AVolumePreservingMapfromCubetoOctahedron

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1 Globl Journl of Science Frontier Reserch: F Mthemtics nd Decision Sciences Volume 18 Issue 1 Version 1.0 er 018 Type: Double Blind Peer Reviewed Interntionl Reserch Journl Publisher: Globl Journls Online ISSN: & Print ISSN: A Volume Preserving Mp from Cube to Octhedron By Adrin Holhoș Universitte Tehnică din Cluj-Npoc Abstrct- Using simple geometric resoning we deduce volume preserving mp from the cube to the octhedron. GJSFR-F Clssifiction: MSC 010: 00A69 AVolumePreservingMpfromCubetoOcthedron Strictly s per the complince nd regultions of: 018. Adrin Holhoș. This is reserch/review pper, distributed under the terms of the Cretive Commons Attribution- Noncommercil 3.0 Unported License permitting ll non commercil use, distribution, nd reproduction in ny medium, provided the originl work is properly cited.

2 A Volume Preserving Mp from Cube to Octhedron Adrin Holhoș Abstrct- Using simple geometric resoning we deduce volume preserving mp from the cube to the octhedron. I. Preliminries Consider the cube C = [ 1, 1] 3 centered t the origin O nd the regulr octhedron K of the sme volume, centered t O nd with vertices on the coordinte xes K = { (x, y, z) R 3, x + y + z }. Let L denote the edge of K. Since the volume of the octhedron K is L 3 /3, nd this is equl to the volume of the cube C, we hve 8 = L 3 /3. Then, the distnce from the origin to ech vertex of K is = L/ = 3 6. (1) We will construct mp U : C K which preserves the volume, i.e. V olume(d) = V olume(u(d)), for ll D C, where V olume(d) denotes the volume of domin D. For n rbitrry point (x, y, z) C we denote (,, ) = U(x, y, z) K Author: Universitte Tehnică din Cluj-Npoc. e-mil: drin.holhos@mth.utcluj.ro

3 A Volume Preserving Mp from Cube to Octhedron 44 1 Figure 1: In the left, the cube C in blck nd the little cube C 1 from the positive octnt in red. In the right, the octhedron K in blck nd the tetrhedron K 1 in red For the construction of U, we split the cube into eight congruent cubes seprted by the coordinte plnes O, O nd O, nd thus, the construction of U cn be reduced to the construction of its restriction to one of these cubes. We will denote C 1 the eight prt of C situted in the positive octnt. We will denote by K 1 the prt of K situted in the positive octnt. The mp U will be constructed in such wy tht C 1 will be mpped in K 1 nd ll the other seven cubes of C will be mpped to the corresponding tetrhedrons of K. II. Construction of the Volume Preserving Mp U We focus on the region C 1 of C situted in the positive octnt I + 0 = {(x, y, z) R 3, x 0, y 0, z 0}, nd we denote the vertices of the cube C 1 s follows: A = (1, 0, 0), B = (1, 1, 0), C = (0, 1, 0), D = (0, 1, 1), E = (0, 0, 1), F = (1, 0, 1) nd G = (1, 1, 1), see Figure (left). We lso consider the following points in K 1 = K I 0 : A = (, 0, 0), B = (/, /, 0), C = (0,, 0), D = (0, /, /), A F E O G B D C A F E O G B D C Figure : The cubicl region C 1 nd its imge K 1

4 E = (0, 0, ), F = (/, 0, /) nd G = (/3, /3, /3), see Figure (right). We split the region C 1 into six tetrhedrons of equl volume: OABG = {(x, y, z) I 0, 1 x y z 0}, OBCG = {(x, y, z) I 0, 1 y x z 0}, OCDG = {(x, y, z) I 0, 1 y z x 0}, ODEG = {(x, y, z) I 0, 1 z y x 0}, OEF G = {(x, y, z) I 0, 1 z x y 0}, OF AG = {(x, y, z) I 0, 1 x z y 0}. They will be mpped onto the tetrhedrons OA B G, OB C G, OC D G, OD E G, OE F G nd OF A G, respectively. We focus on the region OABG nd the corresponding region OA B G OA B G = {(,, ) R 3 0, + + }. Consider point M(x, y, z) in OABG nd the corresponding point M (,, ) through the mp U. Consider the plne π 1 through M prlel with the plne ABG nd P the point of intersection of π 1 with OA. Suppose this plne is mpped in the plne π 1 through M prlel with A B G nd P is the corresponding point on OA, the intersection of the plne π 1 with OA. The rtio of the volumes of the two pyrmids with the sme vertex O nd the bses on the plnes π 1 nd ABG must be equl with the rtio of the volumes of the two pyrmids with the sme vertex O nd the bses on the plnes π 1 nd A B G nd equl with the cube of the rtio OP/OA nd equl with the cube of the rtio OP /OA. We obtin which is equivlent with OP = OA OP, + + = x. () Consider the plne π through M prlel with the plne OBG nd Q the point of intersection of π with OA. Suppose this plne is mpped in the plne π through M prlel with OB G nd Q is the corresponding point on OA, the intersection of the plne π with OA. The rtio of the volumes of the two pyrmids with the sme vertex A nd the bses on the plnes π nd OBG must be equl with the rtio of the volumes of the two pyrmids with the sme vertex A nd the bses on the plnes π nd OB G nd equl with the cube of the rtio AQ/AO nd equl with the cube of the rtio A Q /A O. We obtin A Q = OA AQ, which is equivlent with OQ = OQ. We obtin = (x y). (3) 45 1

5 Consider the plne π 3 through M prlel with the plne OAG nd R the point of intersection of π 3 with AB. Suppose this plne is mpped in the plne π 3 through M prlel with OA G nd R is the corresponding point on A B, the intersection of the plne π 3 with A B. The rtio of the volumes of the two pyrmids with the sme vertex B nd the bses on the plnes π 3 nd OAG must be equl with the rtio of the volumes of the two pyrmids with the sme vertex B nd the bses on the plnes π 3 nd OA G nd equl with the cube of the rtio BR/BA nd equl with the cube of the rtio B R /B A. We obtin 46 1 B R = A B BR, which is equivlent with A R = AR. We obtin = (y z) = (y z). (4) Solving the system of the three equtions (), (3) nd (4), we obtin = x y 6 z = = where the vlue of is specified by (1). y 6 z 3 z, More generl, the equtions for ll eight octnts cn be obtin in the following wy: let M = mx( x, y, z ) m = min( x, y, z ) M = mx(,, ) m = min(,, ) c = x + y + z M m c = + + M m. Using the sme resoning we obtin the following system of equtions + + = M M c = (M c) c m = (c m) with the solution M = M c 6 m c = m = c 6 m 3 m.

6 III. Applictions Ref. A. Holhoș, D. Roșc (017), Are preserving mps nd volume preserving mps between clss of polyhedrons nd sphere, Adv. Comput. Mth., 43, 677{697. A volume preserving mp cn be useful in some sttistics pplictions nd for decomposition of 3D solid into smller elements of equl volume. For exmple, if we strt with uniform distribution of points in the solid cube we cn obtin uniform distribution of points in the solid octhedron (see [1] nd the references therein, for similr ppliction in the cse of plnr domins). For cube is esy to obtin uniform grid, grid with ll the elements hving the sme volume. By pplying the mp U we get uniform grid for the octhedron. Using the mp constructed in [] we obtin uniform grid for the bll. This method to obtin uniform grid in the bll is different thn the method described in [3]. References Références Referencis 1. Holhoș (017), Two Are Preserving Mps from the Squre to the p-bll, Mth. Model. Anl.,, 157{166.. A. Holhoș, D. Roșc (017), Are preserving mps nd volume preserving mps between clss of polyhedrons nd sphere, Adv. Comput. Mth., 43, 677{ D. Roșc, A. Morwiec nd M. De Gref (014), A new method of constructing grid in the spce of 3D rottions nd its pplictions to texture nlysis, Modelling Simul. Mter. Sci. Eng., (17pp). 47 1

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