An ε characterization of the vertices of tetrahedra in the three dimensional Euclidean Space
aa r X i v : . [ m a t h . G M ] M a y AN ǫ CHARACTERIZATION OF THE VERTICES OFTETRAHEDRA IN THE THREE DIMENSIONAL EUCLIDEANSPACE
ANASTASIOS N. ZACHOS
Abstract.
We determine a positive real number (weight), which correspondsto a vertex of a tetrahedron and it depends on the three weights which corre-spond to the other three vertices and an infinitesimal number ǫ. As a limitingcase, for ǫ → Introduction
Let A A A A be a tetrahedron in R . We denote by B i a non-negative number(weight) which corresponds to each vertex A i , A be a point in R , by a ij be theEuclidean distance of the linear segment A i A j , and by ~u ( A i , A j ) is the unit vectorwith direction from A i to A j , for i, j = 0 , , , , i = j. The weighted Fermat-Torricelli problem for A A A A in R states that: Problem 1.
Find a point A (weighted Fermat-Torricelli point) X i =1 B i a i → min. (1.1)Y. Kupitz and H. Martini proved the existence and uniqueness of the weightedFermat-Torricelli point for n non-collinear and non-coplanar points in R m ([2], [1]).Furthermore, a characterization of the weighted Fermat-Torricelli point (weightedfloating and absorbed cases) is given in [1, Theorem 18.37, p. 250]).By setting n = 4 and m = 3 , the following theorem is given for a tetrahedron A A A A : Theorem 1. [1, Theorem 18.37, p. 250] )Let A A A A be a tetrahedron in R . (i)The weighted Fermat-Torricelli point A of A A A A exists and is unique.(ii) If k X j =1 B j ~u ( A i , A j ) k > B i , i = j, for i,j=1,2,3,4, then Mathematics Subject Classification.
Key words and phrases. weighted Fermat-Torricelli problem, weighted Fermat-Torricelli point,tetrahedra. (a) the weighted Fermat-Torricelli point does not belong in { A , A , A , A } (Weighted Floating Case),(b) X i =1 B i ~u ( A , A i ) = ~ (Weighted balancing condition).(iii) If there is some i with k X j =1 B j ~u ( A i , A j ) k ≤ B i , i = j. for i,j=1,2,3,4, then the weighted Fermat-Torricelli point is the vertex A i (Weighted Absorbed Case). The unique solution of the weighted Fermat-Torricelli problem for A A A A iscalled a weighted Fermat-Torricelli tree having one node (weighted Fermat-Torricellipoint) with degree four (weighted floating case). A numerical approach to findthe weighted Fermat-Torricelli tree for A A A A , by introducing a method ofdifferentiation of the length of a linear segment to a specific dihedral angle is givenin [3].The inverse weighted Fermat-Torricelli problem for tetrahedra in R states that: Problem 2.
Given a point A which belongs to the interior of A A A A in R ,does there exist a unique set of positive weights B i , such that B + B + B + B = c = const, for which A minimizes f ( A ) = X i =1 B i a i . The unique solution of the inverse weighted Fermat-Torricelli problem for tetra-hedra in R is given in [3]. We denote by α i,j k the angle that is formulated by theline segment that connects A with the trace of the orthogonal projection of A i tothe plane defined by △ A j A A k and we set α lmn ≡ ∠ A l A m A n , for j, k = 1 , , , l, m, n = 0 , , , , . Proposition 1. [3, Proposition 1, Solution of Problem 2]
The weight B i areuniquely determined by the formula: B i = C k sin α i,k l sin α j,k l k + k sin α i,j l sin α k,j l k + k sin α i,k j sin α l,k j k , (1.2) for i, j, k, l = 1 , , , and i = j = k = l. We need the following two formulas which have been derived in [3] and [4],respectively:
Lemma 1. [3]
The ratio B j B i is given by: B j B i = s k sin α k m − cos α m i − cos α k i + 2 cos α k m cos α m i cos α k i sin α k m − cos α m j − cos α k j + 2 cos α k m cos α m j cos α k j k (1.3) for i, j, k, m = 1 , , , . N ǫ CHARACTERIZATION OF THE VERTICES OF TETRAHEDRA 3
Lemma 2. [4, Proposition 1]
The angles α i,k m depend on exactly five given angles α , α , α , α and α . The sixth angle α is calculated by the following formula: cos α = 14 [4 cos α (cos α − cos α cos α ) ++2 ( b + 2 cos α ( − cos α cos α + cos α ))] csc α (1.4) where b ≡ vuut Y i =3 (1 + cos (2 α ) + cos (2 α i ) + cos (2 α i ) − α cos α i cos α i ) . (1.5) for i, k, m = 1 , , , , and i = k = m. If B , B , B , B satisfy the inequalities of the floating case, we derive a uniqueweighted Fermat-Torricelli tree { A A , A A , A A , A A } , which consists of fourbranches A i A , for i = 1 , , , A . If B , B , B B satisfy one of the inequalities of the absorbed case, we obtain adegenerate weighted Fermat-Torricelli tree { A A , A A , A A } , { A A , A A , A A } and { A A , A A , A A } , { A A , A A , A A } . Assume that: k X j =1 B j ~u ( A , A j ) k ≤ B , i = j. Suppose that we choose B , B , B , such that k X j =1 B j ~u ( A , A j ) k = B , i = j, or B = B + B + B + 2 B B cos α + 2 B B cos α + 2 B B cos α (1.6)or B = f ( B , B , B ) . Thus, we consider the following problem:
Problem 3.
How can we determine the values of B , B , B , such that f ( B , B , B ) gives the minimum value of B that corresponds to the vertex A ? If we set B = 0 or B = 0 or B = 0 in Problem 1, B depends of the values oftwo weights which are determined in [5].In this paper, we determine the value of B by introducing an infinitesimal realnumber ǫ, ( ǫ characterization of A ) such that: α = α − ǫ, α = α + k ǫ,α = α + k ǫ, α = α + k ǫ and α = α + k ǫ, where k , k , ANASTASIOS N. ZACHOS k and k are rational numbers, by applying the solution of the inverse weightedFermat-Torricelli problem for tetrahedra in R . An ǫ characterization of the vertices of tetrahedra in R Let A be an interior point of A A A A in R . We denote by h , the length of the height of △ A A A from A with respectto A A , by α, the dihedral angle which is formed between the planes defined by △ A A A and △ A A A , and by α g the dihedral angle which is formed by theplanes defined by △ A A A and △ A A A . We set α = α − ǫ, α = α + k ǫ, α = α + k ǫ, α = α + k ǫ and α = α + k ǫ, where k , k , k and k are rationalnumbers.We need the following two formulas given by the following proposition which isderived in [3] and expresses a and a as a function w.r. to a , a and α. Proposition 2. [3]
The length a i depends on a , a and α : a i = a + a i − a i [ q a − h , cos( α i ) + h , sin( α i ) cos( α g i − α )] , (2.1) or a i = a + a i − a i [ q a − h , cos( α i ) + h , sin( α i ) cos( α g i − α )] . (2.2) where h , = a a a s − (cid:18) a + a − a a a (cid:19) (2.3) for i = 3 , . By replacing the index 0 → i = 3 in (2.1), we derive the following corollary: Corollary 1. α g = arccos( a + a − a − a q a − h , cos( α )2 a h , sin α ) (2.4) where h , = a a a s − (cid:18) a + a − a a a (cid:19) . (2.5)By solving (2.1) w.r. to α for i = 3 and by replacing the derived formula (2.1)for i = 4 and taking into account (2.4), we derive that a = a ( a , a , a ) . Corollary 2. a = a + a − a [ q a − h , cos( α )+ h , sin( α )(cos( α g ) cos α +sin( α g ) sin α )](2.6) where N ǫ CHARACTERIZATION OF THE VERTICES OF TETRAHEDRA 5 α = arccos( a + a − a − a q a − h , cos( α )2 a h , sin α ) . (2.7) Proposition 3. If α = α − ǫ, α = α + k ǫ, α = α + k ǫ,α = α + k ǫ and α = α + k ǫ, then a = a ( ǫ ) . Proof.
By applying the law of sines in △ A A A , we get: a ( ǫ ) = sin( α − ǫ ) a sin( α + k ǫ ) . (2.8)and a ( ǫ ) = sin( α + ( k − ǫ + α ) a sin( α + k ǫ ) . (2.9)By applying the law of sines in △ A A A , we get: a ( ǫ ) = sin( α + k ǫ ) a sin( α + k ǫ ) . (2.10)By replacing (2.8), (2.9), (2.10) in (2.6) we obtain that a = a ( a ( ǫ ) , a ( ǫ ) , a ( ǫ )) = a ( ǫ ) . (cid:3) Theorem 2.
The weight B i = B i ( ǫ ) are uniquely determined by the formula: B i = C k sin α i,k l sin α j,k l k + k sin α i,j l sin α k,j l k + k sin α i,k j sin α l,k j k , (2.11) where k sin α i,k m sin α j,k m k = s k sin α k m − cos α m i − cos α k i + 2 cos α k m cos α m i cos α k i sin α k m − cos α m j − cos α k j + 2 cos α k m cos α m j cos α k j k (2.12) depends on ǫ, α ( ǫ ) , α ( ǫ ) , α ( ǫ ) , and α ( ǫ ) , such that there exists afunctional dependence between the rational numbers k , k , k and k ,F ( k , k , k , k ) = 0 , (2.13) for i, j, k, l = 1 , , , and i = j = k = l. Proof.
By replacing (2.9) and a ( ǫ ) in the law of cosines in △ A A A , we obtain : α ( ǫ ) = arccos( ( a ( ǫ )) + ( a ( ǫ )) − a a ( ǫ ) a ( ǫ ) ) . (2.14)By replacing (2.10) and a ( ǫ ) in the law of cosines in △ A A A , we obtain : α ( ǫ ) = arccos( ( a ( ǫ )) + ( a ( ǫ )) − a a ( ǫ ) a ( ǫ ) ) . (2.15)By replacing (2.14), α = α + k ǫ, α = α + k ǫ, α = α + k ǫ in (1.4) and by substituting the derived result in the left side of (2.15), we derivea functional dependence F ( k , k , k , k ) = 0 . By replacing the five angles α ( ǫ ) , α ( ǫ ) , α ( ǫ ) , α ( ǫ ) , and α ( ǫ ) , for three given rational numbers k , ANASTASIOS N. ZACHOS k , k and the fourth rational k is determined by (2.13) in (1.2), we obtainthat the weights B i ( ǫ ) are determined by (2.11). (cid:3) Corollary 3.
For ǫ → , we derive a degenerate weighted Fermat-Torricelli tree { A A , A A , A A } . Remark 1.
By substituting B ( ǫ ) , B ( ǫ ) , B ( ǫ ) in (1.6), we obtain B : B == p B ( ǫ ) + B ( ǫ ) + B ( ǫ ) + 2 B ( ǫ ) B ( ǫ ) cos α + 2 B ( ǫ ) B ( ǫ ) cos α + 2 B ( ǫ ) B ( ǫ ) cos α . Thus, k B − B ( ǫ ) k gives an ǫ approximation of the optimal value of B , thatachieves the vertex A . References [1] V. Boltyanski, H. Martini, V. Soltan,
Geometric Methods and Optimization Problems , vol.4. Combinatorial Optimization, Kluwer Academic Publishers, Dordrecht (1999),[2] Y.S. Kupitz and H. Martini,
Geometric aspects of the generalized Fermat-Torricelli problem ,Bolyai Society Mathematical Studies. (1997) , 55-127.[3] A. Zachos and G. Zouzoulas, The weighted Fermat-Torricelli problem for tetrahedra and an”inverse” problem , J. Math. Anal. Appl. , (2009), 114-120.[4] A. Zachos,
The Plasticity of some Mass Transportation Networks in the Three DimensionalEuclidean Space , J. Convex. Anal. , no. 3, (2020), To appear.[5] A. Zachos, An ǫ -characterization of a vertex formed by two non-overlapping geodesic arcs onsurfaces with constant Gaussian curvature , arXiv:2004.14215, (2020), 9 pp. Greek Ministry of Education, Athens, Greece
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