then |∇f (p) − ∇Lf (p)| ≤ RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 56 No. 1 2012
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ACKNOWLEDGMENTS The authors wish to express their gratitude to V. V. Popov for the attention to this work and for useful remarks. REFERENCES 1. S. N. Borovikov, I. E. Ivanov, and I. A. Kryukov, “Modeling of Test Ideal Gas Dynamics Problems Using Tetrahedral Meshes,” Matem. Modelir. 18 (8), 37–48 (2006). 2. S. Korotov, “Some Geometric Results for Tetrahedral Finite Elements,” Proc. Numgrid 2010 (Folium, Moscow, 2010), pp. 41–46. 3. I. G. Pushkina and V. F. Tishkin, “Adaptive Grids from Dirichlet Cells for Mathematical Physics Problems: A Methodology for Grids Generation, Examples,” Matem. Modelir. 12 (3), 97–109 (2000). 4. V. T. Rajan, “Optimality of the Delaunay Triangulation in Rd ,” Discrete Comput. Geom. 12 (2), 189–202 (1994). 5. S. Waldron, “The Error in Linear Interpolation at the Vertices of a Simplex,” SIAM J. Numer. Anal. 35 (3), 1191–1200 (1998). 6. J. Shewchuck, “What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures,” in Proceedings of 11th International Meshing Roundtable (Ithaca, New York, 2002), pp. 115–126. 7. B. R. Gelbaum and J. H. M. Olmsted, Counterexamples in Analysis (Holden-Day, San Francisco, 1964; PLATON, Volgograd, 1997). 8. V. A. Klyachin and E. A. Pabat, “The C 1 -Approximation of the Level Surfaces of Functions Defined on Irregular Meshes,” Sib. Zhurn. Ind. Matem. XIII (2), 69–78 (2010). 9. B. P. Delaunay, “An Empty Sphere. Dedicated to the Memory of Georgy Voronoi,” Transl. from French by A. Yu. Igumnov, in Zap. Semin. “Sverkhmedlennye protsessy” (Volgograd, Volgogr. Gos. Univ., 2006), No. 1, pp. 147–153. 10. V. A. Klyachin, “On a Generalization of the Delaunay Condition,” Vestn. Tomsk. Univ., Matem. i Mekh., No. 1, 48–50 (2008). 11. H. Edelsbrunner, “An Acyclicity Theorem for Cell Complexes in d Dimensions,” Combinatorica 10 (3), 251– 260 (1990). 12. V. A. Garanzha, “Discrete Extrinsic Curvatures and Approximation of Surfaces by Polar Polyhedra,” Zhurn. Vychisl. Matem. i Matem. Fiz. 50 (1), 71–98 (2010).
Translated by O. A. Kashina
RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 56 No. 1 2012
The Delaunay Triangulation for Multidimensional Surfaces and its Approximative Properties V. A. Klyachin* and A. A. Shirokii** Volgograd State University, Universitetskii pr. 100, Volgograd, 400062 Russia Received December 23, 2010
Abstract—We define the Delaunay triangulation for surfaces and prove an analog of the G. Voronoi empty sphere theorem. We also prove a convergence theorem for gradients of piecewise linear approximations constructed on the Delaunay triangulation for functions differentiable on smooth surfaces. DOI: 10.3103/S1066369X12010045 Keywords and phrases: simplex, triangulation, approximation of gradient.
1. INTRODUCTION The use of triangulation in computation and approximation problems is rather wide. See papers and references therein for some results and applications. The Delaunay triangulation is used most often, because the obtained triangles often tend to equilateral ones [5]. This property can be illustrated by the classical Schwarz example ([7], P. 191) of a piecewise linear approximation of the lateral surface of a straight circular cylinder. In this example the areas of the mentioned approximations tend to that of the cylinder, if the location of chosen points on the cylinder satisfies a certain condition. However, in our opinion, the key role here is played by the way of triangulation of the lateral surface of the cylinder, rather than by the mentioned condition. Namely, according to results obtained in this paper, the desired convergence of areas takes place, if the triangulation is neither arbitrary nor specified in the mentioned example, but the Delaunay one. Moreover, we have succeeded in obtaining qualitative characteristics, namely, approximation errors. Let us formulate the obtained results more precisely. Let D ⊂ Rn be some domain. For a function f from the class C 1 (D) we denote by ∇f (x) the gradient vector at the point x ∈ D, and for any vector ξ we do by ∂f ∂ξ = ∇f (x), ξ the derivative of f at x in the direction of the vector ξ. For a collection of points p0 , p1 , . . . , pk , 1 ≤ k ≤ n, in Rn we denote by S = S(p0 , . . . , pk ) the k-dimensional simplex with vertices at these points. We assume that vectors p1 −p0 , p2 −p0 , . . . , pk −p0 are linearly independent. Deleting the point pi , i = 0, . . . , k, from this collection, we can construct the (k − 1)-dimensional simplex; we denote it by Si . This simplex is a (k − 1)-dimensional face of the simplex S(p0 , . . . , pk ). Denote by ηi the unit vector of the outer normal to the face Si that lies in the plane of the simplex S(p0 , . . . , pk ). With respect to the simplex S an arbitrary vector X ∈ Rn can be represented as the sum X = N X + X T , where X T is the orthogonal projection of the vector X onto the plane of the simplex S, and X N is its orthogonal supplement. We also introduce the projection mapping onto the plane of the face of the simplex by setting πi (X) = X T − X, ηi ηi . Consider a simplex S ⊂ D. For f ∈ C 1 (D) we construct an affine function Lf (x) = A, x + a0 such that f (pi ) = Lf (pi ), i = 0, . . . , k, and AN = 0. One can easily prove the following assertion. * **
E-mail: [email protected]. E-mail: [email protected].
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Lemma 1. The affine function indicated above exists and is defined uniquely. For f ∈ C 1 (D) we introduce the following denotations: ωij = ∇f (pi ) − ∇f (pj ), βij =
|f (pi ) − f (pj ) − ∇f (pj ), pi − pj | , |pi − pj |
γij = f (pi ) − f (pj ) − ∇f (pj ), pi − pj ,
i, j = 0, . . . , k.
Lemma 2. For an arbitrary vector Y ∈ Rn and a pair of indices i = m, i, m = 0, . . . , k, the following inequality is valid: |Y, pi − pm | |pi − pm | + |πi (Y )| . (1) |Y, ηi | ≤ |ηi , pi − pm | |pi − pm | Proof. Since Y = Y N + Y T , taking into account that Y T = πi (Y ) + Y, ηi ηi , we get Y, pi − pm = Y, ηi ηi , pi − pm + πi (Y ), pi − pm . Therefore, correlation (1) takes place. Theorem 1. Let a simplex S(p0 , p1 , . . . , pk ) belong to D, k > 1, and let a function f (x) be differentiable at vertices of this simplex. Then for any number i = 0, . . . , k and any pair of numbers j, m = 0, . . . , k different from i the following inequality is valid: ∂f (pm ) ∂Lf (pm ) 1 ≤ βim + |πi (ωjm )| + |πi (∇f (pj ) − ∇Lf (pj ))| ; ∂ηi − ∂ηi sin θim here θim is the angle between the vector pi − pm and the plane of the face Si . If k = 1, then ∂f (p0 ) ∂Lf (p0 ) ≤ β10 , ∂η − ∂η where η = (p1 − p0 )/|p1 − p0 |. Proof. In Lemma 2 we set Y = ∇f (pm ) − ∇Lf (pm ). Then ∂f (pm ) ∂Lf (pm ) ∂ηi − ∂ηi |∇Lf (pm ) − ∇f (pm ), pi − pm | |pi − pm | + |πi (∇f (pm ) − ∇Lf (pm ))| . ≤ |pi − pm , ηi | |pi − pm | Since 1 |pi − pm | = , |pi − pm , ηi | sin θim
|∇Lf (pm ) − ∇f (pm ), pi − pm | = βim , |pi − pm |
and |πi (∇f (pm ) − ∇Lf (pm ))| ≤ |πi (∇f (pm ) − ∇f (pj ))| + |πi (∇f (pj ) − ∇Lf (pj ))| = |πi (ωjm )| + |πi (∇f (pj ) − ∇Lf (pj ))|, we obtain the desired assertion. In the case k = 1 the desired inequality immediately follows from the definition of β10 and the construction of the function Lf (x). Lemma 3. Assume that Π is some k-dimensional plane that forms with the plane of the simplex the angle 0 ≤ α < π. Let the symbol (v)Π stand for the orthogonal projection of the vector v onto this plane, and let (v)⊥ do for the corresponding orthogonal supplement. Then |(∇f (pm ) − ∇Lf (pm ))Π | ≤ |(∇f (pm ) − ∇Lf (pm ))T | + |∇f (pm )| sin α. RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 56 No. 1 2012
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Proof. Since the angle between planes equals the angle between their orthogonal supplements, any vector Y satisfies the inequality |(Y N )⊥ | ≥ cos α|Y N |, whence |(Y N )Π | ≤ sin α|Y N | ≤ sin α|Y |. Taking into account that |Y Π | ≤ |(Y N )Π | + |(Y T )Π |, we obtain |Y Π | ≤ |Y T | + sin α|Y |. Setting Y = ∇f (pm ) − ∇Lf (pm ) and using the fact that by construction [∇Lf (pm )]N = 0, we obtain the desired assertion. Remark 1. If Π is a plane tangent to some smooth surface at the point pm , then the proved inequality and Theorem 1 allow us to estimate the error of the approximation of the gradient of the function in the metric of this surface. Lemma 4. Assume that a function f belongs to C 1 (D) and a function ω(t) is the modulus of continuity of its gradient, i.e., any pair of points x , x ∈ D satisfies the inequality |∇f (x ) − ∇f (x )| ≤ ω(|x − x |). Then |ωij | ≤ ω(|pi − pj |), |pi −pj | 1 ω(t)dt. βij ≤ |pi − pj | 0 Proof. The first inequality is evident. We can obtain the second one by integrating the equality ∇f (x) = ∇f (pj ) + H(x, pj ), where by condition of the lemma |H(x, pj )| ≤ ω(|x − pj |), along the segment connecting points pi and pj . In the terminology of Theorem 1 we have ∂f (pm ) ∂Lf (pm ) T + |πi (∇f (pm ) − ∇Lf (pm ))|. − | ∇f (pm ) − ∇Lf (pm ) | ≤ ∂ηi ∂ηi We get the bound |pi −pm | 1 1 ω(t)dt |(∇f (pm ) − ∇Lf (pm )) | ≤ sin θim |pi − pm | 0 1 (1 + sin θim )(ω(|pm − pj |) + |πi (∇f (pj ) − ∇Lf (pj ))|) . (2) + sin θim T
The following corollaries of Theorem 1 are obtained immediately from bound (2) with the help of Lemmas 3 and 4. Corollary 1. Let k = 2. If m = 0, i = 1, and j = 2, while ϕS and dS denote values of the angle at the vertex p0 and the length of the maximal side of the triangle S, respectively, then dS 1 3 T ω(t)dt + 2ω(dS ) . |(∇f (p0 ) − ∇Lf (p0 )) | ≤ sin ϕS dS 0 Corollary 2. Let k = 2 and let vertices of the triangle S lie on a two-dimensional smooth surface F ⊂ D ⊂ Rn . Assume that the angle between the plane of this triangle and the tangent plane to the surface at the point p0 equals α, 0 ≤ α ≤ π/2. If a function f belongs to C 1 (D), then dS 1 3 ω(t)dt + 2ω(dS ) + sin α sup |∇f (x)|. |∇f (p0 ) − ∇Lf (p0 )| ≤ sin ϕS dS 0 x∈D Here the symbol ∇ denotes the gradient of a differentiable function in the metric of the surface F . Corollary 3. Let k = 3 and m = 0, i = 2, j = 1. Let the angle between the vector p2 − p0 and the two-dimensional face opposite to the vertex p2 equal θS , and let the angle at the vertex p1 of the triangle p0 p1 p3 equal ϕS . Denote by dS the length of the maximal edge of the simplex S. Then the following bound is valid: dS T 8 1 ω(t)dt + ω(dS ) . | ∇f (p0 ) − ∇Lf (p0 ) | ≤ sin ϕS sin θS dS 0 RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 56 No. 1 2012
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Corollary 4. Let k = 3. Assume that vertices of the tetrahedron S lie on a three-dimensional smooth surface F ⊂ D ⊂ Rn . Assume that the angle between the plane of the tetrahedron and the plane tangent to the surface at the point p0 equals α, 0 ≤ α ≤ π/2. If a function f belongs to C 1 (D), then dS 8 1 ω(t)dt + ω(dS ) + sin α sup |∇f (x)|. |∇f (p0 ) − ∇Lf (p0 )| ≤ sin ϕS sin θS dS 0 x∈D Here, as above, the symbol ∇ denotes the gradient of a differentiable function in the metric of the surface F . Note that analogous methods were applied in [8] for formulating conditions of the C 1 -convergence of the piecewise linear approximations of level surfaces of smooth functions. 2. THE DELAUNAY TRIANGULATION OF THE GRAPH OF A SMOOTH FUNCTION Consider an n-dimensional surface F in the space Rn+1 representing the graph of a function xn+1 = f (x1 , . . . , xn ), x ∈ D ⊂ Rn . Define the projection mapping π : F → D, π(x1 , . . . , xn+1 ) = (x1 , . . . , xn ). Let Pi , i = 1, . . . , N , be some set of points Pi ∈ Rn+1 that lie on the surface F such that at vertices of this set any n-dimensional simplex is nondegenerate. Denote by Pi = π(Pi ) the projection of the point Pi onto the domain D. We consider the triangulation of the surface F by n-dimensional simplexes whose vertices belong to the set Pi . We understand a triangulation of the surface as a set T of ndimensional simplexes such that 1) each point Pi of the given set is a vertex of one of simplexes S ∈ T ; 2) each vertex of any simplex S ∈ T is one of points Pi , i = 1, . . . , N ; 3) the interior of the intersection of any two simplexes is empty; 4) the projection of the system of simplexes Sj ∈ T , j = 1, . . . , m, is the usual triangulation of the set of points Pi in Rn . We say that a triangulation is acute-angle, if for each simplex its angles between any two faces are acute. Definition 1. Let S be some n-dimensional simplex in Rn+1 . We understand a circumscribed ball B(S) for S as an (n + 1)-dimensional ball of the least radius that contains vertices of the simplex. Definition 2. We say that a triangulation of a surface F satisfies the Delaunay condition, if for any simplex of the triangulation the interior of the circumscribed ball contains no triangulation vertices. Definition 3. We say that for two n-dimensional simplexes with a common (n − 1)-dimensional face the ball emptiness condition is fulfilled, if the interior of the circumscribed ball of some simplex contains no vertices of another one. From the point of view of the construction of optimal algorithms for triangulating surfaces, as well as the approximation of the gradient in the metric of a surface, the Delaunay triangulation has one important property. Namely, if the Delaunay condition is fulfilled locally, then it is also fulfilled globally [9–11]. The next theorem generalizes this property for the case of triangulation of a smooth surface representing the graph of a function. But as distinct from the plane case, it is necessary that an additional condition should be fulfilled. Below we illustrate this fact with an example. Condition α. Consider an arbitrary (n − 1)-dimensional simplex S of a triangulation. Construct the intersection US of all halfspaces that contain S which are defined by hyperplanes of (n − 1)-dimensional spheres that represent intersections of circumscribed spheres of the simplex S and the neighboring one. The number of such halfspaces does not exceed n for each simplex S. We assume that no vertex of the triangulation belongs to the interior of US for any S. Note that in the plane case, when triangulation vertices lie in one hyperplane, condition α, evidently, is fulfilled. RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 56 No. 1 2012
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Theorem 2. Let a surface F be defined over a convex domain D ⊂ Rn . A triangulation of this surface, for which condition α is fulfilled and any pair of simplexes with a common (n − 1)dimensional face satisfies the ball emptiness condition, is the Delaunay triangulation. Proof. Consider some triangulation of a set of points Pi with the mentioned in the theorem properties. For two neighboring simplexes S and S of this triangulation we construct a hyperplane Π going through the intersection of hyperspheres ∂B(S ) ∩ ∂B(S ). Consider an arbitrary simplex S1 of the given triangulation and some vertex A of an arbitrary simplex of this triangulation different from S1 . Consider the ray A1 A, where A1 ∈ S1 is some interior point of the simplex S1 . Assume that the theorem is not true, i.e., A ∈ B(S). In view of condition α we have A ∈ / US1 , therefore the ray A1 A first intersects one of planes Π constructed above that goes through some face of the simplex S1 . Denote this plane by + Π1 . It also contains one of faces of the neighboring simplex S2 . Denote by Π− 1 and Π1 halfspaces defined − + by the hyperplane Π1 , let A1 ∈ Π1 . Then, evidently, A ∈ Π1 . Note that in view of properties of balls one of the following inclusions is valid: + B(S1 ) ∩ Π+ 1 ⊂ B(S2 ) ∩ Π1
or + B(S2 ) ∩ Π+ 1 ⊂ B(S1 ) ∩ Π1 .
Assume that a vertex C of the simplex S2 does not belong to the plane Π1 . Since by assumption / B(S1 ), but, on the other hand, for the pair of simplexes S1 and S2 the ball is empty, we have C ∈ + + C ∈ B(S2 ) ∩ Π1 . Therefore, the inclusion B(S1 ) ∩ Π1 ⊂ B(S2 ) ∩ Π+ 1 is valid, because otherwise we + ⊂ B(S ) ∩ Π ⊂ B(S ). get a contradiction, namely, C ∈ B(S2 ) ∩ Π+ 1 1 Due to the above inclusion we 1 1 have A ∈ B(S2 ). Let us now construct a sequence of simplexes S1 , S2 , . . . , Sk such that A is a vertex of the simplex Ak . Then simplexes Si and Si+1 are neighboring. Let Πi be a hyperplane containing the (n − 1)-dimensional sphere ∂B(Si ) ∩ ∂B(Si+1 ). To this end we construct a two-dimensional vertical plane containing the ray A1 A. As a result we obtain a polyline connecting points A1 and A, whose segment lies in the corresponding simplexes S1 , S2 , . . . , Sk . The segment of this polyline that lies in S2 intersects the (n − 1)-dimensional face, which is common for simplexes S1 and S2 , and some other face S2 , which is common with the simplex S3 . In view of condition α this segment contains a point A2 , for which the ray A2 A intersects the plane Π2 . Repeating this reasoning, we construct sequences of simplexes Si , hyperplanes Πi , and points Ai such that + Ai ∈ Si , Si ⊂ Π− i , Si+1 ⊂ Πi , A ∈ B(Si ).
By construction, A ∈ B(Sk−1 ), which contradicts the ball emptiness condition for the pair of simplexes / B(S1 ). Since S1 and A are chosen arbitrarily, the theorem is proved. Sk−1 and Sk . Therefore, A ∈ Example. Note that condition α is necessary. The following example was proposed by V. V. Popov. In R3 there exist points A(104, 0, 0), B(134, 40, 120), C(0, 78, 0), D(0, 0, 0), and E(52, −20, 48) with the following properties: a) the ball circumscribed around the triangle ABC does not contain the point D, but contains the point E; b) the ball circumscribed around the triangle ACD does not contain points B and E; c) the ball circumscribed around the triangle ADE does not contain points C and B. Let F be the surface composed of the mentioned triangles. These triangles form a triangulation of F such that any two neighboring simplexes (i.e., those with a common edge) satisfy the ball emptiness condition, and the Delaunay condition is violated. On the other hand, note that the plane containing the intersection of spheres of balls circumscribed around triangles ABC and ACD obeys the equation 3x + 4y + 12z = 312. Then the set US for S = ABC represents the halfspace 3x + 4y + 12z > 312. One can immediately make sure that E ∈ US , therefore, condition α is not fulfilled.
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3. APPROXIMATION PROPERTIES OF THE DELAUNAY TRIANGULATION Let D ⊂ Rn , n > 1, be a domain containing a fixed sequence {Pm } of finite sets of points. For each of these sets we consider its triangulation Tm . We say that a triangulation is acute-angle, if for each simplex its angles between any two faces are acute. A triangulation of a set of points is called the Delaunay triangulation, if for any simplex of this triangulation the interior of the circumscribed ball contains no point from this set. For a simplex S ∈ Tm we denote by dS the length of its maximal edge. We set dm = max dS . S∈Tm
Consider sets of points Pm and their triangulations Tm such that dm → 0 as m → ∞, ∀ε > 0, ∀x ∈ D ∃m0 ∈ N : ∀m > m0 ∃a ∈ Pm : |a − x| < ε.
(3) (4)
The second condition means that Pm is the ε-mesh with all sufficiently great m. Consider some function f (x), x ∈ D, from the class C 1 (D). For any natural m we construct a piecewise affine function fm (x) such that fm (a) = f (a) for any point a ∈ Pm . One can easily prove that under conditions (3) and (4) the sequence fm (x) uniformly converges to the function f (x) on each compact subset K ⊂ D. But these conditions do not guarantee the convergence of derivatives of functions fm (x) to those of functions f (x) (even almost everywhere or in Sobolev spaces 1,p (D)). This fact is proved by the well-known Schwarz example ([7], P. 191). Wloc Below we assume that a function f : D → R is C 2 -smooth in a domain D ⊂ Rn , and all its second derivatives are bounded in D. Then there exists a constant M > 0 such that the modulus of continuity of the gradient ω(t) ≤ M · t. Our goal is to prove that the Delaunay triangulation and those close to it have (in a sense) the mentioned above approximation property. Let us first consider a two-dimensional case. For an arbitrary triangle S with vertices p0 , p1 , and p2 we introduce the value μS = sindSϕS , where ϕS is the value of the maximal acute angle of the triangle. Lemma 5. Let RS denote the radius of the circumference circumscribed around the triangle S. Then μS ≤ 4RS . Proof. Consider two cases. 1) The triangle is acute-angled. Then ϕS is the angle opposite to the longest side of length dS . According to the sine theorem, sindSϕS = 2RS . 2) The triangle is obtuse. Denote by ϕd the angle opposite to the longest side. Evidently, ϕd ≥ π/2 and ϕd + 2ϕS ≥ π, whence | sin ϕd | ≤ | sin 2ϕS |. Consequently, 2dS cos ϕS 2dS dS = ≤ = 4RS . sin ϕS sin 2ϕS sin ϕd
Theorem 3. Consider a finite set of points representing an ε-mesh in a plane domain D ⊂ R2 and assume that the Delaunay triangulation T for this set is given. Assume also that a point p is a vertex of the triangulation T such that for some triangle from T with a vertex at this point the center of the circumscribed circle lies in the domain D. Then |∇f (p) − ∇Lf (p)| ≤ 14M ε.
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Proof. Note that if conditions of the theorem are fulfilled, then the radius of the circle circumscribed around the considered triangle does not exceed ε. Really, assume the contrary, i.e., the radius of the circumscribed circle exceeds ε. Then in view of (4) there exists a vertex p of the triangulation (different from vertices of the mentioned triangle) such that the distance from it to the center of the circle circumscribed around the triangle does not exceed ε. But this contradicts the definition of the Delaunay triangulation. Therefore, taking into account the inequality in Corollary 1, where ω(t) ≤ M · t, we obtain the desired assertion. Remark 2. Note that in [6] one has proved that the error bound expressed in terms of the radius of the circumscribed circle is valid for at least one C 2 -smooth function, i.e., is a lower bound. Therefore, Theorem 3 implies that this bound also is an upper error bound, hence, it is asymptotically exact. The existence of this bound allows one to apply the Delaunay triangulation for approximating derivatives. Note that in the paper [12] one proposes a method for approximating the discrete curvature of surfaces based on the use of locally polar polyhedral surfaces. For studying cases of the triangulation of domains D ⊂ R3 we need the following lemma. Lemma 6. Let numbers 0 < λi ≤ π/2 + δ, i = 1, 2, 3, be given, and 0≤ δ ≤ π/2. If λ1 + λ2 + λ3 ≥ π, then at least two of these numbers are not less than λ0 = 12 π2 − δ . Proof. Assume the contrary, i.e., some of these two numbers are less than λ0 . Then for their sum we have 3
λi < 2λ0 +
i=1
π + δ = π, 2
which contradicts the condition of the lemma. Setting δ = 0 and taking into account that the sum of angles of plane and spherical triangles is not less than π, we obtain the following assertion. Corollary 5. Any acute-angled tetrahedron has the following properties: 1) at any vertex there exist two dihedral angles not less than π/4; 2) in any face there are two plane angles not less than π/4. Consider some acute-angled tetrahedron with vertices at points p0 , p1 , p2 , p3 ∈ R3 . According to the corollary, there exists a vertex, for example, p0 , with a plane angle not less than π/4. Evidently, the sum of two other plane angles at this vertex is not less than this angle. Therefore one of them is not less than π/8. Denote the obtained angles by ψ1 , ψ2 ≥ π/8. In addition, note that there also exist two dihedral angles at the same vertex, whose values are not less than π/4. Denote them by δ1 , δ2 ≥ π/4. These considerations imply that one of faces 0 p1 p2 or p0 p1 p3 forms with the face p0 p2 p3 an angle that equals δ1 or δ2 ; moreover, it also has a plane angle that equals ψ1 or ψ2 . Then, if θ is the angle between the vector p1 − p0 and the plane of the face p0 p2 p3 , then elementary geometric considerations give the equality sin θ = sin δi sin ψj ≥ sin π4 sin π8 with certain i, j = 0, 1. In the face p0 p2 p3 there exists one of plane angles ϕ not less than π/4. Therefore, in Corollary 3 we can estimate the multiplier as follows: 16 8 ≤ . sin ϕS sin θS sin π8 Thus, the following assertion is valid. Theorem 4. Consider a finite set of points representing an ε-mesh in a domain D ⊂ R3 . Assume that for this set an acute-angled triangulation T is given, and a point p is a vertex of the triangulation T . Then |∇f (p) − ∇Lf (p)| ≤ RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 56 No. 1 2012
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ACKNOWLEDGMENTS The authors wish to express their gratitude to V. V. Popov for the attention to this work and for useful remarks. REFERENCES 1. S. N. Borovikov, I. E. Ivanov, and I. A. Kryukov, “Modeling of Test Ideal Gas Dynamics Problems Using Tetrahedral Meshes,” Matem. Modelir. 18 (8), 37–48 (2006). 2. S. Korotov, “Some Geometric Results for Tetrahedral Finite Elements,” Proc. Numgrid 2010 (Folium, Moscow, 2010), pp. 41–46. 3. I. G. Pushkina and V. F. Tishkin, “Adaptive Grids from Dirichlet Cells for Mathematical Physics Problems: A Methodology for Grids Generation, Examples,” Matem. Modelir. 12 (3), 97–109 (2000). 4. V. T. Rajan, “Optimality of the Delaunay Triangulation in Rd ,” Discrete Comput. Geom. 12 (2), 189–202 (1994). 5. S. Waldron, “The Error in Linear Interpolation at the Vertices of a Simplex,” SIAM J. Numer. Anal. 35 (3), 1191–1200 (1998). 6. J. Shewchuck, “What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures,” in Proceedings of 11th International Meshing Roundtable (Ithaca, New York, 2002), pp. 115–126. 7. B. R. Gelbaum and J. H. M. Olmsted, Counterexamples in Analysis (Holden-Day, San Francisco, 1964; PLATON, Volgograd, 1997). 8. V. A. Klyachin and E. A. Pabat, “The C 1 -Approximation of the Level Surfaces of Functions Defined on Irregular Meshes,” Sib. Zhurn. Ind. Matem. XIII (2), 69–78 (2010). 9. B. P. Delaunay, “An Empty Sphere. Dedicated to the Memory of Georgy Voronoi,” Transl. from French by A. Yu. Igumnov, in Zap. Semin. “Sverkhmedlennye protsessy” (Volgograd, Volgogr. Gos. Univ., 2006), No. 1, pp. 147–153. 10. V. A. Klyachin, “On a Generalization of the Delaunay Condition,” Vestn. Tomsk. Univ., Matem. i Mekh., No. 1, 48–50 (2008). 11. H. Edelsbrunner, “An Acyclicity Theorem for Cell Complexes in d Dimensions,” Combinatorica 10 (3), 251– 260 (1990). 12. V. A. Garanzha, “Discrete Extrinsic Curvatures and Approximation of Surfaces by Polar Polyhedra,” Zhurn. Vychisl. Matem. i Matem. Fiz. 50 (1), 71–98 (2010).
Translated by O. A. Kashina
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