Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. ei White Denis gnSaeUiest,Corvallis, University, Labora- State Research egon Environmental Agency Protection gen- assess ronmental to is program 1991). monitoring a Overton such and of Linthurst, goal (Messer, The re- program is monitoring (EPA) Agency Protection Environmental U.S. The assess- in focus wide-area a for need the reemphasize ple, spoesro ttsisi h eateto ttsis Oregon Statistics, of Department the in statistics of professor is States United the across environment the of health the ecological the of States erally long-term a United knowledge planning the by sound that and environment view the of shared comprehensive state widely more a needs to sponding model. global priate approaches several analyze first and we review paper we this Then In concerns. environment. the of quality the ing tt nvriy Corvallis, University, State is tory, appro- an of a of surface the the properties on implement the placed to describe grid used we be sampling Finally could these regular that to design. envi- geometry statistical responsive global is designed a that to statistically a program of monitoring aspects salient ronmental the discuss rfso fgorpyi h eateto esine,Or- Geosciences, of Department the in geography of professor 0 .W. S. 200 T ehat fteevrneti niseo global, of issue an is environment the of health he neeg rdcinadciaecag,frexam- for change, trends climate Recent and concern. local production and energy in regional, national, niomna oioigt Meet to Monitoring Environmental atgahcadGoercCmoet faGlobal a of Components Geometric and Cartographic apigdsg ae pnasseai rdcnaeutl sestecniino aytpso resources of types many of condition the assess adequately can grid systematic a upon based design sampling A n fteshrclPaoi ois n esdsoto nsaeoe h xeto aewe sdfra for used when face a of extent the over ways. shape several in in addressed distortion less and solids, Platonic spherical the of any apig h emtyo h raglrgi rvdsfrvrigtedniy n onso h rdcnbe can grid the on points for and density density, appropriate the an at the varying grid covers for icosahedron triangular provides a grid truncated into the triangular decomposed of the be face can of hexagon and geometry A The States, projection. United sampling. azimuthal Lambert conterminous the entire by surface projection cshdo,heacia rdgeometry. grid hierarchical icosahedron, KEYWORDS: necessary ABSTRACT. n eanfeiiiyfradesn e susa hyaie h admzto fti rdrqie that requires grid this of approaches existing of randomization review The After arise. earth. the they of as surface the issues on projected new when cells addressing equal-area retain for and regular flexibility retain and otegoe hsgoercmdlhsls eito nae hnsbiie sashrcltselto than tessellation spherical a as subdivided when area in deviation less has model geometric This globe. the to ocntutn eua udvsoso h at' ufc,w rps h eeomn ftesmln rdon grid sampling the of development surface the earth's propose the we of surface, projection earth's equal~areamap the of azimuthal Lambert subdivisions the regular constructing to sarsac egahrwt E! ..Envi- U.S. MET!, with geographer research a is 5hSre,Corvallis, Street, 35th apigDsg o niomna Monitoring Environmental for Design Sampling atgah n egahcIfrainSystems, Information Geographic and Cartography ei ht,A o ieln,adW ct Overton Scott W. and Kimerling, Jon A. White, Denis to ayNeeds Many Introduction rvd siae ftesau f n hne rted n h odto feooia resources. ecological of condition the in, trends or changes and of, status the of estimates provide opeesv niomna oioigpormbsdo on ttsia einis design statistical sound a on based program monitoring environmental comprehensive A lblsmln ein apiggi,mppoetos oyerltselto,truncated tessellation, polyhedral projections, map grid, sampling design, sampling global R97331. OR R931 W. 97331. OR R97333. OR A. ct Overton Scott o Kimerling Jon apigdsg o igersuc.Tedsg should design The resource. single a for design sampling , ilgclrsos osrs.Ti betv rsrbssome prescribes of objective This indicators stress. to among response association biological the determining is jective ilb ape.Tenihoho fec on ilbe collec- will A point criteria. each land-use of and ecological neighborhood by based The is characterized objectives sampled. these be meet will to chosen strategy design The confidence. of degrees known with ino apiguiso ahrsuc yei h neigh- the in type resource each of units type sampling of resource tion ecological each which at points of grid a upon time. and space in sampling in coincidence ob- of degree additional important biolog- An broad for various States. of health its United the of status, of the in indicators of description trends physical regions or and the are chemical, changes ical, types and resource status, ecological accommodate vidual to flexible, popu- be also well-defined but for conventional samples resources, a of in probability be lations explicit would to lead there as available, frame pling ore fevrnetlsrs,epsr osrs,and stress, to exposure stress, environmental of sources re- ecological stresses. of kinds environmental of different kinds emerging many and of sources sampling the sam- explicit and and an lakes grasslands, having and forests, without of deserts ecosystems) types estuaries, as agricultural wetlands, (such streams, resource environment the ecological of the of condition estimates the comprehensive in, sound a entire trends obtain the upon to and possibly of, based order and be in status will design America, program statistical North of The parts globe. other and htcnsml n ptal itiue n identifiable and distributed spatially any sample can that pcfcojcie ftemntrn rga o indi- for program monitoring the of objectives Specific design a requires program monitoring the Developing o.1,N.,1992, No.1, 19, Vol. to ttsia einStrategy Design Statistical h aeo rnae icosahedron truncated a of face the pp. 5-22 it fit be Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. onsi o etitdt h eore dniidapriori. a identified ~esources the to restricted not is double the points infor- in the data the of to relevance sample the that on and data sample the 1 in Tier mation the of ness well-being. be to sample, 1 Tier a constitute will point each of borhood grid. response in time later a at sampled be may is resources strategy Other design this of advantage major a But sample. visual samples, water of analyses chemical example, for the of attributes other of sampling field for used be will from ple obtainable measures and geophysical pattern, or landscape geometric sur- shape, other their dimension, units, linear resource of area, face numbers the include properties ovnetycvrtecnemnu ntdSae,adbe and States, United conterminous the cover cyclical conveniently regular implementing for useful are sam- a density in increased the grid for ing the useful of are density pattern features. the any hierarchical with landscape enhancing regular aligned for physical be methods or not Also, cultural it that spaced and regularly units, the sampling criteria select achieves Additional best criteria. projection equal-area map and equal-area randomization an of plane sampled, being location. of its probability of same sampling regardless any the the that has assures preserve area also fixed-size to property equal-area required of The is projection probabilities. the projection by map followed area plane, a in trans- grid most random The the single of a randomly. lation by positioned is be randomization requires grid straightforward sample sampling probability the a that of the of specification of properties The set geometric a design. and to cartographic lead the strategy for criteria design and objectives program The design sampling primary the The is capability adaptive health. this their implement to regarding device issue grid new a the to around areas of characterization 1 Tier the that extensive- the from part large in derive sample double environmental the of indicators other composi- and species wetlands, forests, of to tion damage foliage of include, will symptoms measurements Field design. consti- the 1977)will of 2 (Cochran Tier sample tute double sam- This 1 Tier resources. this of subsample A imagery. sensed remotely structural The types. regional these the of of properties populations structural national the and estimate to used rahst hspolm ic ehv pcfe nequal- an specified have we Since problem. this to proaches an·appropriately select the to of important areas is large it Over Therefore, large. both. or distortion shape or to area potentially, and, world. country the of the of parts can parts other that other system to grid extended a want we Finally, time. reduc- in for sampling methods and resources, ecological rare of pling to which within areas compact provide grid the that are equal- An surface. earth's the onto grid randomized the 6 implement to which on earth the of piece shaped and quite sized become can distortions these of magnitudes the earth, h apiggi.W eiwblwanme fap- of number a below review We grid. sampling the h fiin ttsia rpriso siae aefrom made estimates of properties statistical efficient The hs eua ytmtcgi fsmln onsi the in points sampling of grid systematic regular a Thus, rjcigfo h at' ufc oapaerslsin results plane a to surface earth's the from Projecting ainl o atgahcadGeometric and Cartographic for Rationale rpriso Design of Properties eto noasur ya rapeevn udaue and, quadrature, preserving area an by square a into jection Lambert the proposed (1986) Chen and Tobler vision. eaint aiueadlniue n qa-rasubdivi- equal-area and longitude, and latitude to hierarchical relation regular principles: several satisfy to attempts co- map-projection Existing Approaches. Map-Projection systems. subdivision adaptive the of surface the subdividing to approaches of number A oth- in distortions area analyze and review to (or useful also shape is with is concern primary the projection, map area nevl n h hrzna"gi ie r aall spaced parallels are lines grid "horizontal" the and intervals "vertical" The result. the upon grid square a placing finally, rec- the transforming then projection, cylindrical Lambert's direct systems, other to conversion easy structuring, data quadtree a for basis the as projection subdi- equal-area cylindrical for frameworks potential provide systems ordinate and as tessellations, classified be polyhedral can analysis data approaches, approaches global The or map-projection area suggested. large been for way have regular a in earth approaches. attractive it erwise However, surface. projection the across distortion scale) eaern(ue,othdo,ddchdo,adico- and dodecahedron, octahedron, (cube), hexahedron to approaches of set Another Tessellations. Polyhedral zones. the of edges lesser the It and at regions, pieces. polar discontinuities equal-area the in discontinuitiesin of surface major has disadvantage earth's also our major the ad- the For representing tree has quad not system representation. linear UTM-based the as computer use, for implemented into convenient be patch dresses then each can and dresses patches, of grid square subdivided regular further a then is into subzone cell Each necessary.) ultimate be the also upon square are (Depending that plane. subzones projection south the to in the north divide into to Mer- propose 60UTMzones Transverse They Universal system. the (UTM)coordinate cator upon the the based system at of poles. storage the square shape at nearly triangles in from isosceles acute earth, variation to the equator the disadvan- on application: cells major au- one subdivision our the has for meeting system While tage this function. requirements, regular sine thors' at the spaced to meridians proportional are system this of lines grid pro- cylindrical the in from earth derived the representation representing tangular first by achieved is This sions. solution Their system. storage and 1984)referencing (Samet utraytinua eh(T) h caernis octahedron The called (QTM). system a mesh for triangular basis world the pro- a quaternary as has of octahedron a basis example, an the for using (1989), as posed Dutton sphere These a system. 1978). in subdivision Pearce inscribed be and can Pearce solids 1948; (Coxeter tetrahedron, sahedron the polyhedra, three-dimen- regular five Pla- of the The Greece. are ancient solids knowledge tonic in developed uses first geometry, world sional the subdividing ad- cell and Patch storage. data for cells of grid square a might subzones east-west into division base, data the of size akadLuo 18)pooe nadesn and addressing an proposed (1985) Lauzon and Mark atgah n egahcIfrainSystems Information Geographic and Cartography eiwo lentv Geometries Alternative of Review lblFramework Global Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. h iihe ewr sas nw steVrnior Voronoi the as known also is network Dirichlet The great single the from displaced slightly vertex a at joined and latitude the of p'oints cardinal the with aligned initially atgah n egahcIfrainSystems Information Geographic and Cartography are loci whose polygons net- into This 1985). spheroid Fingleton the separates and work (Upton network Thiessen areal of surface a tessellates and of hybrid earth a the upon on or objects earth, of the to decompositions fit upon map or polyhedra of plane, regular a ap- decompositions to of earth regular preceding the of upon two projections based The are Systems. proaches Subdivision following Adaptive the (see tessellation octahedral the analysis). by duced sub- recursively and icosahedron, the of vertices the ver- with whose triangles as modeled are surfaces spherical ued Dutton interest. of features the resolve to enough small is 90, - the 90, of face 0, at triangular Each equator longitude. the are of along vertices degrees six and 180 the poles and is, the at that placed system; coordinate longitude h ewr.Teol eso onskont eutin result to known points defined of that sets points only original of The set network. the the to closest points all spheroid. a on network Dirichlet the (1987)describes creating Lukatela for system property. a some upon based units recur- equal-areacells, with produce tessellation not icosahedral subdivisiondoes triangular the sive QTM, with As recursive ing. fourfold The minimum. prescribed a location) resolving below in necessarily is (not values surface the ing Single-val- sphere. the on distributions data randomization representing simple a of use in the irregularities small prevent the which is application shape, our for advantage equal create to as so midpoints, the connecting arcs arc circle circle great equal-length two with midpoints edge ing do- the of pentagon each subdivide initially They work. are thus property. and cells, equal-area into subdivide not do UTM, vertices, edges, between distance the the of or faces, midpoints the of the size at the subtri- vertices until four placing into by subdivided angles recursively then is octahedron hs.A lentv prahsat ihagvnpattern given a with starts approach alternative An these. pro- that than less is variation shape and area the address- though and storage data quadtree, for planar quadtree, the sphere of the adaptation called an allows subdivision represent- in error the until models QTM initiated the is in as model divided The sphere. the onto projected are tices dis- scheme. primary The sphere. maintains the system on this Thus, property equal-area shape. the in figures, six-sided triangular and to four- close are created polygons The areas. connect- by obtained is pseudo-tri- subdivision four equal-area into An triangles angles. then these and subdivide triangles, net- recursively spherical isosceles five sampling into global decahedron a creating for (1974) Edvarson and equal-area the specifying design sampling like a a for systems, suitable provides QTM not The and system. level, coordinate QTM any longitude the and at and latitude between subtriangles conversion straightforward four system numbering the different a for (1989),uses Good- Shiren by and proposed QTM, child 21levels on after variation A obtained decomposition. is of resolution I-meter that calculates eee(90hsue h cshdo samdlfor model a as icosahedron the used (1990)has Fekete Elvers, Wickman, by used been has dodecahedron The esmlryaayetetuctdicosahedron. truncated the analyze similarly We ln.W upr u rpslwt naayi fthe of analysis an with proposal our is support propose We we plane. projection map The model. this of hexagon faces States); United conterminous the to including enough America, large faces objectives: following polyhedra the semiregular meeting and in regular the among compromise iey a rjcin pntefcso hs polyhedra. these of alterna- faces the and, upon polyhedra projections map regular tively, of tessellations a spherical of tangent center a with and projection equal-area vertices azimuthal the Lambert decompo- the by regular formed a as grid defined the be of then sition (the systematic can The familiarity grid soccerballs). of construct sampling (hex- degree to used shapes often face and.some similar pattern pentagon); projections and and map compact agon in percent; 2 distortions scale North about to keep central to as enough small (such subcontinents significant cover central a best the about achieves solid pen- This symmetry 12 five-way pentagons. and two a connecting hexagons axis in 20 with arranged tagons polyhedron 32-face a ron, for adaptations possible with but equal- polyhedron, an chimedean) upon based system new a for propose we appropriate Therefore, be to States, United the for configured tion specific of coverage for optimized projections hierarchical map a existing and sub- distortion, shaped scale compactly minimal and with equal divisions area, equal of sub- objectives global for proposals existing of survey our a upon Based as specified be alternatively could function area constraint. objective arbitrary of an an with Equality of sphere the placement lan- on the the cells in of is stated optimi- number optimization, problem, (quadratic) Our of is sphere. guage knowledge, a to our applied to zation literature, other the some in than in compact less grid therefore regular grid and equal-area these quadrilateral, an sphere,' for the provide not on do subdivision systems equal-area obtain- an While both. ing combining of cases several investigated sphere the on cells equal-area obtain to meridians and cells. lels shape and size regular have of approach level next level. preceding the the of creating the vertices the and with Dirichlettessellationfrom starting dodecahedron decomposition of·a a vertices example, for nication), of centers the are tessellations Dirichlet equal-area perfect ra fitrs.Ti ytmue h rnae icosahed- shaped truncated noncompactly the or uses system of large This covering parts interest. for different of fit areas or best earth to the Ar- polyhedron (or the semiregular repositioning a elsewhere. of extend plates projectiol1-onto to map difficult area but areas, specific projec- those conic .equal-area Albers standard the as regard such also areas, We reduction. and enhancement the for of structure all meet none that conclude we systems, division shape. of compactness and area of equality specifying explored not approach adaptive cells'are possible the Another Furthermore, shapes. cell. a within except plane, the (1973) Paul meridians. the only or parallels the only pacing this does dodecahedron original the of level the at Only h ae ftersetv ltncsld i otesphere commu- the (personal to described fit solids has Platonic Lukatela 1969). respective (Coxeter the of faces the o peod.Bie 15)otatdtetocsso res- of cases two the (1956)contrasted Bailey spheroid). (or w rvossuishv osdrdrsaigparal- respacing considered have studies previous Two e prahadRationale and Approach New A 7 Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. n peecnb aet oc l etcso ahpoly- each of vertices all touch to made be can sphere ing al .Goerccaatrsiso h ltncsolids. Platonic the of characteristics Geometric 1. Table 8 polyhedra. regular five the of tessellations spherical The 2. Figure points starting of set ideal of equally an Vertices are provide area. characteristics hence, surface and, these spaced, sphere's with the polygons exam- icosahedron of spherical for an 1/20 solid; from Platonic cover entire projected the will the triangle in spherical to as each relative same ple, the the polygon 2). be (Figure spherical in will equal each sphere regular are of angles are area interior and and The sides sphere all that the sense cover pentagons great- or completely squares, as triangles, that sphere spherical lines of geodesic the edges These the onto form lines. geodesic projected chords called as be segments, of circle can thought that be can circles edges of polyhedron circumscrib- and A 1. hedron, Table in properties their Figure in with listed shown are and 1 polyhedra Platonic, or Polyhedra. regular, Regular five of The Tessellations Spherical of Analysis iue1 h ierglrplhda rPaoi solids. Platonic or polyhedra, regular five The 1. Figure rprino total of proportion rai ahface each in area ubro vertices of number edges of number faces of number aepolygon face tetrahedron hexahecton erhdo hexahedron tetrahedron (a.tle) triangle 114 4 4 6 qaetinl pentagon triangle square 116 12 8 6 ntesm anr n oprn h ra n interior and areas the comparing and manner, same the in ei osrcini aldteatraemto (Gasson method alternate the called is construction desic pol- triangulating all if suggests which helps degree, It same the formed. of are polygons ygons smaller be the can of angles polyhedron shape each of face and con- spherical size a of is subdividing in pro- by distortion studied Distortions polyhedron shape which and interest. of size siderable overall question least the the size both, duces in or unequal shape polygons or spherical produce will polyhedra face one portrays 3 Figure further. globe the subdividing for htcmrs edscdms(ok 1968). networks (Popko triangular domes the· to geodesic similar comprise that network a into faces the azimuthal Lambert the projection. using tessellation, spherical each of h otpplrmto ftinl udvso ngeo- in subdivision triangle of method popular most The spherical five these of any of subdivision further Since octahedron atgah n egahcIfrainSystems Information Geographic and Cartography 1/8 12 8 6 dodecahedron 1112 30 20 12 icosahedron triangle 1120 30 20 12 Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. eniiilydvddit oradfv rage fequal of triangles five and four into divided initially been oc- Tetrahedron, angles. interior equal two only have but eigta h peia ueadddchdo ae have faces dodecahedron and cube spherical the that bering dode~a- and Cube 4. Figure in shown are polyhedra ical method, alternate the In triangle. spherical different each for grid. geodetic longitude, 30° latitude, atgah n egahcIfrainSystems Information Geographic the and that Cartography conjecture a dif- prompting a Average data, reaching triangles. four-frequency triangle, initial small and starting pro- infinitely in the for of zero decrease area of generally the ference to triangles among portion and differences area two- Remem- for area polyhedron. average each of the of subdivisions interior trigon~metI?" percentages and four-frequency as area spherical 2 in sphere through Table deviation have on"the the determined and triangles be and can equal, shape, angles four-frequency be and all and size cannot two- in these identical of are five each and for four faces into subdivided first be must faces agam hedron the procedure In subdivision. quartering triangle. subdivisi~n this "four-frequency" parent a at Performing "two-frequency" the a vertices produces of triangle. called creating a is edge of by this each field, of constructed geodesic lines are midpoint edge triangles the geodesic the new to four directly applied be can and 1983), 3. Figure eitosi raas pert etesm o h two- the for same the be to appear also area in deviations that demonstrate numbers the respectively, shape, and size angles interior the and and areas The triangular, angles. are interior faces identical three shape, icosahedron and size and in identical tahedron, are that respectively, triangles, h eitosi peia rageaesaegVnm gIVen are areas triangle spherical in deviations The spher- the of subdivisions four-frequency and two- The n aeo ahshrcltselto iha with tessellation spherical each of face One '. ,. tetrahedron ~, - hexagon icosahedron tnn::ated -~-,-."" -., I .( -'. .. "\\. -,-,. : ~ , ~ --,/ .--- / Ic:osahoctcn pentagon tnn::ated eae'n(o.bl) hexahea'on 15° ito ag hntetofeuny ute investigation Further two-frequency. the than range viation nraet aiu f3 ecn o oh hsi pri- is This both. for percent 30 of maximum a to increase approaches range deviation the if determine to needed is tessellation. spherical each of face The 4. Figure rnae cshdo eslain udvdn h ico- the Subdividing and Tessellation. area in Icosahedron small cases. being both Truncated in triangles equilateral starting slowly to the should sub- close to range due increasing marily deviation with The subdivi- the stable ideal. and be to equilateral dodecahedron frequencies, the appears division from and least deviation listed the average icosahedron ranges deviate The clearly deviation sions 2. and Table deviations in angle interior average tet- the of icosahedron. tessellations and consid- spherical octahedron, the from triangles, from points rahedron, spherical except sample successive shaped obtained have by to similarly formed of sam- desirable triangles also defining network of is for a It vertices point the starting quartering. as the points as ple triangles subdivision. initial small increasing with limit maximum a subdivi- fre- two-frequency three all subdivision for approximately of is deviation deviation average area independent the of is times range The deviation quency. of amount aesraeae.Ti esr a eetne oall to the the extended giving of be subdivision, can triangle four-frequency measure or equilateral two- This an a in of area. triangles those surface from same angles be terior cannot network triangle equilateral initial an that ering de- wider a have subdivisions four-frequency and sions, n esr fsmlrt stedvaino ragein- triangle of deviation the is similarity of measure One using of benefits the out point data deviation area The tetrahedron hexagon icosahedron tn.rcated w-adfu-rqec udvso fone of subdivision four-frequency and two- •••.••.•, - ~ pentagon 9 Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. iue5 h rnae cshdo,a oyernadas and polyhedron as icosahedron, tessellation. truncated spherical The 5. Figure icosa- a truncated 60 polyhedra, the to Platonic fit the be can with As sphere icosahedron. circumscribing the from ernvrie,adte9 de a epoetdonto projected be can edges 90 the and a vertices, trunca- of form hedron 20 size to the one of only truncated is remainder There the vertices in 12 faces. hexagons the triangular ico- 20 of original leaving truncated each The pentagon, with polygon. regular ron, of kind one than Archime- 14 more the of one polyhedron icosahedron, semiregular the truncated of the characteristic called a are hexagons sam- many for large unacceptably are deviations maximum 10 hexagons and pentagons regular precisely creates that tion icosahed- an as conceptualized be have can may 5) but in (Figure faces, con- 14th as have sahedron the polygons solids regular Greece, These 1963]). and ancient vertices [Lyusternik in gruent Such century discovered 20th (13 the triangulated. solids is trian- dean hexagon equilateral the nearly when that formed so smaller, are faces, requires gles deviations shaped these hexagonally Minimizing triangles systems. pling spherical of vertices at points sample produce will htaeapoiaeyeuli ieadsae u the but shape, and size method in equal alternate the approXimately are using that dodecahedron and sahedron rmtePaoi solids. Platonic the from 2. Table eito range deviation deviation avg. %o v.it angle) int. avg. of (% v.itro angle interior avg. %o v.it angle) int. avg. of (% (degrees) %o avg. of (% range deviation avg. of (% v.deviation avg. v.tinl area triangle avg. %o sphere) of (% eito nae n neirage o w-adfu-rqec udvsoso h nta peia triangles spherical initial the of subdivisions four-frequency and two- for angles interior and area in Deviation A A area) area) 100.0 133.0 ~ ~ 0040.46 40.0 5063.75 75.0 6.25 50.0 tetmhedron 226.93 136.43 50.0 1.56 66.57 22.17 91 19.14 19.12 ~ hexahedron 2560.625 62.5 .266 20.19 6.65 6.32 .40.26 1.04 28 52.24 72.86 84 53.84 28.46 ~ ~ rmpaetinls h etcso hs rage r likely are triangles these of vertices The triangles. plane from udvso,adrahsamXmmo .3preta the as percent 7.13 of maXimum a reaches and subdivision, increases range the deviation the solids, and Platonic range spherical gradually. deviation frequencies, the the with vision than in As less dodecahedron of the better. improvement over the even deviation is and improvement average threefold The subdivision, a 3. is equilateral Table be percent in to marized appear what into subdivision frequency On triangle. from icosahedral the reduced of percent correspondingly 28.2 is to triangle percent one 33.3 in pentagons peia rage eoegoerclyindistinguishable further geometrically with become little sub- triangles increases in spherical range four-frequency the decrease deviation for icosa- The fourfold degrees the to 62 division. to to 58 threefold relative a approXimately angles with interior coupled in hedron, deviation subdi- average increased with constant app~ars sum- deviation as average shape, and four- area in and these slightly two- since a deviate misperception, slight shows do a is which triangles This 4, triangles. Figure spherical in seen large- a be for can point grid. icosahedron con- starting sampling excellent truncated the an triangular The cover area MeXico. therefore, to is, northern enough hexagon and large is Canada three hexagon the the of pieces globe, ico- the the of spherical a area of total percent The 71.8 triangle. but triangle, sahedral icosahedron an globe. the for 5, Figure h eito ag.Ti rnltst nlsrnigfrom ranging angles to translates This range. deviation the southern plus States, United the of s'tates 48 terminous h peet omte"ocrbl"pten sse in seen as pattern, ball" "soccer the form to sphere the vnmr tiigi h eryfvfl eraei the in decrease fivefold nearly the is striking more Even faces hexagonal the of properties geometric favorable The h eao fti oe s6. ecn fteae of area the of percent 66.7 is model this of hexagon The 18.91 .2 0.78 3.125 67.5 octahedron atgah n egahcIfrainSystems Information Geographic and Cartography 61.875 69.06 77.54 20.19 i.!'ml 19.4 dodecahedron 27.1l .1 0.104 0.417 ~ .68.37 8.36 61.0 .59.1 7.15 2.34 60.25 29.06 !fIN .47.25 2.34 17 27.9 21.78 ~ ~ 19.33 icosahedron 63.0 .47.84 7.34 1.25 60.75 0.312 25.64 7.25 1 Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. efo 01t 20dges n o h w mle angles smaller two the for and degrees, 72.0 to 70.1 from be atgah n egahcIfrainSystems Information Geographic and Cartography spherical by defined grid face sampling icosahedron triangular A truncated a a Faces. of produces Polyhedral subdivision of Tessellation high-frequency Surface Map-Projection will angle larger the for range maximum the since however, rm5. o5.9dges hs ausrpeettelimit subdivision. the increasing represent with values angles These small, these degrees. is of 55.69 to angles 54.0 but from isosceles individual equilateral, plane the being is degree 72-54-54 within This not a of and Deviation hexagon. triangles shape the the deviation triangle. for pentagon nearly than angular the rather greater the to far but is due range subdivision, deviation pentagon the triangles. hexagon the of division udvdn nti anrgvstinls2. percent 21.5 triangles gives hexagon, spherical area manner the the of of this division percent in initial 78.5 is the in subdividing triangles triangle five a these of of each and Since deviation average the both 3, Table in seen divided As be method. can pentagon Each pentagons. 12 the within mle naeaeta hs ntesm rqec sub- frequency same the in those than average on hexagon. smaller spherical the corresponding four-fre- of the and two- than subdivision the less the of slightly from areas alternate are the triangles the in four-, triangles two-, range using quency into deviation 4), the subdivisions, partitioned (Figure further triangles be higher-frequency can isosceles falling and which surface spherical of earth's each identical the five of into percent across States. 28.2 have United sphere will the conterminous hedron the subdividing of size recursively the on areas based grid, al .Dvaini raaditro nlsfrto n orfeunysbiiin fteiiilshrcltriangles spherical initial the of subdivisions four-frequency and two- for angles interior and area in Deviation 3. Table ob h lss oa qa ra qa-hp sampling equal-shape area, equal an to closest the be to rmtehxgn n etgn ftetuctdicosahedron. truncated the of pentagons and hexagons the from h vrg neiragei locoe o6 ere for degrees 60 to closer also is angle interior average The lblsmln ytmbsdo h rnae icosa- truncated the on based system sampling global A %oag n.angle) int. ofavg. (% eito range deviation angle) int. avg. of (% vrg deviation average angle interior average (degrees) (% eito range deviation deviation average %o v.~area) ~ avg. of (% area triangle average %o sphere) of (% favg. of 11 area) 03960.0898 60.359 .460.0374 0.1496 eao triangles hexagon ro4fe freq 2 freq 4 freo 2 6.21 .22.79 2.22 0.82 0.82 1.56 ernfc.Ti rjcinpoue h hp distortion shape the produces projection This face. hedron itrinwl eut u otesmer nec poly- each in symmetry shape in the to range cen- minimal due the a from result, with away shape will surface pattern center, projection distortion circular a projection a and in the ter, vary made will used is The center distortion was 6. face Figure projection polyhedral in map illustrated equal-area are azimuthal faces Lambert icosahedron cated iceo ie aisfo h rjcincne.I the If center. projection any the along from same the radius is given distortion a of shape circle because throughout coordinates. geographic the the that can is as sphere point, the sample every subdividing directly over advantage sphere the on triangles equal-area define surface projection hp itrini rda,adcnb optdesl for easily computed The be globe. can and the equal on gradual, of is shape, triangles distortion varying shape define surface continuously along scale projection but map-projection map the area, the the on in angles distorting accomplished differentially the is on by This equal-area triangles equations, an spheroid. equilateral onto or that project so to the surface on ability the positions map-projection including longitude which several, and upon latitude trans- surface, are into sphere map-projection retransformed the on planar a faces into polyhedron formed the Here, bility~ h aall n eiin ota dnia qiaea tri- equilateral identical that so meridians and parallels the are approach this to and plotted advantages be The can spheroid. vertices or triangle sphere equilateral of grid a ocmuea h raglrntokdniyi increased is density and network faces. tedious slightly, of triangular become differ the thousands characteristics triangles as to all distortion compute that to and in vertices their suffers approach rage eryeuli raadsae oee,this However, shape. and area in equal nearly triangles qa-ramppoetoso ltncsld n trun- and solids Platonic of projections map Equal-area a-rjcinsraetselto saohrpossi- another is tessellation surface map-projection A .129.17 6.91 .611.87 1.56 021 60.07045 60.2818 0.1174 etgntriangles pentagon 2.20 0.67 0.66 0.02935 29.79 freo 4 11.87 2.50 11 Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. peia tessellation. spherical 12 Figure .TeIpretsaedsoto otusfo h abr zmta qa-rapoeto foefc feach of face one of projection equal-area azimuthal Lambert the from contours distortion scale I-percent The 6. k •••••• octahedron = 12.6% tetrahedron ••• k lila hexagon truncated icosahedron 25•• 22.5 dodecahedron k mill = 5.6% atgah n egahcIfrainSystems Information Geographic and Cartography icosahedron truncated pentagon k....= eaern(OJbe) hexahedron k 1.6% -- = 12.6% ••• k icosahedron IIlIlI 5.6% Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. eto ufc o h erhdo peia aehsthe has face spherical tetrahedron the for surface jection circles These 6. Figure in circles concentric the by indicated r h otuso itrinaogteprle kdistor- (k parallel the along distortion of contours the are raetdsoto,wt h a cl ln h aallat parallel the along scale map the with distortion, greatest increments. percent 1 in tion) ewe h lb etradhxgneg etxo the on vertex edge hexagon and center globe the between sub- recursive by introduced distortion the than range in by and faces, polyhedra spherical of subdivision recursive United conterminous The meridian. the along smaller percent size small and its of cube Because the faces. of faces icosahedron and projected the cahedron on edge less (h face is a smaller at distortion percent larger distortion) projection, 18-plus (h percent be equal-area 22 meridian an a then In than along must center. scale more face the map being at the scale edge the face than map-pro- farthest The the sizes. face decreasing with distortion shape ewe h aetovrie ntehxgnprojection. hexagon the on vertices two multi- same be the must between pentagon each for coordinates projection to modified slightly being polyhedron the to due is faces Fig- frequency. subdivision same the for triangles hexagon smaller and compute, equal-area to Lambert easy the systematic, by is introduced projection map distortion the on shape The defined and area triangles The in significantly. different slightly deviating others however, are, more triangles stated The be now can faces spherical the of projection circle. distortion 2.2 scale a and is percent 2 edge the parallel, face the within falls along farthest these easily all the larger States of at percent scale 2.2 distortion of map maximum the maximum lowest dode- Here, the the figures. has of face projections agon the on less still shape and maximum The center. octahedron, face zero-distortion the at than atgah n egahcIfrainSystems Information Geographic and Cartography pentagon the on triangles per- of of areas the creation face. computing for when count required between enlargement distance the small-scale if This constant scaling 1.006 differ- a The this by surface. plied transform projection same projection the Lambert for pentagon the than adjacent for longer the distance formulas percent the on 2.65 each but is of vertex globe, surface center the the projection touches that pentagon means hexagon and This and globe. hexagon difference pentagon the the percent circumscribe on 0.6 The distortion parallel. scale the maximum along in percent 1.6 of penta- each on triangles isosceles degree 72-54-54 smaller sphere. the on directly globe. division the on triangles spherical and equal-area ideal, represent equilateral the and to close area being the some in with regular a shape, nearly is produces that face points spherical map the sample of of equal-area an network subdivision on Direct clearly. triangles equilateral of subdivision hex- icosahedron truncated the shape, circular roughly and etymthn oyernvrie utb ae noac- into taken be must vertices distance polyhedron the matching match fectly y) to is (x, projection the the that on vertices meaning edge two percent, 0.6 to distance in ence vertex edge each than at area distortion in scale smaller maximum a percent shows 20.5 6 ure be will These face. gon area in equal are hand, other the on locations. projection, point Lambert the define that triangles the of shape and hs itrinmp hwteepce erae in decreases expected the show maps distortion These iia udvso rcdr ilcet rdo ever of grid a create will procedure subdivision similar A by points sample determining between difference The = 11k = 1/1.225) etaiuhlpoeto.Ec eao' oto fthe of portion hexagon's Each projection. azimuthal bert geo- The plane. projection the on vertices pentagons transforms and projection hexagons map azimuthal Lambert The eao r xeddb n o notesliver. the into row the one by in extended hexagons are adjacent hexagon two across grid randomized A randomized A Top: (a) the in purposes). hexagons illustration for plane width between in slivers Spherical 7. Figure great the and line, straight a in (greatly meeting boundary. sliver sliver the hexagons how the two of adjacent to with two illustration the clipping schematic of this a for one sliver, is 7a from the method grid Figure into One triangular grids. the hexagons be- sampling extend formed to adjacent is are two re- between respectively, edges, km, 64.7 pentagon and and 72.4 of hexagon polygon widths the spherical for bowed widths slightly imum the and edges polygon sam- planar desired a the of to edge equally according of each by density. along only triangles pentagon, pling or of points created be hexagon sample vertices can the of regular be on These row to a shape. p~ints pentagons spacing projec- and for This and the area on equal required hexagons center. slight points are as regular projection sample plane since rather of the surface, but from generation tion lines, the poly- outward straight spherical complicates as each bowing projected bound curves not that regular are of vertices gon vertices between into lines desic pentagon and hexagon spherical a of gn ssoni hnln.Sml onsfo each from points Sample line. hex- thin the a in between arc shown great-circle is The agons to projection. edge hex gnomonic straight the from Lam· half·lune the a in as shown shown each is sliver hexagons, adjacent (exaggerated two across projections grid gnomonic and azimuthal Lambert op- the into arc an as projection projected Lambert sliver the the in dividing appears circle width) in exaggerated maximum with slivers max- km "lune-like" 28.5 that and km meaning (36.2 spectively), narrow are planar slivers the These between edges. created "slivers" the within points sample r ewe h eaostu per wc.()Bottom: (b) twice. appears great-circle thus The hexagons hexagons. the the between separating arc arc circle great the edges. hexagon-pentagon and hexagon-hexagon tween hspoeueitoue h rbe fhwt set to how of problem the introduces procedure This n ouini opaeadtoa onsi h sliver the in points additional place to is solution One nefcsBtenAjcn Faces, Adjacent Between Interfaces n etgnGrids Pentagon and • 13 Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. eto,t ecle h qa-raPlhda Projection. pro- Polyhedral a Equal-Area such the called devised be recently to very jection, (1991)has Snyder John fe eiigti rjcin efudta rigFisher Irving that found he projection, this deriving After eto a lob ple otefv ssee triangles isosceles five the to applied be also can jection equilateral six the of one is examining idea by Snyder's understood analysis. distortion best and formulas, inverse icosahedron, the for design a in concept same the used had map-projec- the modifying by polygon spherical projected sdn o h i eao de,alhexagon-to-hexagon all this edges, Since hexagon six percent. re- 3.3 the of for triangle, done distortion the is scale of maximum a center and to edge spacing the the from adjusting by meridians area equal of made to is faces. projected projection adjacent The be sliversbetween to eliminatingthe base. base line, posi- triangle straight triangle of a great-circle the coordinates the of geographic along center tions the progressively the allows to been reduction edge has This the direction from meridional reduced the in scale polyhedra, has other Snyder include to (1946),but further, Bradley derivation in the footnote carried a as reported pentagons. and hexagons regular produce to equations tion way. similar pen- a hexagon a in and the handled hexagon and a be can line between interface tagon straight The arcs. a as as edges vertices circlebetween hexagon gno- great a the two in represents situation which the projection, monic illustrates 7b Figure hexagon. posite de ilmtheaty h qa-raPlhda Pro- Polyhedral Equal-Area The exactly. match will edges degrees 3.75 of deformation angular maximum a in sulting The 8). (Figure hexagon pole-centered a forming triangles iue8 h abr zmta qa-raadteEulAe oyerl(eindb onSye)mpprojections map Snyder) John by (designed Polyhedral Equal-Area the and Equal-Area Azimuthal Lambert The 8. Figure foesxho rnae cshdo eao,cnee nteNrhPole. North the on centered hexagon, icosahedron truncated a of one-sixth of 14 oeeeatslto st ubw aheg fthe of edge each "unbow" to is solution elegant more A eao etcsa Starting as Vertices Hexagon Between Drawn Line Straight iefrTinua Sampling Triangular for Line qa raProjection Area Equal abr Azimuthal Lambert NP uvdGetCircle Great Curved tHxgnEdge Hexagon at eodpsiiiyi oetn h ieajcn hexagon adjacent five the extend to is possibility second A rd noteajiigtinl ftepnao,o,in or, pentagon, the of triangle adjoining the into grids bad9) ssonaoe hssciie h o distor- low the (Figures sacrifices solid this model above, the shown As as 9c). icosahedron and 9b the use to effect, those 9a). (Figure to grid close hexagon are the triangles of triangles the equilateral of the areas of and points subdivide tween and surface triangles, projected isosceles five equal-area into the pentagon divide the to of is equilateral solution An One issues. additional introduces icosahedron lsa xetteedo h luincano islands of chain Aleutian the of of all coastal covers end nearby hexagon the neighboring and except a states that Alaska, such conterminous and the waters, in po- area have we land States, United sampling a conterminous to the tessellation for global this network of application the For eaosadpnaoso h rnae icosahedron. projected truncated the the on of achieved pentagons be and can hexagons that characteristics tion be- spacing the that such triangles congruent into these pentagon. a of vertices the include cannot grid triangular 10 in part 1 match within edges to Pentagon-hexagon pentagon. each composing l eao,ohrprso h ol a aual eless be naturally may world the sin- to of a parts with icosahedron America other truncated hexagon, North gle central the of coverage placing the In within optimize fall hexagon. Islands Hawaiian third the a that note 10).Also (Figure the of all covers hexagon one that such solid the sitioned h lcmn fagi napnao fatruncated a of pentagon a on grid a of placement The datgsadDsdatgso Different of Disadvantages and Advantages atgah n egahcIfrainSystems Information Geographic and Cartography oyerlProjection Polyhedral lblConfigurations Global tagtGetCrl Line Circle Great Straight 5 qa Area Equal ftelnt faside. a of length the of tHxgnEdge Hexagon at NP Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. atgah n egahcIfrainSystems Information Geographic coor- and standard Cartography some to conversion the as continuous long or as discrete metries, Mapping known. are designs ious ouaindsrbtosfrtegoe rfrlreaesof areas large for or globe, the for distributions population esti- among scale coherence of lack continental the or attributes, single global population no of making solu- mates For Third, adaptive an alternatives. used. rather, be specified; be could of need parts configuration different configuration in cific grids address- sampling for of possible are placement the alternatives Several ing covered. well h lb,as sntafce ydfeetsmln geo- sampling different by affected not var- is the also of globe, structures the not probability is the designs as long sampling as different problem, from a geometries different used. be could interest of area less-spe- any a to Second, suited tion globe. the across extends it as uration the of pentagon the on grids regular of Placement 9. Figure h ol.Oe fcus,i oueti atclrconfig- particular this use to is course, of One, world. the triangles. pentagon adjacent two on triangle. meets one grid (larger pentagon on grid the hexagon hexagon to placed the The grid clipped Middle: (b) Sample triangle) triangles. Top: (a) isosceles the of icosahedron. truncated eea rtraaentiprati hoigoeo these of one choosing in important not are criteria Several •••••••0••• • 0.0. 0 0 ••• 0 •••••• 0 ••• ••• • • •• •0••0••••••••••••••• 0 ••• 0 •• •0"••00• 0." 0 •• 0.0 ••• 00" •• •0.0.00.0 •• 0 • •0•••• 0 •• • 0 •• 00" ••• 0 •••••• o (c) otm How Bottom: h he eua oyo eslain ntepae(Cox- plane the in tessellations polygon regular three The qae n raglrgiswoepit r h centers the are points tessella- grids-whose hexagonal triangular and and square, square, triangular, the 1969), eter pentagons adjacent adjacent and between icosahedron. problem hexagons truncated seam the between on the and to hexagons, similar is problem as such systems reference standard other to relates best in,hv orsodn ulpitgis-tehexagonal, the - grids dual-point corresponding have tions, This designs. such between cen- seams reconciling different creating on for an for Such developed mechanisms or designs distortion. from ters, requires grids Lambert between map-projection the overlapping approach compromise most, and with good not a adaptive coverage faces if of provide many, config. icosahedron area In other projection best by truncated the azimuthal projections. possibly by or the each faces, covered associated cases, view, is and icosahedron this earth models In the truncated of on best advantages. since uration interest as and of certain area approach has sampling, geometric map-projection configurations probability this for equal-area of compro- is an a objective as earth specify config- we primary the icosahedron of the optimal Since no truncated model the and model. for mise figure, the pattern in tessellation uration shown optimal is known Figure in configuration This projection Europe. Hammer-Aitoff and re- polar Australia north the as perhaps such and areas Study chauvinism, of by earth latitude-longitude. element the any the of avoids and axis poles, uration rotational the the at about placed are fixed is pentagons pattern two of centers three in shown is Japan divides configuration and This west, and hexagons. east Unfortu- into two America. Europe South into splits central it and nately subcontinent Indian common a to converted are coordinates location whose utdfrti betv,a dpieplc oadglobal toward policy adaptive an objective, this for suited as such hexagon, single a by many well how- including covered faces, placement; be several this require could areas with that other symmetrically many ever, covered are gion between edges meridional the of the one case, locating this the In is arbitrarily it sphere. as rotational configuration, a for polar the orientation labeled be natural can 11. sahedron Figure in views the orthographic notably most world, cov- the of good the parts of provides other and advantage several takes for development erage Hawaii and cartographic existing Alaska, America, North observations sampling pop- . individual system aggregate as as or either estimates system, ulation scale georeferencing common continental a or global into designs sampling different w eaosaogteGenihmrda.Ti config- This meridian. Greenwich the along hexagons two into placed be by can limited data not since likewise geometries, is from sampling models data of different geophysical incorporation or the ecological And known. is system dinate mlmnigaSmln rdi h Plane the in Grid Sampling a Implementing h ia lentv st eonz gi htteei no is there that again recognize to is alternative final The ico- truncated the of configuration fixed alternative An xedn h ofgrto o odcvrg fcentral of coverage good for configuration the Extending faTuctdIoaernHexagon Icosahedron Truncated a of ytmtcPitGisi h Plane the in Grids Point Systematic 12. 15 Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. hoet s h raglrgi otk datg fcom- of advantage freedom. We take directional to application. grid greater our for and triangular point tri- the The advantage hexagonal pactness use little to application. have corresponding choose to easy its appear and and grid frequent tessellation most angular of advantage 16 Fur- isotropic. and the compact have most tessellation the triangular The hexagonal being of three network. the corresponding of advantage sampling its Any 13). and equal-area (Figure grid an polygons define can corresponding the of 95 la18;Mtr 96.Tesur rdhsthe has grid square (Switzer The 1986). properties the to Matern geometric accrue 1984; its of Olea advantages because 1975; statistical grid modest triangular some thermore, Alaska. and America North central covering icosahedron truncated the of Hexagons 10. Figure irrhclDcmoiino raglrGrids Triangular of Decomposition Hierarchical h hoyfrteedcmoiin scnandi the in The 1967). ways. contained Hudson is 1964; several (Dacey in decompositions groups of decomposed these theory be for algebraic can theory grid The triangular The • '\ Q .' .~/'. ··": ...~.~ . ~... :: " ··d· .: ·····C. ···········C ~.;:: ...... ". ' :::. : ~:::.::.< : ~': : . .... '.' ::' '. :.:: : :~':' :. ~ h loihsaeapidt h ntfgr associated figure unit the points. to additional the applied containing are figure unit algorithms a of The plane the in aea ragewt n etx orsodn oapoint a to corresponding vertex, one the with triangle presentation, lateral this For point. grid original transformation each a algorithm as with each described is description, consistent enhancement a density provide for to order he xsi aallt h xso h atsa grid. its cartesian of the one of that axis x such the aligned to be parallel to is axes assumed three is grid original In density. higher a to grid a transforming hex- of for describes tessellations gorithms 1954)}. dual geography Losch the 1966; literature to (The economic taller [Chris 7. applied in as and agons in 4, theory hierarchies 3, result of place these and factors central by randomization, structure, of density under in identity triangular the probabilities the increases from uniform maintain (aside hence that decompositions decomposition) lowest-order three ' :' .•... o h hefl nacmn,teui iuei nequi- an is figure unit the enhancement, threefold the For al- develop can we decompositions, these implement To . ' .. .. atgah n egahcIfrainSystems Information Geographic and Cartography : ~"::~~ ~_., '.' . ":'~"\" . . ": . ... ". . . Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. o pia oeaeo h otriosUie States, projections. United orthographic three conterminous in the shown of Hawaii and coverage Alaska, optimal for a ocnetaieti sb lcn nadtoa point additional an placing by is and this bring grid, conceptualize enhancements to original threefold way two the grid, in triangular a points in metry between by distance rotated the is nteoiia rd tteoii,ascn on t( V2, (- at point second a origin, the at grid, original the on globe the on placed as icosahedron truncated The 11. Figure iue1.Teplrcniuaino h rnae cshdo hw nteHme-iofprojection. Hammer-Aitoff the in shown icosahedron truncated the of configuration polar The 12. Figure alternative An alignment. original the into back grid the by scaled are figure unit the of points third atgah n egahcIfrainSystems Information Geographic and Cartography y3/2), - n h hr at third the and -rr/6 Fgr 4.Sneteei he-a sym- three-way is there Since 14). (Figure 12 Y3/2). - (1/2, h eodand second The dly3, hr d where u ihoevre,crepnigt on nteorig- the on point a to corresponding vertex, one with bus etcsae(,) -Y/,/) -Y/,- Y3/2, (- Y3/2,1/2), (- (0,1), are vertices grid. An 15). original (Figure the in enhancement fourfold the in rotation no is [1988]). Mandelbrot eao sacnein iuefo hc ocet a create to which from figure convenient a is hexagon A the and In resources, rare density. for in sampling of increase 7/4 density a the in increase results reduction appro- fourfold algorithms the with grid, original an of alternating by reductions grid original -arctan the by with realign rotated can and hancements above, as defined d whose d/7, and by grid scaled original the on point a is center ad- whose an agon points placing neighboring by of is pairs all this between halfway conceptualize y3/2), point to - ditional way (-1/2, alternative at point second a origin, the at grid, in inal example, for (as, grid original the in points neighboring raglrgi,snetevrie n h etrfr a form between center compromise the suitable and A vertices grid. the starting since seven-point grid, triangular time. in sampling to of cycles used be periodic typically for a will reductions with enhancements the enhancement en- application, many sevenfold of our achieve a to factors Thus, combined basic be factors. may these additional reduction and Furthermore, hancement inverted. priately rotation. of angle the of sign Roessel the van and [1961] Richardson (See 16). (Figure (1/3y3) - (Y3/2, last There The above. (1,0). as at defined fourth d the d/2, and by scaled y3/2), are - points (1/2, three at third a three by formed triangle each of center geometric the at 18]frrltddsusos)Atraigsvnoden- sevenfold Alternating discussions.) related for [1988] h emti oi fehneetapisa elto well as applies enhancement of logic geometric The hex- a is figure unit the enhancement, sevenfold. the For rhom- a is figure unit the enhancement, fourfold the For V2), Y/,2.Tesxpit ftehxgnare hexagon the of points six The (Y3/2,V2). aeGisfrSampling for Grids Base V2), 0 -1), (0, 17 Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. nqeyietfigpit ntesmln rdi nec- is grid sampling the on points identifying Uniquely evr lsl elzdwt eopsto ftehex- the of decomposition a with realized closely very be o hsbs apigdniycnann 797points. 27,937 containing density corre- sampling 9,216, base of this factor for a by is density in increase The 4. rxmtl 7klmtr,hsbe ugse.Ti can This suggested. been has kilometers, 27 proximately 640 sayfrasgigatiue otepit.Tesimplest The points. the to attributes assigning for essary grid triangular a parts each obtain equal to 96 Thus, into 96. connected of divided points be factor can a derived by the hexagon and large increase the linear of a edge to sponding gno h rnae cshdo yafco f3 of factor a by ap- of icosahedron points truncated the of sampling agon between distance of a approximately an to density application, the sponding our For determine grid. can sampling base resources a financial available with tessellation corre- Square with Middle: (b) plane grid. the in hexagonal with tessellations Regular 13. Figure 18 by enhancements five by followed 3 by enhancements two h eie pta eouino apigadteprojected the and sampling of resolution spatial trian- desired with the tessellation Hexagonal grid. tessellation Bottom: (c) gular Triangular grid. Top: (a) square grids. dual-point sponding qaeknhxgntselto bu h ons corre- points, the about tessellation hexagon krn square · · · drsigSystems Addressing · · · · · · · · · · · · 2 • 5 4, or lctsti prah eurn nuhdgt fprecision of digits enough (USGS requiring used base scale approach, convention effectively large one this be these plicates than numbering for could the Survey more quadrangles Geological no quad, than U.S. be topographic less the any can 7.S-minute is by in a there States adopted in point hence United occurs sampling and conterminous that particular, kms, the In 20 distance in point. the maximum longitude quadrangle specify point and the uniquely Each latitude since standard to in the precision longitude. numbers is ficient of and pair these a of latitude assigned of useful be can most system The attention. avoid geographic to merit systems. addresses, systems numbers dinate or awarded addressing Enhancements of other numbers, grid. history number Two a the to additional and of duplication. and axes require sequence, rows the the density in of by one grid of sequentially in rows number within to is hierarchy. system decomposition Middle: (b) threefold the rotation. Top: of (a) before levels grid; by Three reduction. unit rotation to and After scaled enhancement figure Unit Threefold 14. Figure 95.Teptnilfrehneeto est locom- also density of enhancement for potential The 1985). suf- with number) single interleaved or concatenated a (or h is sacosrfrnesse iheitn coor- existing with system cross-reference a is first The (-112,-{312) • (-112{3,-112) atgah n egahcIfrainSystems Information Geographic and Cartography • • (O,-1/{3) • • TT/6; .-----~ (0,0) nacmn opee c Bottom: (c) complete. enhancement , • , • , , , II , , , , , • • (112{3,-112) • • (112,-{312) (112,-112{3) • • • • (1,0) • Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. et scmuaie ota hr smr hnoead- one than more is there that so commutative, 18a. is Figure ments in hierarchy fourfold level, atgah n egahcIfrainSystems Information Geographic and for Cartography digits quaternary by nonan- five a achieved by into be followed may digit, enhancements 9) points threefold (radix these two ary for the hier- addresses combining of 18b. sation fourfold-by-threefold Figure radices. in two-level, same the shown a the is indicating mixed with lev- archy have the subscripts successive in example can with of An points addresses concatenation decomposition for of two left-to-right els is notation no possible but environment One point, a address. two- for two- a in A dress point level. a for coarser shown the is on radix) point (without single address a quaternary, digit from three- ternary, digit the a of the derived level hierarchy, 17, finer sevenfold Figure any or At In fourfold, digit respectively. fold, specified. a be and systems, dimen- to 1967; enfold number need (Lindgren base higher or radix for radix (and hierarchies (1982) a that mixed in both Lucas plane of 1982), and system the Woldenberg our Gibson in In and sions). hierarchies plane analogous the also hexagonal is in it systems; hierarchies polyhedral sys- respective their identification for hierarchical single a construct to ometry en- needed, rotation no Top: grid; (a) unit reduction. to and scaled figure enhancement Unit Fourfold 15. Figure h iefufl nacmns ihpstoa re as- order positional With enhancements. fourfold five the points the identifies uniquely sev- respectively, digit, and septenary fourfold, or threefold, the for given are conventions square for (1982) Gargantini and (1966) (1989), Morton Dutton (1974) of of others that that and to to Wickman and (1989), analogous Shiren is and Goodchild approach This tem. points. grid enhanced for fication four- the of levels hierarchy. Three Bottom: (b) decomposition fold complete. hancement ob eevdi dac,o h s fadtoa identi- additional of use the or advance, in reserved be to hnmxn irrhe,tecmoiino enhance- of composition the hierarchies, mixing When ic h aegi sfxdi u plcto,aconden- a application, our in fixed is grid base the Since ge- decomposition the of use makes system second The • (-1/2,·{312) (-1/4,-{314) • • • • • •---•4 • - - - •• 00 (112,0) (0,0) • • , , , , 11------• • • , . (1/4,-{314) . . • (112,-{312) • • • • • (1,0) • h orrobss rprs r ubrda ntefour- the in as numbered are parts, or rhom- Adding rhombuses, part four or hexagon. The American rhombuses, North four the the of of (triangles), one buses from position fe oainb -tan- by rotation After ihnrw n otes osuhetars os This rows. across southwest to northeast and joining row, rows hexa- within grid the the of the center induding the example, not of) but for to south the 19. Therefore, extending (to rhombus Figure gon, in system. beneath initial shown the rhombus is for enhancement scheme address This fold the seven. of of total front a the makes on digit a hierarchy. ntfgr cldt ntgi;bfr oain b Middle: (b) rotation. Top: before (a) grid; reduction. unit to and scaled enhancement figure Unit Sevenfold 16. Figure ierobssnmee o8nrhett southeast to into and northwest subdivided 8 is south, to It 0 3. southeast, number numbered is vertices rhombuses center, the hexagon nine of large southwest three the decom- six-digit the a for only necessary is required, points digits baseline radix the no for decomposition thus address and sevenfold signed the of levels Three Bottom: -I4-31),"_ ~ __ " , (-5I14,-{3114) -32f,127 ' -~({31217,-112.f1) _--~ ~'" (-f3l2.f1,-11217) ••••••••••••••• (-f3l2{7,112{7) ..•...... •..... (-1/2,-{312) ....•...... •...... •...... •. .•...... •...... (-217,f3l7) ...•...... •...... •...... •.. ••••••••••••••• • (-1/14,-3{3114) • 1 ~ l31) nacmn opee (c) complete. enhancement (l/3y13); I • I I -"'" - r ~";(~Of'~ • : I (1/14,3/3114) (0,1/17) (O,-11{7) 11- .. I " (217 ({312{7,112.f1) (5114,{3114) • • (112,·{312) , .{3f7) • (1,0) • 19 Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. erpeetdb w eaeia iis ihasingle a with digits, hexadecimal two by represented be 20 could 20. Figure enhancements in fourfold shown five is the five of hexagon four American base- quartered the North Alternatively, successively in the of then grid are enhancements threefold Rhombuses two grid. the line for accounts utraydgta h n,frhrcnesn h ad- the condensing further end, the at digit the on quaternary address example An point. single a identify to times Sevenfold Bottom: (c) Threefold Top: system. (a) Fourfold systems. system. Middle: (b) addressing three system. The 17. Figure ...... •...... •...... •... •••••••• •• • • • • • • • a oaayrte hnqaenr oe,requiring nodes, algorithms. searching quaternary a level and for access second than tree basis the quad the to rather where form modifications system, nonanary could storage system has tree quad addressing modified This dress. from area. hexagon a one for in scheme shown addressing Baseline 19. Figure h rnae cshdo.Tefv ihs-re iisare digits highest-order five The icosahedron. truncated the atgah n egahcIfrainSystems Information Geographic and Cartography 3 • ~

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Copyright (c) Cartography and Geographic Information Society. All rights reserved. .•.: a rjcinuo hm h ln rdi lcdi a in placed is grid plane The them. upon projection map projection map equal-area an of plane the environ- on of placed cations variety environmental wide the of a sam- a assessment of address the especially geometry can issues, and that mental cartography design the pling described have We face. the specify to necessary be would addressing of level atgah n egahcIfrainSystems Information Geographic and Cartography EPA's encour- of for McKenzie also Program Dan We Assessment and projections. and Messer, Monitoring map Jay Environmental on Linthurst, advice John Rick thank and to thank inspiration and geometry, for global in Snyder issues the our of to MichaelGoodchild Tobler, understanding Waldo and of Lukatela, Hrovoje contributions the GeoffreyDutton, points. acknowledge to the the like would varying We addressing for for ACKNOWLEDGMENTS options geometry several regular with a and of the with of density, areas an pattern characteristics provide surface triangular distortion large-enough model the to geometric small-enough between fit construed the and be model of compromise coverage can faces geometric The plane appropriate regular The earth. lo- a the of earth. sampling of face the of fun- of the grid is as surface the systematic design a from This upon resources. based ecological major damentally of quality necessary. identi- be would icosahedron, also different a truncated configuration from the the derived would of of also points fication faces For two additional configuration an between global treatment. seam special the model, in need icosahedron introduced Points truncated the of is faces address the condensation. for point's hexadecimal darkened addressing The 339EOusing with or hexagon. grid 3321320, American sampling North Baseline 20. Figure -\ '. o dniyn apepit nqeyars adjacent across uniquely points sample identifying For Summary ak .. n ..Luo.18."prahsfrQuadtree- for "Approaches 1985. Lauzon. J.P. and D.M., Mark, 1986. B. Matern, and Creases without Landscapes "Fractal 1988. B.B. Mandelbrot, usnJc 97"nAgbaceainbtenteLshand Losch the between AlgebraicRelation 1967."An Hudson,J.c. State Oregon CR816721with agreement cooperative and Inc., Kilke1ly. 68-03-3439with METI, 68-C8-0006with contracts EPA by oc,A 1954. A. Losch, Hier- Hexagonal Mixed of Geometry 1967."The C.E.S. Lindgren, Data HierarchicalSpatial 1989."A Shiren. Y. and M.F., Goodchild, Quadtrees." 1983. P.C. Gasson, Represent to Way Effective "An 1982. I. Gargantini, Icosahedron." the on Projection 1946."Equal-Area A.D. Bradley, Sur- Entire the Divide that Systems Grid 1956."Two Bailey,H.P. University. supported been has research This work. our sponsoring and aging ysei,LA 1963. Ov- L.A. An Lyustemik, Model: Geopositioning "Hipparchus 1987. H. Lukatela, isn . n .Lcs 92 SailDt rcsigUsing Processing Data "Spatial 1982. Lucas. D. and L., Gibson, to Structure Data New A Quadtrees: "Sphere 1990. G. Fekete, Hierar- via Uncertainty Locational 1989."Modeling G.H. Dutton, Hex- a of Properties Number Some on Note 1964."A M.F. Dacey, 1969. Coxeter,H.S.M. 1948. H.S.M. Coxeter, 1977. W.G. Cochran, 1966. W. Christaller, REFERENCES Resources Technical 68-CO-0021with Associates, Environmental Jh ie n Sons. and Wiley -John eec nPtenRcgiinadIaeProcessing, Image and Recognition Pattern on ference mrcnGohsclUnion, Geophysical American ae egahcIfrainSsesa otnna rGlobal or Continental at Systems Information Geographic Based nTertclGorpyN..Lbrtr o Computer for Laboratory No.7. Geography Theoretical in 133-135. pp. 3, no. 19, vol. Technology. and Scales." Springer-Verlag. Berlin: 36. no. Statistics, in Rivers." with Publications. Dover York: New Smith. T.J. erview." YaleUniversity Connecticut: Haven, New Stolper. W.F. 10mand adUniversity. vard Papers Harvard PlaceTheory." Central of Context the in archies Networks." Place Central Christaller Barbara. Information Santa Geographic California, of for University Center Analysis, and National 89-5, Paper nical 905-910. pp. 12, no. Science Imaging Electronic on Symposium SPIE/SPSE the of ceedings chicalTessellation," Sons. and Wiley York:John Hall. Prentice Jersey: Cliffs,New Englewood Baskin. C. by lated Review, Geographical EqualArea," of Quadrilaterals into Earth the faceof ap es) e ok Springer-Verlag. York: New (eds.). Saupe Press. rpisadSailAayi,Gaut colfDsg,Har- Design, Schoolof Graduate Analysis, Spatial and Graphics Tech- Systems." Information Geographic Global for Structure Ternary." Balanced Limited. EllisHorwood Machinery, Computing for Association the of Communications Data." Distributed Spherically of Visualization the Support Taylor London: (eds.). Gopal S. and 63-67. pp. 2, no. 5, Lattice." HierarchicalPlane agonal Ltd. Company rceig,Auto-Carta Proceedings, rceig,Auto-Carta Proceedings, h cnmc fLocation, of Economics The h cec fFatlImages, Fractal of Science The pta Variation, Spatial emtyo pta Forms. Spatial of Geometry apigTechniques, Sampling h eta lcso otenGermany, Southern of Places Central The Introduction rceig fteIE optrSceyCon- Society Computer IEEE the of Proceedings cuayo pta Databases, Spatial of Accuracy o.3,n.1 p 101-104. pp. 1, no. 36, vol. ovxFgrsadPolyhedra, and Figures Convex eua Polytopes. Regular o.3,n.5 p 628-635. pp. 5, no. 37, vol. to ,p.355-364. pp. 7, ,p.87-96. pp. 8, eodeiin etr Notes Lecture edition. second Geometry, ora fRgoa Science, Regional of Journal h rfsinlGeographer, Professional The & rnltdb ..Wog- W.H. by translated hr dto.NwYork: New edition. third Francis. odn ehe and Methuen London: ..PignadD. and Peitgen H.C. hcetr England: Chichester, eodeiin New edition. second ..Goodchild M.F. p 566-571. pp. rnltdby translated Transactions, o.25, vol. trans- Pro- vol. 21 Delivered by Publishing Technology to: Oregon State University IP: 128.193.164.203 on: Wed, 09 Nov 2011 18:53:57 Copyright (c) Cartography and Geographic Information Society. All rights reserved. otn .16." optrOine edtcDt ae and Base, Data Geodetic Computer-Oriented "A 1966. G. Morton, ndr ..19."nEulAe a rjcinfrPolyhedral for Projection Map Equal-Area "An 1991. J.P. Data Snyder, Hierarchical Related and Quadtree "The 1984. H. Samet, esr .. ..Lnhrt n ..Oetn 91 A EPA "An 1991. Overton. W.S. and Linthurst, R.A. J.J., Messer, ndr ..1987. J.P. Snyder, Contiguity." of Problem "The 1961. L.F. Richardson, 1968. E. Popko, al ..17."nCmuaino qa raBlocks." Area Equal of Func- Computation Spatial "On for 1973. M.K. Optimization Paul, Design "Sampling 1984. R.A. Olea, ere . n .Pac.1978. Pearce. S. and P., Pearce, 22 oia uvyPoesoa ae 1395. Paper Professional Survey logical veys, nentoa uiesMcie,Cnd,Ltd. Canada, Machines, Business International otadRihl Company. Reinhold Nostrand lbs"Mnsrp umte o publication. for submitted Manuscript Globes." Structures." e ehiu nFl eunig"Itra memorandum, Internal Sequencing." File in Technique New a Grodesique, tions." Trends." and Assessment, and Status Ecological Monitoring mental Monitoring for Program tems, o.1,n.2 p 187-260. pp. 2, no. 16, vol. o ,p.139-187. pp. 6, no. ahmtclGeology, Mathematical o 0,p.73-84. pp. 107, no. soito o optn ahnr optn Sur- Computing Machinery Computing for Association Geodesics. a rjcin- okn Manual. Working Projections-A Map eri:Uiest fDtotPress. Detroit of University Detroit: o.1,n.4 p 369-392. pp. 4, no. 16, vol. oyer Primer. Polyhedra o 7 p 67-78. pp. 17, no. e ok Van York: New eea Sys- General ..Geo- U.S. Environ- Bulletin a ose,JW 98 Cneso fCreinCodntsfrom Coordinates Cartesian of "Conversion 1988. J.W. Roessel, van odneg ..17." eidcTbeo pta Hierarchies." Spatial of Table of Periodic System "A 1979. "A M.J. 1974. Woldenberg, Edvarson. K. and Elvers, E. F.E., Wickman, Topo- to "Indices dates. other and 1985 Survey. Geological U.S. Infor- 1985. Global Fingleton. for B. and G., Quadtree "A Upton, 1986. Chen. Z-t. and W., Maps." Tobler, Qualitative of Accuracy the of "Estimation 1975. P. Switzer, hlspyi Geography, in Philosophy h ehrad:D edlPbihn Company. Publishing Reidel D. Netherlands: The Problems." Sampling A, Global Series for Domains Addresses." Sensing, Ternary Remote 1570. Balanced and Engineering Generalized metric Mapping to National and state). (by Coverage" Map Division. Other and graphic Data. Sons. Quantitative and and Wiley Pattern Point Sons. 371. and Storage." Wiley mation John York: New (eds.). Cullagh ipa n nlsso pta Data, Spatial of Analysis and Display atgah n egahcIfrainSystems Information Geographic and Cartography o.5,ns -,p.201-212. pp. 3-4, nos. 56, vol. egahclAnalysis, Geographical .Gl n .Oso es) Dordrecht, (eds.). Olsson G. and Gale S. pta aaAayi yExample: by Analysis Data Spatial hcetr nln:John England: Chichester, ..DvsadMJ Mc- M.J. and Davis J.e. o.5,n.1,p.1565- pp. 11, no. 54, vol. o.1,n.4 p 360- pp. 4, no. 18, vol. egaikrAnnaler, Geografisker Photogram-