Generating Topologically Correct Schematic Maps 1 Introduction

Generating Top ologically Correct Schematic Maps Silvania Avelar Matthias Muller Department of Computer Science Swiss Federal Institute of Technology Zurich Switzerland favelarmuellermginfethzch Abstract This pap er studies the creation of schematic maps from traditional vectorbased carto graphic information An algorithm is prop osed to mo dify p ositions of lines in the original input map with the goal of pro ducing as output a schematic map that meets certain geomet ric and aesthetic criteria Sp ecial emphasis is placed here on preserving top ological structure of features during this transformation The known existing metho ds for preserving top ol ogy during map transformation generally involve computing several constrained Delaunay triangulations MolGol JBW Rua The algorithm prop osed here computes a transformation which preserves top ological relations among linear features using simple ge ometric op erations and tests Keywords Schematic maps digital cartography top ology computational geometry Intro duction Schematic maps are linear cartograms designed to convey only essential features of network routes Mon They indicate imp ortant top ological information on transp ortation or utility maps such as connectivity and stops while preserving only a general sense of the original geom etry Dor As long as essential top ological information is preserved the length and shap e of routes in schematic maps need not b e faithful to reality This cartographic exibility is p ossible b ecause geometric accuracy is considered to b e less imp ortant than linkages adjacency and rel ative p osition Mon Bra Because schematic maps do not have the mass of details usually presented in conventional top ographic maps they present the essential information more clearly Figures and illustrate the top ological nature of schematic maps Generating schematic maps on demand can b e seen as a geometric constraint problem that should b e solved algorithmicallyininteractive time In general to create schematic maps automatically rst geometric line detail should b e removed via smo othing or ltering Then the new straightened line network should b e adjusted to improve legibili ty of imp ortant information This line displacementfollows a set of constraints dened by some common sense geometric and aesthetic criteria for the schematic map Direction and distance are only approximately preserved for recognition purp oses while top ological information of the road network is kept The dicult asp ect of generating schematic maps automatically is to rep osition the network routes of the input map to lo cations which b esides fulllin g the set of constraints also preserve the map top ology Elr By preservation of map top ology we mean the following three prop erties i no absence of line crossings that were present in the input map ii no line crossings that were not present in the input map iii cyclic order of outgoing connections around any no de agrees with the ordering of con nections in the input map In this pap er we prop ose a technique to preserve the map top ology during the transforma tion pro cess The technique is based on simple geometric op erations and tests To the b est of our knowledge there do es not seem to b e such a public generally acknowledged solution for preserving top ological relations while moving p oints Figure Example of a public transp ortation map of Zurich Switzerland Figure Example of a bus map of Porto Portugal Some suggestions and theory concerning the automation of schematic maps can b e found in the literature Morrison Mor studies public transp ort maps of various cities and prop oses rules to govern the mapping metho d based on characteristics of transp ort services Elroi de scrib es his exp erience in generating schematic maps using GIS Elr Our contribution here is to pro duce schematic maps on demand while taking into account geometric and aesthetic constraints in suchaway that map top ology is preserved Characteristics of schematic maps Routes on schematic maps are usually drawn as straight lines with cartographic micro crenu lations removed Direction of lines usually varies only via xed stylized angles and degrees or and degrees are common though some schematic maps have only simplied lines with arbitrary but few directions Network lines with overlapping routes are separated by a minimum legibili ty distance This distance can either b e zero or a constantchosen for the map Sometimes adjacentschematized lines have smo oth artistic circular arcs around b ends these preserve the graphic proximity for the greatest length p ossible see for example Figure top left corner where the lines curvebetween the two white line connectors We also observe that the straightnetwork lines are generally not to o long A small number of breaks or changes in the line direction can b e incorp orated to add a sense of the original geometry and to provide a b etter visualization of all routes together In some schematic maps all lines are presented in one color and the route of a particular network is traced bynumber only In other maps contrasting colors dierentiate the various transp ortation lines Diering geometric and aesthetic criteria can b e used to design a schematic map Although it is easy to nd dierences in style eg compare Figure and common goals are graphic simplicity the retention of network information content and presentation legibility Generating schematic maps The purp oseful deformation of a geographic map is a common op eration in cartographyTwo typ es of map deformations can b e considered one for bringing two maps together suchas combining a satellite image and a geographic map via rubb ersheeting or minor geometric ad justmen t and the other to highlight quantities other than geographic distance and area such as in cartograms DEG Our schematic maps can b e thought of as the rst typ e of defor mation but we use techniques for transforming lines similar to HK Dor for constructing continuous area cartograms We assume the network lines to b e schematized are provided as input to our algorithm to b egin the automatic generation of schematic maps Wechose to create straight lines whichcan b e horizontal vertical or at angles of degrees For readability and aesthetic considerations wehave set a userdened maximum length for a straight line and a minimum distance b etween the lines The cartographic data structure contains p oints and p olylines As a prepro cessing step to remove line crenulations we p erform p oint and line ltering simplication to reduce the numb er of visualization p oints that constitute the detailed map The DouglasPeucker algorithm DPwas used for this study this algorithm is quickto implement and exible with a user denable threshold that controls the amount of simplication After the simplication long straight lines are cut according to the userdened maximum length The next step is to nd a new lo cation for eachpointofeach presimplied line according to the other constraints of the schematic map We used a dynamic iterative algorithm driven by the map constraints Mue HK The network conformation is improved iteratively in order to satisfy all map constraints The user decides when the schematic network reaches an acceptable app earance to stop the pro cess Tochange the actual conformation the algorithm visits all p oints p p of the simplied map in turn If the current p oint p violates one or more n i m constraints a b etter lo cation p for p is computed For each constraint c fc c g that i i aects p p is the lo cation nearest to p that satises c The new lo cation p is chosen to b e the i arithmetic mean of all p s Instead of computing a simple mean of the displacementvectors it would also b e feasible to evaluate a weighted average of the vectors if certain roads are more imp ortant than others j j For the constraintofstylized angles say c the new lo cation p is chosen among the lo cations that t the line segment in one of the allowed schematic directions The nearest distance p determines the new lo cation for p see pq in Figure a The constraints evaluation is carried out for all line segments in which p exists The nal lo cation p for p will b e obtained by the arithmetic mean of all required displacementvectors for all of its lines segments in Figure b three constraints are considered j p (2) p p p (1) p p p f q (3) p p (a) (b) pq to the nearest schematic direction and b nding the Figure a Fitting a line segment new lo cation p for p We then check the minimum distance constraint The nal found lo cation p should keep the sp ecied minimum distance to the other linear features of the map Adjustments in the arithmetic mean of the nal displacementp vector are p erformed when necessary The f line segments may b e stretched or shortened to reach the new lo cations whichwe refer to as schematic lo cations However the top ology of the map has to b e preserved during this mo dication pro cess Preserving the map top ology The previous section describ ed how to compute a b etter lo cation p b etter according to the aesthetic and geometric constraints for a given p oint p Before displacing the p oint from its original p osition p to the new lo cation p a test must b e p erformed to detect situations that can lead to a change in the map top ology that is to detect if one of the prop erties of Section might b e violated We recall them i no absence of line crossings that were present in the input map ii no line crossings that were not present in the input map iii cyclic order of outgoing connections around any

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