SOME DISTANCE PROPERTIES OF TWO SPACES ...2018/10/12 · Some distance properties of two spaces induced by dual convex polyhedra 13 A polyhedron is called convex if a line segment

INTERNATIONAL JOURNAL OF GEOMETRYVol. 7 (2018), No. 2, 12 - 24

SOME DISTANCE PROPERTIES OF TWO SPACESINDUCED BY DUAL CONVEX POLYHEDRA

ZEYNEP ÇOLAK and ZEYNEP CAN

Abstract.Dual convex spaces have an important place in mathematics.In geometry, some polyhedra are associated into pairs called duals, wherethe vertices of one correspond to the faces of the other. Arranging regularpolyhedra into dual pairs needs some simple operations but it is not thateasy to do this for semi-regular polyhedra. In further studies it has seenthat there are some relations between polyhedra and metrics. For example,it has been shown that cube, octahedron and deltoidal icositetrahedron aremaximum, taxicab and Chinese Checker�s unit spheres, respectively. In thispaper some distance formulae of two spaces which are induced by two dualconvex polyhedra has been researched by a new method di¤erent from [1]and [8], and by this method distance formulae in Minkowski geometries canbe generalized. This polyhedra are cuboctahedron which is an Archimedeansolid and rhombic dodecahedron a Catalan solid which is the dual of cuboc-tahedron.

1. Introduction

Molecules, galaxies, sculpture, viruses, crystals, architecture and more,polyhedra are always in nature and art. It is at the same time hands-on, mind-turned-on introduction to one of the oldest and most fascinatingbranches of mathematics. Although it is so ancient it is not that easy todescribe what a polyhedron is. To describe a polyhedron the terms faces,edges and vertices would be used. Polygonal parts of the polyhedron arecalled faces, line segments that along faces come together are called edgesand points where several edges and faces come together are called vertex.So, simply, a polyhedron can be de�ned as a closed, three-dimensional �gurewhich faces are polygons. For example, any closed box is a polyhedron, forits faces are rectangles.� � � � � � � � � � � � �Keywords and phrases: Polyhedra, Dual polyhedra, Convex polyhe-

dra, Metric geometry, Rhombic Dodecahedron, Cuboctahedron, DistanceFormulae.(2010)Mathematics Subject Classi�cation: 51P99, 60A99Received: 27.07.2018. In revised form: 30.09.2018. Accepted: 20.08.2018.

Some distance properties of two spaces induced by dual convex polyhedra 13

A polyhedron is called convex if a line segment joining any two pointsin the interior of the �gure lies completely within the �gure. There aresome classi�cations of convex polyhedra. The regular polyhedra (for eachof these polyhedra, every face, every vertex and every edge is like everyother) are also known as the "Platonic Solids" because the Greek philoso-pher Plato (427-347 B.C.E.) immortalized them in his dialogue Timaeus.Platonic solids are only �ve. An other class of polyhedra is Archimedeansolids because they were �rst discovered by Archimedes (287-212 B.C.E.).Vertices of the Archimedean polyhedra are all alike, but their faces, whichare regular polygons, are of two or more di¤erent kinds. For this reasonthey are often called semiregular [13]. And irregular polyhedra are de�nedby polygons that are composed of elements that are not all equal and theyare called Catalan solids or Archimedean duals. There are thirteen Cata-lan solids just like Archimedean solids. They are named for the Belgianmathematician, Eugéne Catalan, who �rst described them in 1865. Twopolyhedrons called dual if either of a pair of polyhedra in which the facesof one are equivalent to the vertices of the other. Starting with any regularpolyhedra, the dual can be constructed in the following manner:(1) Place a point in the center of each face of the original polyhedron;(2) Connect each new point with the new points of its neighboring faces;(3) Erase the original polyhedron.This is an operation "of order 2" meaning that taking the dual of the dual

of x gives back the original x. For example, take the dual of the octahedronand see that it is a cube and take the dual of the cube and see that is anoctahedron. The relative sizes of the two dual polyhedra can be adjusted asshown here, so that their edges are the same distance from their commoncenter, and so cross through each other.In geometry, polyhedra are associated into pairs called duals, where the

vertices of one correspond to the faces of the other. The dual of the dual isthe original polyhedron. The dual of a polyhedron with equivalent verticesis one with equivalent faces, and of one with equivalent edges is anotherwith equivalent edges. So the regular polyhedra � the Platonic solids andKepler-Poinsot polyhedra, � are arranged into dual pairs. But the samesimple operations are not valid for semi-regular polyhedrons. Because of thecenters of surfaces around a corner is not on the same plane. Thereby wehave to put the semi-regular polyhedra into sphere and make a plane tangentto each hill. The dual polyhedra of a Platonic solid or Archimedean solidcan be also drawn by constructing polyhedra edges tangent to the midspherewhich are perpendicular to the original polyhedra edges.

14 Zeynep Çolak and Zeynep Can

Figure 1 is lightly rotated, showing the edges of rhombic dodecahedron (yellow),octahedron (green) and cube (blue).

To measure the distance between two points in a space commonly Euclid-ean distance is used. Despite of it is so popular and ancient, sometimes itis not practical in the real world. For example taxicab distance is usefulinstead of Euclidean distance in an urban life where streets are planned par-allel or perpendicular to each other, as it is in Manhattan. Especially themaximum distance is a very useful model in the real world and its applica-tions are so much. For example, urban planning at the macro dimensionscan be used for nearly calculations. Minkowski geometry is a non-Euclideangeometry in a �nite number of dimentions. Here the linear structure is sameas the Euclidean one but distance is not uniform in all directions. That is,the points, lines and planes are the same, and the angles are measured inthe same way, but the distance function is di¤erent. Instead of the usualsphere in Euclidean space, the unit ball is a general symmetric convex set[15]. In studies about metric space geometry it had seen that unit spheresof these geometries are associated with convex polyhedra. When a metricis given it is easy to �nd its unit sphere in the related space geometry. In[9 and 4] authors found cuboctahedron and rhombic dodecahedron metricsby a contrary question as if sphere is known how the metric would be foundand by these metrics new geometries are induced.For two points P1 = (x1; y1; z1) and P2 = (x2; y2; z2) in R3, Cuboctahe-

dron and Rhombic Dodecahedron metrics are de�ned as

dCO(P 1; P 2) =max

�jx1 � x2j ; jy1 � y2j ; jz1 � z2j ;

1

2(jx1 � x2j+ jy1 � y2j+ jz1 � z2j)

�and

dRD(P1; P2) = max fjx1 � x2j+ jy1 � y2j ; jx1 � x2j+ jz1 � z2j ; jy1 � y2j+ jz1 � z2jg

respectively. R3CO , R3RD and R3E are denote analytic 3-spaces which arefurnished by Cuboctahedron metric, Rhombic Dodecahedron metric andEuclidean metric, respectively. Linear structure of the R3CO , R3RD are al-most the same of R3E , there is only one di¤erence, this di¤erence is thatits distance function. It is so important to work on notions concerned to


distance in geometric studies, because change of metric can bring out in-teresting results. In this study formulae of distance of a point to a plane,distance of a point to line and distance between two lines in two spaceswhich are induced by two dual convex polyhedra has been researched by anew method which is di¤erent from further calculations done in [1] and [8],and which enables generalizing distance formulae for Minkowski geometries.This convex polyhedra are cuboctahedron which is an Archimedean solidand rhombic dodecahedron a Catalan solid which is the dual of cuboctahe-dron.

2. DISTANCE FORMULAE OF CUBOCTAHEDRON ANDRHOMB·IC DODECAHEDRON

CuboctahedronThe cuboctahedron is a uniform polyhedron bounded by 6 squares and

8 triangles. A cuboctahedron has 14 faces, 12 identical vertices, with twotriangles and two squares meeting at each, and 24 identical edges, each sep-arating a triangle from a square. As such it is a quasiregular polyhedron,i.e. an Archimedean solid, being vertex-transitive and edge-transitive. Itsdual polyhedron is the rhombic dodecahedron. It is edge-uniform, and itstwo kinds of faces alternate around each vertex, so it is also a quasi-regularpolyhedron. It is, also called the heptaparallelohedron or dymaxion. Thedistance between the center of a cuboctahedron and each vertex is equal tothe length of its edges. The regular hexagon and the cuboctahedron are theonly uniform polytopes of their respective dimensions with this property.The cuboctahedron has the Oh octahedral group of symmetries. Accordingto Heron, Archimedes ascribed the cuboctahedron to Plato (Heath 1981;Coxeter 1973, p. 30). The Cartesian coordinates of vertex of the cuboctahe-dron, centered on the origin and having edge length 2, are all permutationsof coordinates and changes of sign of (0; 1; 1):

Figure 2 Cuboctahedron

De�nition 1. Let P1 = (x1; y1; z1) and P2 = (x2; y2; z2) be two points inR3. The cuboctahedron distance between P1 and P2 is

dCO(P1; P2) = max

�jx1 � x2j ; jy1 � y2j ; jz1 � z2j ;

1

2(jx1 � x2j+ jy1 � y2j+ jz1 � z2j)

�In accordance with dCO-metric, the shortest way between points P1 and

P2 is a line segment that is parallel to a coordinate axis or 12 times of sum

of three line segments each of them is parallel to a coordinate axis.


Figure 3 the shortest way between points P1and P2

Lemma 2.1. dCO : R3xR3 ! [0;1) de�ned by

dCO(P1; P2) = max

�jx1 � x2j ; jy1 � y2j ; jz1 � z2j ;

1

2(jx1 � x2j+ jy1 � y2j+ jz1 � z2j)

�where P1 = (x1; y1; z1) and P2 = (x2; y2; z2) forms a metric space, that isR3CO := (R3; dCO)

Lemma 2.2. Let l be a line through the points P1 and P2. If l has directionvector (p; q; r) then

dE(P1; P2)pp2 + q2 + r2

=dCO(P1; P2)

max�jpj ; jqj ; jrj ; 12 (jpj+ jqj+ jrj)

The following theorem is given from without proof [12]

Theorem 2.1. The Cuboctahedron distance is invariant under all transla-tions in analytic 3�space. That is T : R3 ! R3 de�ned by T (x; y; z) =(a+x; y+ b; z+ c) for every a; b; c 2 R does not change the distance betweenany two points in R3CO:

Proposition 2.2. dCO distance of a point P = (xo; yo; zo) to a plane P :Ax+By + Cz +D = 0 is

dCO(P;P) =jAxo +Byo + Czo +Dj

max fjA+Bj ; jA+ Cj ; jB + Cj ; jA�Bj ; jA� Cj ; jB � Cjgwhere it is apparently each of the mentions in the denominators is not zero.

Proof. Set of points at equal distance from a point P is a sphere. Thissphere is Cuboctahedron in our space. If we enlarge this P -centered sphere,then the �rst intersection point of the sphere and the plane P is the closestpoint of plane P to the point P .In the CO-space R3CO, the distance from a point P to a plane P would

be de�ned asdCO(P;P) = min fdCO(P;X) j X 2 Pg

To �nd this distance the Cuboctahedron which center is P would be con-sidered. The sphere consists of equidistant points from the �xed point. SodCO(P;P) = dCO(P;Q) where Q is the intersection point of the plane andthe cuboctahedron while enlarging the cuboctahedron, and Q must be a cor-ner of the cuboctahedron. Thus if we consider lines li, i = 1; 2; 3; 4; 5; 6 pass-ing through P and a corner of the cuboctahedron, each of li has a directionvector which is an element of � = f(1; 1; 0) ; (1; 0; 1) ; (0; 1; 1) ; (1;�1; 0) ;(1; 0;�1) ; (0; 1;�1)g. Let Pi = li \ P; i = 1; 2; 3; 4; 5; 6. So,


dCO(P;P) = min fdCO(P; Pi) : i = 1; 2; 3; 4; 5; 6g. For example let(p; q; r) = (1; 1; 0) be the direction vector l1. Thus, any point on the line l1which is passing through P is

x� xo1

=y � yo1

= t1; z � zo = 0

So coordinates of the any point P1 = (x; y; z) on the line l1 is x = x0 + t1;y = yo + t1; z = z0. Also this point provides plane equation, since it is onthe plane. Thus,

A(x0 + t1) +B(y0 + t1) + C(z0) +D = 0

) t1 =Ax0+By0+Cz0+D

�(�A�B)

Similarly, If (p; q; r) = (1; 0; 1) ; (0; 1; 1) ; (1;�1; 0) ; (1; 0;�1) or (0; 1;�1)then

dCO(P; Pi) =Ax0 +By0 + Cz0 +D

�i

where �2 = A + C, �3 = B + C, �4 = A � B, �5 = A � C, �6 = B � C.Therefore the required distance from the point to the plane is obtained.

Proposition 2.3. CO-distance of a point P = (xo; yo; zo) to a line l givenby

x� ap

=y � bq

=z � cr

is

dCO(P; l) =max f�1; �2; �3; �4; �5; �6g

�

where �1 = jr (x0 � a)� r (y0 � b)� (p� q) (z0 � c)j, �2 = jq (x0 � a)� (p� r) (y0 � b)� q (z0 � c)j,�3 = j(q � r) (x0 � a)� p (y0 � b) + p (z0 � c)j, �4 = jr (x0 � a) + r (y0 � b)� (p+ q) (z0 � c)j,�5 = jq (x0 � a)� (p+ r) (y0 � b) + q (z0 � c)j, �6 = j(q + r) (x0 � a)� p (y0 � b)� p (z0 � c)jand

� =

8>>>>>><>>>>>>:

max fjp� q + rj ; jp� q � rjgmax fjp+ q � rj ; jp� q � rjgmax fjp+ q � rj ; jp� q + rjgmax fjp+ q + rj ; jp+ q � rjgmax fjp+ q + rj ; jp� q + rjgmax fjp+ q + rj ; jp� q � rjg

if �1 � �i , i = 2; 3; 4; 5; 6if �2 � �i , i = 1; 3; 4; 5; 6if �3 � �i , i = 1; 2; 4; 5; 6if �4 � �i , i = 1; 2; 3; 5; 6if �5 � �i , i = 1; 2; 3; 4; 6if �6 � �i , i = 1; 2; 3; 4; 5

Proof. We would consider a cuboctahedron which center is a point P to�nd distance of a point P = (xo; yo; zo) to a line l given by x�a

p = y�bq = z�c

r :

If we enlarge this P -centered sphere, then the �rst intersection point of theline l and the cuboctahedron is the closest point of the line l to the pointP . If we enlarge the radius of the cuboctahedron, then the cuboctahedrondistance between P and l is same with the cuboctahedron distance between


P and the intersection point of the cuboctahedron and the line l.

Figure 4 Intersection of the P-centered cuboctahedron and the line l

For example let�s take the cuboctahedron which center is point P = (xo; yo; zo)and the length of the radius of this cuboctahedron be k when the cubocta-hedron and the line l intersects while enlarging the radius of the cubocta-hedron. So eight of the cuboctahedron�s vertices are P1 = (xo + k; yo; zo +k); P2 = (xo � k; yo; zo + k); P3 = (xo + k; yo � k; zo); P4 = (xo � k; yo �k; zo); P5 = (xo�k; yo; zo�k); P6 = (xo�k; yo+k; zo); P7 = (xo+k; yo+k; zo)and P8 = (xo+k; yo; zo�k). Line l and the cuboctahedron would intersect atan edge that is on a line which direction vector is (1; 1; 0) ; (1; 0; 1) ; (0; 1; 1) ;(1;�1; 0) ; (1; 0;�1)or (0; 1;�1), and passing through P1; P2; P3; P4; P5; P6; P7or P8. For example let�s consider the edge on the line that direction vec-tor is (1; 1; 0) and passing through P2. Hence this edge is on the line x =x0�k+�; y = y0+�; z = z0+k and line l is x = p�+a; y = q�+b; z = r�+c.Now we can �nd the intersection point of line l and the cuboctahedron Sowe obtain k = r(x0�a)�r(y0�b)�(p�q)(z0�c)

p�q+r as the cuboctahedron distance be-tween P and l.Similarly the other cases can be easily found.

Proposition 2.4. CO-distance between any two lines given by

l:::x� ap

=y � bq

=z � cr

= �

l�:::x� a�p

=y � b�q

=z � c�r

= �

in R3CO can be expressed as follows:If l is parallel to l�, then dCO(l; l�) = dCO(A; l�);where A = (a; b; c): If l is

not parallel to l�thenin R3CO can be expressed as follows:If l is parallel to l�, then dCO(l; l�) = dCO(A; l�);where A = (a; b; c): If l is

not parallel to l�then

dCO(l; l�) =j(rq�� r�q) (a� a�) + (pr�� rp�) (b� b�) + (p�q � pq�) (c� c�)jmax fj�+ j ; j�� j ; j�+ �j ; j�� j ; j� + j ; j� � jg

where � = p�q � pq�, � = pr�� p�r and = rq�� r�q.Proof. dCO(l; l�) = min dCO(X;X�) where X 2 l, X�2 l�:If l is parallel to l�then it can be taken as (p; q; r) = (p�; q�; r�) and P = (a�+ �p; b�+ �q; c�+ �r)


for any point P on l�, without loss of generality. Then it can be eas-ily computed using the formula given by proposition 2 that dCO(l; P ) =dCO(A; l) where A = (a; b; c):If l is not parallel to l�; then at least one ofpq�� qp�; qr�� rq�; rp�� pr�is not zero. Otherwise these lines would be parallelto each other. Let�s consider the points P = (p�+ a; q�+ b; r�+ c) and P�=(p��+ a�; q��+ b�; r��+ c�) which are on l and l�respectively. Thus if dCO(P; P�)is minimum then direction vector of the line l��which passes through P and P�is an element of f(1; 1; 0) ; (1; 0; 1) ; (0; 1; 1) ; (1;�1; 0) ; (1; 0;�1) ; (0; 1;�1)g.For example let directon vector of l��be (p��; q��; r��) = (1; 1; 0). Since

p�+ a� p�� a�1

=q�+ b� q�� b�

1=r�+ c� r�� c�

0

then

� =r (a� a�)� r (b� b�)� (p� q) (c� c�)

r (p�� q�)� r�(p� q)

and

� =r�(a� a�) + r�(b� b�)� (p�� q�) (c� c�)

r (p�� q�)� r�(p� q)

So

dCO(P; P�) = max

�jp�+ a� p�� a�j ; jq�+ b� q�� b�j ; jr�+ c� r�� c�j ;

12 (jp�+ a� p�� a�j+ jq�+ b� q�� b�j+ jr�+ c� r�� c�j)

�=

�� (rq��r�q)(a�a�)+(pr��rp�)(b�b�)+(p�q�pq�)(c�c�)(p�q�pq�)+(rq��r�q)

��Other cases can be given by the similiar way. And as a subcase if (p�; q�; r�) isan element of f(1; 0; 0) ; (0; 1; 0) ; (0; 0; 1)g then

dCO(l; l�) =j(rq�� r�q) (a� a�) + (pr�� rp�) (b� b�) + (p�q � pq�) (c� c�)j

max fjp�q � pq�j ; jpr�� rp�j ; jrq�� r�qjg

Rhombic DodecahedronThe rhombic dodecahedron is a very interesting polyhedron. It has 12

faces, 14 vertices, 24 sides or edges. Its sometimes also called the rhomboidaldodecahedron. The rhombic dodecahedron can be built up by a placing sixcubes on the faces of a seventh.The rhombic dodecahedron is a zonohedronand a space-�lling polyhedron (Steinhaus 1999, p. 185). The vertices aregiven by (+/-1, +/-1, +/-1), (+/-2, 0, 0), (0, +/-2, 0), (0, 0, +/-2).


Figure 5 Rhombic Dodecahedron

De�nition 2. Let P1 = (x1; y1; z1) and P2 = (x2; y2; z2) be two points inR3. The rhombic dodecahedron distance between two points P1 and P2 isdRD(P1; P2) = max fjx1 � x2j+ jy1 � y2j ; jx1 � x2j+ jz1 � z2j ; jy1 � y2j+ jz1 � z2jgIn accordance with dRD-metric, the shortest way between points P1 and

P2 is union of two line segments that each one is parallel to a coordinateaxis

Figure 6 the shortest way between points P1and P2

Lemma 2.3. Let l be a line through the points P1 and P2. If l has directionvector (p; q; r) then it is easy to see that

dE(P1; P2)pp2 + q2 + r2

=dRD(P1; P2)

max fjpj+ jqj ; jpj+ jrj ; jqj+ jrjgThe following theorem is given from without proof [14]:

Lemma 2.4. Let l be a line through the points P1 and P2. If l has directionvector (p; q; r) then it is easy to see that

dE(P1; P2)pp2 + q2 + r2

=dRD(P1; P2)

max fjpj+ jqj ; jpj+ jrj ; jqj+ jrjgThe following theorem is given from without proof [14]:

Theorem 2.5. The Rhombic dodecahedron distance is invariant under alltranslations in analytic 3�space. That is

T : R3 ! R3 � T (x; y; z) = (a+ x; y + b; z + c); a; b; c 2 Rdoes not change the distance between any two points in R3RD:


Proposition 2.6. RD� distance of a point P = (xo; yo; zo) to a planeP : Ax+By + Cz +D = 0 is

dRD(P;P) =jAxo +Byo + Czo +Dj

max fjAj ; jBj ; jCj ; jA+B + Cj ; jA+B � Cj ; jA�B + Cj ; jA�B � Cjgwhere it is apparently each of the mentions in the denominators is not zero.

Proof. In our space sphere is Rhombic Dodecahedron. The similar way toCuboctahedron; if we enlarge this P -centered sphere, then the �rst intersec-tion point of the plane P is the closest point of P plane to the point P . Inthe RD-space R3RD, the distance from a point P to a plane P is de�ned as

dRD(P;P) = min fdRD(P;X) j X 2 PgThe rhombic dodecahedron is the sphere of the rhombic dodecahedron spacegeometry and the sphere consists of equidistant points from the �xed point.So dRD(P;P) = dRD(P;Q) where Q is the intersection point of the planeand the rhombic dodecahedron while enlarging the rhombic dodecahedronand Q must be a corner of the rhombic dodecahedron. Thus if we considerlines li, i = 1; 2; 3; 4; 5; 6; 7 passing through P and a corner of the rhom-bic dodecahedron, each of li has a direction vector which is an element of� = f(1; 0; 0) ; (0; 1; 0) ; (0; 0; 1) ; (1; 1; 1) ; (1; 1;�1) ; (1;�1; 1) ; (1;�1;�1)g.Let Pi = li\P; i = 1; 2; 3; 4; 5; 6; 7 So, dRD(P;P) = min fdRD(P; Pi) : i = 1; 2; 3; 4; 5; 6; 7g.For example let (p; q; r) = (1; 1; 1) be the direction vector of l1. Thus, anypoint on the line l1 which is passing through P is

x� xo1

=y � yo1

=z � zo1

= t1

So coordinates of the any point P1 = (x; y; z) on the line l1 is x = x0+t1; y =yo + t1; z = z0 + t1. Also this point provides plane equation, since it is onthe plane. Thus,

A(x0 + t1) +B(y0 + t1) + C(z0 + t1) +D = 0

) t1 =�(Ax0+By0+Cz0+D)

A+B+C

Similarly, If (p; q; r) = (1; 0; 0) ; (0; 1; 0) ; (0; 0; 1) ; (1; 1;�1) ; (1;�1; 1) or (1;�1;�1)then

dRD(P; Pi) =Ax0 +By0 + Cz0 +D

�iwhere �2 = A, �3 = B, �4 = C, �5 = A + B � C, �6 = A � B + C,�7 = A�B�C. Therefore the required formula for distance from the pointto the plane is obtained.RD� distance of a point P = (xo; yo; zo) to a planeP : Ax+By + Cz +D = 0 is


max fjAj ; jBj ; jCj ; jA+B + Cj ; jA+B � Cj ; jA�B + Cj ; jA�B � Cjgwhere it is apparently each of the mentions in the denominators is notzero.RD� distance of a point P = (xo; yo; zo) to a plane P : Ax + By +Cz +D = 0 is


max fjAj ; jBj ; jCj ; jA+B + Cj ; jA+B � Cj ; jA�B + Cj ; jA�B � Cjgwhere it is apparently each of the mentions in the denominators is not zero.


Proposition 2.7. RD-distance of a point P = (xo; yo; zo) to a line l givenby

x� ap

=y � bq

=z � cr

is

dCO(P; l) =max f�1; �2; �3; �4; �5; �6g

�where �1 = j(q + r) (x0 � a)� (p+ r) (y0 � b)� (p� q) (z0 � c)j,�2 = j(q � r) (x0 � a)� (p+ r) (y0 � b) + (p+ q) (z0 � c)j,�3 = j(q + r) (x0 � a)� (p� r) (y0 � b)� (p+ q) (z0 � c)j,�4 = j(q � r) (x0 � a)� (p� r) (y0 � b) + (p� q) (z0 � c)j and

� =

8>><>>:max fjp� qj ; jp+ rj ; jq + rjgmax fjp+ qj ; jp+ rj ; jq � rjgmax fjp+ qj ; jp� rj ; jq + rjgmax fjp� qj ; jp� rj ; jq � rjg

if �1 � �i , i = 2; 3; 4if �2 � �i , i = 1; 3; 4;if �3 � �i , i = 1; 2; 4if �4 � �i , i = 1; 2; 3

Proof. If we enlarge this P -centered sphere, then the �rst intersectionpoint of the line l and the sphere is the closest point of line l to the pointP . We would consider a rhombic dodecahedron which center is point aP to �nd distance of a point P = (xo; yo; zo) to a line l given by x�a

p =y�bq = z�c

r : If we enlarge the radius of the rhombic dodecahedron, then therhombic dodecahedron distance between P and l is same with the rhombicdodecahedron distance between P and the intersection point of the rhombicdodecahedron and the line l.

Figure 7 Intersection of the P � centered rhombic dodecahedron and the line lFor example let�s take the rhombic dodecahedron which center is the pointP = (xo; yo; zo) and the length of the radius of this rhombic dodecahedron bek when the rhombic dodecahedron and the line l intersects while enlargingthe radius of the rhombic dodecahedron. So six of the rhombic dodecahe-dron�s vertices are P1 = (xo; yo; zo + k); P2 = (xo; yo + k; zo);P3 = (xo; yo � k; zo); P4 = (xo + k; yo; zo); P5 = (xo � k; yo; zo) andP6 = (xo; yo; zo�k). Line l and the rhombic dodecahedron would intersect atan edge that is on a line which direction vector is (1; 1; 1) ; (1; 1;�1) ; (1;�1; 1)or (�1; 1; 1), and passing through P1; P2; P3; P4; P5 or P6. For example let�sconsider the edge on the line that direction vector is (1; 1;�1) and passingthrough P1. Hence this edge is on the line x = x0+�; y = y0+�; z = z0+k��and line l is x = p� + a; y = q� + b; z = r� + c. Now we can �nd the


intersection point of line l and the rhombic dodecahedron So we obtaink = (q+r)(x0�a)�(p+r)(y0�b)�(p�q)(z0�c)

p�q as the rhombic dodecahedron distancebetween P and l.Similarly the other cases can be easily found.

Proposition 2.8. RD-distance between any two lines given by

l:::x� ap

=y � bq

=z � cr

= �

l�:::x� a�p

=y � b�q

=z � c�r

= �

in R3RD can be expressed as follows:If l is parallel to l�, then dRD(l; l�) = dRD(A; l�);where A = (a; b; c): If l is

not parallel to l�then

dRD(l; l�) =j(rq�� r�q) (a� a�) + (pr�� rp�) (b� b�) + (p�q � pq�) (c� c�)jmax

nj�j ; j�j ; j j ; j�+�+ j2 ; j��+�+ j2 ; j��+ j2 ; j�+�� j2

owhere � = p�q � pq�, � = pr�� p�r and = rq�� r�q.Proof. dRD(l; l�) = min dRD(X;X�) where X 2 l, X�2 l�:If l is parallel to l�then it can be taken as (p; q; r) = (p�; q�; r�) and P = (a�+ �p; b�+ �q; c�+ �r)for any point P on l�, without loss of generality. Then it can be eas-ily computed using the formula given by proposition 2 that dRD(l; P ) =dRD(A; l) where A = (a; b; c):If l is not parallel to l�; then at least one ofpq�� qp�; qr�� rq�; rp�� pr�is not zero. Otherwise these lines would be parallelto each other. Let�s consider the points P = (p�+ a; q�+ b; r�+ c) and P�=(p��+ a�; q��+ b�; r��+ c�) which are on l and l�respectively. Thus if dRD(P; P�)is minimum then direction vector of the line l��which passes through P and P�is an element of f(1; 0; 0) ; (0; 1; 0) ; (0; 0; 1) ; (1; 1; 1) ; (�1; 1; 1) ; (1;�1; 1) ; (1; 1;�1)g.For example let directon vector of l��be (p��; q��; r��) = (1; 1; 1). Since

p�+ a� p�� a�1

=q�+ b� q�� b�

1=r�+ c� r�� c�

1then

� =(q � r) (a� a�) + (r � p) (b� b�) + (p� q) (c� c�)

(p�q � pq�) + (pr�� p�r) + (rq�� qr�)and

� =(r�+ q�) (a� a�) + (r�� p�) (b� b�) + (p�� q�) (c� c�)

(p�q � pq�) + (pr�� p�r) + (rq�� qr�)So

dRD(P; P�) = max

8<: jp�+ a� p�� a�j+ jq�+ b� q�� b�j ;jp�+ a� p�� a�j+ jr�+ c� r�� c�j ;jq�+ b� q�� b�j+ jr�+ c� r�� c�j

9=;= 2

�� (rq��r�q)(a�a�)+(pr��rp�)(b�b�)+(p�q�pq�)(c�c�)(p�q�pq�)+(pr��p�r)+(rq��r�q)

��Other cases can be given by the similiar way.


References

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[2] Coxeter, H. S. M., 1961, Introduction to Geometry, John Wiley&Sons Inst., 443p.[3] Cromwell, Peter R., Polyhedra, Cambridge University Press,1-7p, 1997.[4] Erç¬nar G.Z., 2015, Birim Küresi Rhombic Dodecahedron Olan Metri¼gin Arast¬r¬l-

mas¬, Master Thesis, Eskisehir Osmangazi University.[5] Ermis, T. and Kaya, R. �On the Isometries of 3 Dimensional Maximum Spaces,�

Konuralp Journal of Mathematics, vol. 3, no. 1, pp. 103�114, 2015.[6] Eugène Catalan Mémoire sur la Théorie des Polyèdres. J. l�École Polytechnique

(Paris) 41, 1-71, 1865.[7] Gelisgen, Ö. and Kaya, R., 2006, On ��Distance in Three Dimensional Space, Ap-

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[9] Gelisgen Ö., Can Z., On The Family of Metrics for Some Platonic and ArchimedeanPolyhedra, "Konuralp Journal of Mathematics ", 4, (2016), s.25-33.

[10] George E. Martin: The fundation of Geometry and Non-Euclidean Plane, Springer,1998.

[11] Milmann, R. S. and Parker, G. D., 1991, Geometry a Metric Approach with Models,Springer, 370p.

[12] Salihova, S. , On The Geometry Of Maximum Metric , ESOGÜ, PhD Thesis, 2006.[13] Senechal, M. and Fleck, G., Shaping Space, Birkhäuser, Boston, 1988,[14] Tian, S., 2005, Alpha Distance-A Generalization of Chinese Checker Distance and

Taxicab Distance, Missouri Journal of Mathematical Sciences, 17, 1, 35-40.[15] Thompson, A.C., Minkowski Geometry, Cambridge University Press, 346p, 1996.[16] http://mathworld.wolfram.com/Cuboctahedron.[17] html.http://mathworld.wolfram.com/RhombicDodecahedron.html.[18] https://en.calameo.com/read/003359034cdaca7f0d9a7.

DEPARTMENT OF ECONOMETRY,CANAKKALE UNIVERSITY,17270, CANAKKALE, TURKEY.E-mail address: [email protected]

DEPARTMENT OF MATHEMATICS FACULTY OFSC·IENCE AND LETTERS, AKSARAY UNIVERSITY,68000, AKSARAY, TURKEY.E-mail address: [email protected]

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SOME DISTANCE PROPERTIES OF TWO SPACES ...2018/10/12 · Some distance properties of two spaces induced by dual convex polyhedra 13 A polyhedron is called convex if a line segment