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 no de agrees with the ordering of con

nections in the input map

In this section we describ e such a test and also how to adjust if necessary the new lo cation

p to avoid change in top ologyWe explain the test rst and prove correctness later The test

ts whichhave p as an endp oint It will b e sucient to consider a will involve all line segmen

single suchsegment say pq to describ e the test Consider the triangle T T ppq to p erform

the test Wehave to nd out whether there is any line segment of the map crossed bythe

b oundary edge pp of the triangle T Wealsohavetocheck whether the triangle T contains

inside it anypointAve

If the top ology would change the moveofp must b e smaller than p to avoid the intersection

It is imp ortant whether a p ointmoves at all If a p ointmoves this will aect other p oints in

the p olyline when the algorithm tries to nd a b etter lo cation for them

We distinguish b etween the following cases

Thereisnopoint inside the triangle T and no line segment crossing edge pp

In this case map top ology will not b e changed The moveof p to p is allowed see Figure

q5

q4

p

p q

q 1 3

q 2

Figure Point p can b e moved to p

pp Thereisatleast one line segment intersecting edge

In this case map top ology will change see Figures a and b A new lo cation for p has

to b e obtained The new p is taken to b e the nearest intersection p oint say utop plus

or minus the predened minimum distance d measured along pp see Figure a The

paramater d is an aesthetic constraintforthemap

Thereisatleast one point v inside the triangle T

In this case map top ology mightchange see Figures c and d To calculate the new

lo cation for p dene a straightline l through v and q and calculate the intersection p oint

pp of T Take p to b e the nearest intersection p oint u to p plus or u of l and the edge

minus a predened minimum distance d see Figure b

The cases and as well more cases of each one can o ccur together In such situations

the new lo cation for p will also b e calculated taking the nearest intersection p ointtop

Note that after the ab ovechecks have b een p erformed the newly adjusted lo cation p will

pq In other words the situation in Figure b e such that case holds for all line segments

holds

pq then moving the point with location p to Lemma If case holds for al l line segments

the new location p wil l preserve properties i ii and iii

p

p

p

q

p q

(a) (b)

p p

p p

q q

(c) (d)

Figure Illustration of p ossible cases for which top ology changes

p

p

d

d

u

u

v

l

p q

p q

(a) (b)

Figure Top ology checking

Pro ofWe consider each prop erty in turn

p Since only p oint p is moved we need only consider crossings with line segments of the

pq Assume there is a line segment ab which crosses such a segment pq Because by form

assumption neither p oint a nor b lies inside triangle T ppq segment ab must cross also pq

See Figure

p Note that triangle T ppq coincides with triangle T ppq and line segment pp coincides

with segment pp The result now follows from prop erty i and from symmetry by considering

the transformation of the p oint with new lo cation p back to the old lo cation p

p For no des of typ e p see Figure Now consider no des of typ e q Supp ose that

qp qp qp qp g is the set in cyclic order of all outgoing connections from no de q See f

 k

Figure Provided that angle is less than the cyclic order will b e preserved but this must

hold since otherwise condition would not b e satised This ends the pro of

We remark that although condition is sucient to preserve map top ology it is not neces

sary Consider Figure Since there are no line crossings in pq neither in pq so the top ology is

preserved We consider this a p ositive feature of the triangle test since wewant whenever p os

sible to avoid changes in the sidedness relations among geographic elements as represented

p

a

q

p

b

Figure Segments cross preserving top ology

p

p

k

p

4 p

q p

p 1

3 p

2

Figure Cyclic order around no de is preserved

by pq and v The test of p oint inside triangle can also detect some of suchchanges among

geographic features as shown in Figure p is at one side of the lake and p at the other side

Extensions of the algorithm could check the neighb ourho o d along pp out of the triangle p

street A

lake

v

p

q

Figure In this case top ology is preserved but not sidedness relations

Implementation

We next present the algorithm as develop ed so far

Pro cedure SchematicMap map maxSegment tolerance

Input N p oints of the conventional map the maximum line segment the simplication tolerance

Output n p oints for the schematic map

b egin

simplify lines with DouglasPeucker algorithm using given tolerance

cut long line segments according to maxSegment

repeat

for all p oints p p of the map do

n

test constraints and calculate a new schematic lo cation p for p

i

if there is any segment collision with p p then

i

calculate new lo cation for p

for eachpoint q that makes a line segment with p do

i

if there is anypoint inside triangle p pq then

i

calculate new lo cation for p

end for

change lo cation of p to p

i

end for

until reached desired app earance

end

As stated the search starts with the original conguration The structure is iteratively

improved by p erforming optimization steps based on dierenttyp es of moves in the inner lo op

We do not need to test all line segments and p oints in the map to lo ok for geometric intersec

tions of segments and p oints inside triangle Only adjacent regions of the p oint b eing analysed

are considered The no des and segments are stored in a uniform grid that divides the plane into

nonoverlapping regions Line segments are distributed among the regions where their center

point lies The grid spacing is chosen in suchaway that p otentially intersecting segments can

only b e found in either the same or two neighb our cells Spatial partioning algorithms like this



ASB are sucient b ecause we do not movepoints to o far and the line segments are at

um the size of a cell maxim

Results and Conclusion

Wehavepresented in this pap er a new metho d for preserving top ological relations among linear

features while generating schematic maps The metho d could also b e applied when schematic

maps are generated using grid tting algorithms Elr

The main ideas of the schematic map generator were implemented with encouraging rst

results Figure shows an example of the results obtained with our prototyp e using a database

containing all streets of Zurich For this study map lab els and cartographic features were not

included

The Porto bus map in Figure has less aesthetic treatment and relatively less geometric sim

plication than the public transp ortation map of Zurich in Figure For this reason wehoped

to achieve in early stages a map of comparable quality to Figure Preliminary insp ection of

our results indicates our schematic maps have a similar design app earance and feel to Figure

Our primary fo cus here was to preserve map top ology but there are also other features that are

desirable to b e preserved in the resulting map such as side relations among geographic features

and the graphical continuity of main streets

Figure Example of streets in a standard map and a derived schematic map

For the line simplication we used the classical DouglasPeucker algorithm However there

could b e some top ological and selfintersection errors intro duced by using this algorithm Vari

ants of the DouglasPeucker algorithm which take top ology into account Saa or other line

simplication metho ds dBvKS could b e tried instead

Further research on the presented topic is still necessary Wehopetocontinue this work

by studying and dening other formulation of aesthetic and geometric constraints to generate

automatic schematic maps and determining a measure of how go o d a resulting map is We plan

also to extend the prototyp e to add cartographic features to the visualization

Acknowledgments

The authors of this pap er are grateful to Ambros Marzetta and Matthieu Barrault for helpful

comments Sp ecial thanks go to Arne Storjohann and Frank Brazile for valuable suggestions on

pro ofreading this pap er

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