Automated placement of supplementary contour lines

Abstract

Supplementary contour lines are placed between regular contour lines to visualize small but important forms that regular contour lines are unable to show. On topographic maps, typical forms are hillcrests, depressions, saddles, terraces, banks, and levees. No automated method for the selection of supplementary contour lines has been described so far. We document cartographic design principles for the selection of supplementary contour lines for topographic maps, and present an automated method for their placement. Results of the automated method are similar to manually placed supplementary contour lines. Our method helps map authors to create contour line maps that more effectively illustrate relevant small details in maps showing terrain elevation or other scalar fields.

Full text

RESEARCH ARTICLE
Automated placement of supplementary contour lines
Timofey Samsonov
a
, Sergey Koshel
a
, Dmitry Walther
a
and Bernhard Jenny
b
a
Department of Cartography and Geoinformatics, Faculty of Geography, Lomonosov Moscow State
University, Moscow, Russia;
b
Faculty of Information Technology, Monash University, Melbourne, Australia
ABSTRACT
Supplementary contour lines are placed between regular contour
lines to visualize small but important forms that regular contour lines
are unable toshow. On topographic maps, typical forms are hillcrests,
depressions, saddles, terraces, banks, and levees. No automated
method for the selection of supplementary contour lines has been
described so far. We document cartographic design principles for the
selection of supplementary contour lines for topographic maps, and
present an automated method for their placement. Results of the
automated method are similar to manually placed supplementary
contour lines. Our method helps map authors to create contour line
maps that more effectively illustrate relevant small details in maps
showing terrain elevation or other scalar fields.
ARTICLE HISTORY
Received 7 January 2019
Accepted 20 April 2019
KEYWORDS
Supplementary contour line;
intermediate contour;
isoline; isarithm; isohypse
1. Introduction
Cartographic contour lines show two-dimensional scalar fields, such as elevation or other
types of continuous 2.5D surface data. Cartographers must make a compromise when
choosing the interval for a contour map. With a large interval, the map contains a small
number of contour lines, and interference with other map elements is minimized.
However, the sparse contour lines are unable to show the topography of nearly flat
areas or the details of small features, such as terraces, saddles, hillcrests, depressions, or
levees. A large interval also results in large horizontal spacing between contours, which
makes the interpretation of terrain difficult, because the form of the terrain cannot be
derived from isolated contour lines. With a small interval, contour lines show small details,
but the lines can become excessively dense in steep areas, to the point where they
coalesce in very steep areas and cover a slope entirely (Figure 1, left and centre). Dense
contour lines are also more likely to graphically interfere with other map features (Imhof
1982, Keates 1989). The use of supplementary contour lines is an elegant solution to this
dilemma. Supplementary contour lines are short contour line segments with the same
interval as regular contour lines, but with the base level offset by half the interval. They are
placed where small but important terrain features are not shown by regular contour lines
(Figure 1,right),andinflat areas where the horizontal distance between regular contour
lines is large. Supplementary contour lines are commonly represented as dotted, dashed,
or thin bright lines to differentiate them from dark and solid regular contour lines.
CONTACT Timofey Samsonov tsamsonov@geogr.msu.ru
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE
https://doi.org/10.1080/13658816.2019.1610965
© 2019 Informa UK Limited, trading as Taylor & Francis Group

Supplementary contour lines are historically used in many topographic national map
series (Collier et al.2003), and cartographic textbooks invariably recommend their use for
terrain maps (Imhof 1982, Keates 1989, Robinson et al.1995,Field2018). It is therefore
surprising that to date, no digital method for automatically placing supplementary contours
on maps has been discussed in the literature or is available in commonly used geospatial
software. Supplementary contour lines, therefore, have become rare on maps that use
contour lines derived from digital elevation models. For example, the recently redesigned
map series of the Swiss Federal Office of Topography swisstopo no longer includes supple-
mentary contour lines, as their contour lines are now algorithmically derived from an
elevation model. This article addresses this shortcoming and makes two contributions: We
compile cartographic design principles for the placement of supplementary contour lines
from the literature, and we introduce an automated method for placing supplementary
contour lines that takes these design principles into account.
Because contour lines are mainly used for visualizing topography, cartographers have
focused on formalizing and documenting design principles for contour lines on topographic
maps. This article also focuses on contour lines for topographic maps, but the design
principles for supplementary contour lines and the presented digital method also apply
to mapping other types of two-dimensional scalar fields (Dent et al.2009,Imhof1972).
2. Terminology
Supplementary contour lines, the term used in this article and in various cartographic text-
books and scholarly publications (Keates 1989, Robinson et al.1995, USGS 2005, Kimerling
et al.2016,Field2018, and others), are occasionally called intermediate contour lines, most
notably by Imhof in his seminal work on terrain mapping (Imhof 1982). Others, such as
Mackaness and Steven (2006), Kimerling et al.(2016), and the USGS (2005), use the term
intermediate contour lines to refer to regular contour lines in order to distinguish them from
index contour lines (which are labelled and drawn with a wider stroke). This article uses index
contour line, regular contour line,andsupplementary contour line as illustrated in Figure 2.
Figure 1. Small contour interval (left), appropriate contour interval (centre) with supplementary
contour lines (right). Index contour lines are shown by labelled thick lines. (From Imhof 1982.).
2T. SAMSONOV ET AL.

3. Design principles for supplementary contour lines
We identify the following design principles for supplementary contour lines from the
cartographic literature.
Interval and base level: As shown by Imhof (1982), placing supplementary contour lines at
values that are not exactly in-between neighbouring regular contour lines creates confusing
maps that can be impossible to interpret correctly. For the same reason, Imhof also argues
against placing multiple supplementary contour lines between neighbouring regular con-
tour lines. Nevertheless, rare examples of maps with multiple neighbouring supplementary
contours exist. For example, the USGS quadrangle map standard allows for multiple
supplementary contours between regular contour lines ‘when necessary’(USGS 2003).
Centrality: A supplementary contour line is drawn when its distance to the next
contour line on one side considerably differs from the distance to the contour line on
Figure 2. Terminology used in this article: index contour lines (thick lines with labels), regular
contour lines (thin lines), and supplementary contour lines (dotted lines). (Map from Imhof 1982).
Figure 3. Supplementary contour lines clarify the topography of terraces (left), passes (centre), and
small hills (right) (Swiss Society of Cartography 1987).
Figure 4. Superfluous supplementary contour lines that should be avoided. They do not convey
useful information, because their distances to neighbouring higher and lower contour lines are
identical (Imhof 1982).
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 3

the other side. This is particularly important for terraces (Figure 3 left), and edges of flat-
topped ridges, plateaus, or mesa tops. If the next upper and next lower contour lines are
approximately at the same distance, no supplementary contour line is drawn, as it would
not convey any additional information (Figure 4, Imhof 1982, USGS 2003).
Density: Supplementary contour lines must keep a minimum distance to neighbour-
ing contour lines. They are therefore not added to steep slopes where they would be
very close to, or even coalesce with, regular contour lines and thereby become difficult
to interpret (Imhof 1982).
Flat areas: Supplementary contours are added to flat terrain where the horizontal
distance between regular contour lines is large. Imhof (1982) recommends a distance
threshold of 10–20 mm, whereas the USGS (2003) standard for quadrangle maps uses an
area threshold of 33 ×33 mm (or 0.5 ×0.5 miles at a scale of 1:24 000).
Small forms:Knolls,hollows,saddles,dunes,smalllandslipdeposits,morainesides,and
other small geomorphological forms often require supplementary contour lines (Figure 3
centre and right) (Imhof 1982). The USSR guide for cartographic works (Military Topographic
Service 1978)makesaspecificcaseforsaddles:Supplementary contours should be drawn on
both sides of a saddle.
Extent: In general, supplementary contour lines are extended such that they start and
end close to the centre between neighbouring contour lines. Views on where to start
and stop supplementary contours differ. The USGS (2003) requires supplementary con-
tours to reach the centre one-third between neighbouring contours. The Military
Topographic Service (1978) of the USSR required them to reach the centre between
the two neighbouring contour lines. The Swiss mapping agency swisstopo extends
supplementary contours, such that they follow the centre between neighbouring con-
tours for a short distance. This increases readability and avoids short line fragments
(Figure 5, swisstopo 2008).
4. Related work
A variety of contouring algorithms were developed in the early history of digital
cartography, and Tobler (1965) reported the existence of several dozen contouring
applications by the mid-sixties. Contour lines are either extracted from two-
dimensional scalar fields, such as digital elevation models in grid format, or from
scattered points. For regular scalar fields, the marching squares algorithm is commonly
Figure 5. Extension of supplementary contour lines to the centre between neighbouring contour
lines as recommended by swisstopo (2008).
4T. SAMSONOV ET AL.

used, which is a two-dimensional version of the marching cubes algorithm for extracting
isosurfaces from three-dimensional scalar fields (Lorensen and Cline 1987).
No digital method for creating supplementary contour lines has been described in
the literature. The only documented method for placing supplementary contour lines
was mentioned by Jaara and Lecordix (2011), who discuss the digital generation of
contour lines for the French national map series. They detect regions where supple-
mentary contour lines are needed, but they neither provide a description of the method
nor visual examples.
An alternative approach to placing supplementary contour lines is to reduce the
contour interval and either remove or displace regular contour lines that are too dense
in steep areas. The Ordnance Survey and the Canadian national mapping agency
remove contour lines to avoid touching or overlapping contour lines in steep areas
(Keates 1989, Mackaness and Steven 2006). Another example is the 1:25 000 map series
by swisstopo, where regular contour lines are removed from areas with rock drawings,
as contour lines would otherwise coalesce in these generally very steep areas (Jenny
et al.2014). Displacing contour lines is essential for avoiding coalescing contour lines at
smaller scales (Imhof 1982). Gaffuri (2009) introduced an automated method to increase
the spacing between dense contour lines in steep areas. He deforms contours with
a constraints-based algorithm that preserves the general shape of the surface.
5. Automated placement of supplementary contour lines
The computation of supplementary contour lines consists of the following steps: First,
contour lines at intervals between regular contour lines are computed; these are
candidate supplementary contour lines. Each candidate contour is then analysed to
decide whether any of its segments qualify as supplementary contours that should be
shown on the map.
We first identify candidate supplementary contours that do not enclose any other
contour line. They are the lowest contours for depressions or the highest contours for
peaks. They are either drawn or removed as a whole, based on their length and the
average width of the enclosed region.
All other candidate contours undergo a vertex-by-vertex filtering process. First, the
relevance of each contour vertex is analysed. An individual vertex is relevant if it meets (a)
aregion width constraint and (b) a centrality constraint. The region width constraint ensures
there is sufficient space between regular contour lines to place a supplementary contour.
The centrality constraint ensures a supplementary contour does not closely follow the
geometric centre between neighbouring contour lines, as illustrated in Figure 4. Vertices
that meet both constraints are connected. We then ensure that resulting line segments
are sufficiently long and not separated by small gaps. Finally, supplementary contours are
extended if needed, such that they reach the central areas between neighbouring con-
tours as illustrated in Figure 5.
Algorithmic details are provided in the next subsections. First, we introduce methods
for computing region width (Section 5.1) and centrality (Section 5.2), and then describe
filtering of vertices based on region width and centrality (Section 5.3). Section 5.4
describes filling short gaps and removing short contour segments, and Section 5.5
discusses the extension of supplementary contour lines towards the centre between
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 5

neighbouring contours. Section 5.6 provides an overview of the workflow and para-
meters, and also discusses the filtering of supplementary contour lines for depressions
and peaks that do not enclose any other contours.
All processing stages are illustrated with an elevation model, shown in Figure 6. This
elevation model was obtained by geodetic survey of a coastal area near N.A. Pertsov
White Sea Biological station of Lomonosov Moscow State University, Karelia, Russia (66°
34‘N, 33°08‘E) during student field training in 2015.
5.1. Region width
Regular contour lines and the border of the map divide the mapped area into a set of
regions. Each region can potentially contain a supplementary contour line. A random
example region is shaded in Figure 6.
Region width is used to (a) ensure there is enough space to place a supplementary
contour and (b) identify excessively wide regions in flat areas where supplementary
contour lines are included, even if they are close to the centre of the region. Since
a supplementary contour can be located anywhere within a region, an estimate of
region width is required for any point within a region.
In our approach, region width Wp;ri
ðÞat point plocated inside region riis defined as
the diameter of the largest circle cthat contains pand is entirely located within
ri:Wp;ri
ðÞ¼max
p2c$ri
dcðÞ;where dð%Þ is a function that calculates the diameter of a circle.
This largest circle is called a dominant circle for the point p, and the centre of this circle
(named p0) is called a dominant circle-reachable neighbour of point p.Figure 7(a)illustrates
the circle-based definition of region width. For clarity of illustration, it shows only a limited
number of example points pand p0, and the corresponding dominant circles.
Computation of continuously changing region width can be effectively automated
via map algebra in raster mode. We replace the search for exact dominant circles
within a region with per-cell computations in a regular grid. First, for each cell of the
Figure 6. Two neighbouring contour lines and the map border define a region (shaded in grey) that
may contain supplementary contour line segments. The rectangle indicates the area in Figure 7.
6T. SAMSONOV ET AL.

raster field, the Euclidean distance to the closest regular contour line is calculated and
stored (Figure 7(b)). This corresponds to the FocalProximity spreading map algebra
operation (Tomlin 2012). Next, an output raster field with the same spatial resolution
and coverage is allocated and initialized with zero values. We now propagate the
doubled value of each cell of the distance raster to the output cells that are covered by
the circle neighbourhood of the corresponding size (Figure 7(c)). The resulting value is
determined by the following rule: If a pixel is empty or has a value smaller than the
Figure 7. Circle-based region width computation: (a) Circle-based definition of region width
(brighter circles represent larger region width). Points p0are dominant circle-reachable neighbours
of example points p. Their circles A, B, C and D are drawn in white. (b) A scalar field with the
distance to the nearest contour. (c) Propagation of the doubled distance to the circle neighbourhood
of the corresponding size (brighter pixels represent larger circle diameters). Pixel size is 2.5 m.
Figure 8. Computation of region width and centrality: (a) distance to nearest contour, (b) region
width, (c) distance to the central line of each region, (d) centrality. Bright shades correspond to large
values. Darkest shades in (c) and (d) correspond to areas where centrality is undefined.
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 7

doubled value of the distance raster, then its value is replaced with the doubled value
of the distance raster; otherwise, it remains unchanged. This is illustrated in Figure 7(c),
where pixels of the lower left part of circle C are overridden by circle B, which has
a larger radius.
AcompleteexampleofthecomputationofregionwidthisshowninFigure 8.Figure 8(a)
shows a raster field with the Euclidean distance to the closest regular contour, and Figure 8(b)
represents the derived region width raster.
5.2. Centrality
A supplementary contour is generally informative if it is clearly not equidistant to
contour lines that define the borders of its region. To estimate this property, we employ
the notion of centrality. The centrality of a point that is exactly in-between two regular
contour lines has the maximum value of 1; the centrality of a point that is close to
a regular contour line is close to the minimum value of 0.
Centrality is not measured orthogonally with respect to contour lines. To derive the
calculation of centrality we analyse different situations, which are presented in Figure 9.
The straightforward way to calculate centrality is to obtain the ratio of the distances to
the first and second closest contours: C12 ¼d1=d2. This formula works effectively in
simple cases like point A in Figure 9, where the contours follow each other (C12 &0:4),
but it fails for complex configurations. For example, point B is located in a wide isolated
part of the region, where a supplementary contour would be very informative. However,
point B has a high centrality value according to the formula introduced above
(C12 &0:8) and it would, therefore, be filtered out in further processing.
A more robust measure of centrality can be obtained based on the central line
between two contour lines. In vector mode, this line can be obtained by constructing
the Voronoi diagram of the regular contour lines and following the boundary between
Voronoi cells. With the distance dcto the central line, the centrality can be calculated as
C¼d1
d1þdc. This formula still works for point A, and effectively handles the case of
point B (C&0:3).
Instead of constructing and analysing a Voronoi diagram, we compute centrality with
map algebra in raster mode. The nearest contour distance d1can be extracted from the
Euclidean distance surface previously constructed for computing region width (Figure 8(a)).
Figure 9. Computation of centrality: d1is the distance to the first closest contour, d2is the distance
to the second closest contour, dcis the distance to the central line between the first and the second
closest contours (depicted by the grey line).
8T. SAMSONOV ET AL.

To calculate the value ofdc,thecentrallinesareobtainedbyaEuclidean allocation operation
(also called FocalNeighbor by Tomlin (2012)) applied to regular contours. The categorical
raster resulting from this operation contains the identifier of the closest contour line in each
pixel and is a raster-based approximation of the sought-after Voronoi diagram. The bound-
aries between the obtained raster zones are the desired central lines, shown in Figure 8(c)as
thin lines between contours.
Unlike d1, which is always a straight-line distance, the value of dccan be curvilinear if the
straight connection between the pointand the central line is blocked by the closest contour.
Therefore, to find dcwe (a) construct a supplementary cost surface that prohibits crossing
regular contours during shortest path computation and (b) apply a cost distance map
algebra operation (also called FocalProximity in FRICTIONLAYER by Tomlin (2012)) that uses
the cost surface as a friction parameter. The cost surface is constructed by nullifying pixels
that are intersected by regular contours, and assigning the value 1 to all other pixels. The
surface resulting from the cost distance operation is shown in Figure 8(c).
For our purpose, centrality is undefined for regions with only one contour along their
border (which includes contours for depressions and peaks that do not contain any
other contour line). For such regions, centrality is set to 0. These regions are shown with
the darkest shade in Figure 8(c).
After finding the distance to the nearest contour d1and the distance to the central
line dc, we can estimate the centrality of each pixel with a map algebra expression using
the formula for C.Figure 8(d) shows the resulting raster surface.
5.3. Vertex filtering by region width and centrality
Each vertex has to pass two filtering constraints: The vertex is removed, if (a) the region
at the vertex position is too narrow or (b) the vertex is close to the central line.
Removing a vertex results in splitting a candidate line into two new segments, which
are separated by a gap.
When filtering by region width and centrality, we distinguish between wide, narrow,
and intermediate regions.
Wide regions: According to cartographic design principles, supplementary contours
should be placed where a large distance separates regular contour lines. Therefore, we
introduce the Wmaxthreshold parameter for detecting excessively wide regions. If the
region width Wat a vertex is greater than Wmax, then the vertex is not removed, and the
vertex is not filtered by the centrality constraint.
Narrow regions: To ensure supplementary contour lines are not placed where neigh-
bouring regular contour lines are close to each other, we remove a vertex if its region
width Wis smaller than Wmin, the minimum region width parameter.
Intermediate regions with centrality filtering: A vertex with a region width between Wmin
and Wmax is removed if it is too close to the central line between neighbouring regular
contours. We filter the vertex by its centrality value: if the centrality value is greater than
a centrality threshold, then the vertex is removed. Because narrow regions should only
contain supplementary contours that are significantly closer to one of their neighbouring
contours, we vary the centrality threshold with region width. The centrality threshold could
vary linearly between Cmin (the centrality for Wmin)and1(thecentralityforWmax). However,
our experiments showed that linear interpolation is not sufficiently flexible, and a non-linear
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 9

mapping provides better results. The rest of this section provides details about this non-
linear mapping between region width and the centrality threshold.
The non-linear mapping of region width to the centrality threshold splits the range of
region width between Wmin and Wmax by the optimal centrality Wopt parameter into two
classes. As illustrated in Figure 10, class A maps region width between Wmin and Wopt to
Cmin;Copt
!"
and class B maps region width between Wopt and Wmax to Copt;1
!"
. We set
Cmin to a default value of 0.4, and Copt to a default value of 0.8. Most supplementary
contours are generated in class B areas. No centrality restriction is applied, when region
width is larger than Wmax.
The three grey shades in Figure 11(a) indicate the classes of region width defined by
Wmin,Wopt , and Wmax . We set Wmin,Wopt , and Wmax to 25%, 50%, and 75% of the largest
of all region width values in the elevation model.
Figure 10. Mapping region width to the centrality threshold. No supplementary contours are placed
in the darkest area; vertices are filtered by centrality when in class A or B; no centrality filtering is
applied to vertices in the white area.
Figure 11. (a) Region width, (b) centrality threshold obtained from (a) with the mapping in Figure 10.
10 T. SAMSONOV ET AL.

The three grey shades in Figure 11(b) show the centrality threshold values created by
applying the mapping in Figure 10. Supplementary contour lines can be placed any-
where, except in the darkest area in Figure 11(b).
Figure 12(a) shows the result of filtering vertices by region width and centrality. It can
be seen that supplementary contour lines are interrupted by numerous gaps, and some
lines are too short. These issues are addressed in the following section.
5.4. Filling gaps and removing short line segments
We improve the visual continuity of supplementary contour lines by filling gaps and
removing short contour segments. We start with the original entire supplementary
contour line and mark each vertex that has been removed in the previous step with
aflag F¼0 and each vertex that has not been removed with F¼1. Two operations
are then applied in the following order (see Figure 13(a,b)) to flag each vertex with
F¼1 to include the vertex in the supplementary contour line, or F¼0toexclude
the vertex:
Figure 12. Supplementary contours (dotted lines) (a) after filtering vertices by region width and
centrality threshold, (b) after filling gaps, removing short line segments, and extending line segments.
Figure 13. Creating continuous supplementary contours by filling gaps, removing short line segments,
and extending segments. (For illustrative simplicity, a constant distance between vertices is assumed.).
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 11

(1) Short gaps between supplementary contour line segments are filled. Gmin is the
minimal possible gap length parameter. Vertices are flagged with F¼1, if they
are part of a gap that is shorter than Gmin.
(2) Short supplementary contour line segments are removed. Lmin is the minimal
required length of a supplementary contour. All vertices of contour line segments
shorter than Lmin are flagged with F¼0.
5.5. Extending supplementary contour lines
This stage of the workflow aims at satisfying the design principle that supplementary
contours should reach the centre between two neighbouring contour lines (Figure 5). As
there is no guarantee that a supplementary contour line ever reaches the centre
between its neighbouring contours, we analyse the geometry at the start and end of
each supplementary contour line segment.
We introduce the extension centrality Cext and extension length Lext parameters.
Processing is performed individually for the start and end vertex of each segment.
Beginning at the start vertex, we traverse vertices that are not currently part of the
supplementary contour line segment. We compute the cumulative distance until the
cumulative distance becomes larger than Lext. Then, the vertex plwith the largest
centrality Clamong the traversed vertices is identified. If Cl(Cext then all traversed
vertices between the start vertex and plare flagged with F¼1(Figure 13(c)). Otherwise,
the flags are not changed. The same procedure is then performed starting at the end
vertex of the segment (Figure 13(d)).
A side effect of the extension procedure is a shortening of the gaps between line
segments, which requires an additional gap filling after the extension (Figure 13(e)). As
afinal step, contiguous sequences of vertices with F¼1 are converted to vector lines
(Figure 13(f)).
Figure 12(b) illustrates the combined effect of filling gaps, removing short line
segments, and extending line segments. In comparison to Figure 12(a), it can be seen
that in Figure 12(b) small line segments have been removed or merged into longer
continuous segments, and many contours are reaching a near-central position.
5.6. Workflow, default parameter values, and implementation
The entire workflow is represented in Figure 14. The final result is a combination of (a)
regular contour lines, (b) closed supplementary contour lines for depressions and peaks
that do not encircle any other contour line, and (c) supplementary contour lines
generated by the described method.
The set of parameters controlling the processing is represented in Table 1.An
explanation of each parameter is given in the Description column of Table 1. The
Value range column of Table 1 represents the possible range of values.
Optimal centrality Copt is the only fully independent parameter. The contour interval
h, the base level Hbase, the optimal region width Wopt , and the minimum length Lmin
depend on the elevation range, map scale, and spatial extent of the digital elevation
12 T. SAMSONOV ET AL.

Figure 14. Contour generation workflow. Raster fields are represented by white ovals (the DEM oval
is the start of the workflow), vector lines are represented by grey ovals, and procedures are
represented by grey rectangles. Parameter symbols from Table 1 are included.
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 13

Table 1. Parameters for generating supplementary contours.
Symbol Name Description
Default
value Value range
hContour interval Vertical interval between regular contour lines. –0;þ1
ðÞ
Hbase Base contour level Level from which the contour interval his added. 0 )1;þ1ðÞ
Wmin Minimum region width If region width is narrower than Wmin, vertices are filtered. 0:5%Wopt 0;Wopt
!"
Wopt Optimal region width Optimal width of a region at which supplementary contours should be constructed. –0;þ1½Þ
Wmax Maximum region width If region width is wider than Wmax, vertices are not filtered by the centrality threshold. 1:5%Wopt Wopt;þ1
!#
WeMinimum average region width for empty
contours
An empty supplementary contour is only included if its average region width is greater than
We.
0:25 %Wopt 0;Wopt
!"
Cmin Minimum centrality Maximum allowable centrality for Wmin. 0.4 0;Copt
!"
Copt Optimal centrality Maximum allowable centrality for Wopt. 0.8 0;1½+
Cext Extension centrality Minimum centrality that must be reached for extending a supplementary contour line segment. 0.8 Copt;1
!"
Lmin Minimum length Supplementary contours shorter than Lmin are removed. Wopt 0;þ1½Þ
Lext Extension length Supplementary contour line segments are extended on both sides by Lext.Wopt 0;Lmin
½+
Gmin Minimum gap length A gap between supplementary contours that is shorter than Gmin is filled. Wopt 0;Lmin
½+
14 T. SAMSONOV ET AL.

model. All other parameters are restricted by the values of Wopt ,Lmin, and Copt . Default
values were derived from experimental work described in the next section.
Empty supplementary contour lines –supplementary contours for depressions and
peaks that do not encircle any other contour line –are included if they are longer than
Lmin and if their average region width is greater than We. We do not filter empty
contours by their enclosed area, as filtering by area could falsely retain contours that
enclose long but narrow regions.
The workflow was implemented in Python as a freely available toolbox for Esri ArcGIS
that can be downloaded from a GitHub repository (Samsonov and Walther 2019). The
toolbox provides a graphical user interface for parameter selection, and generates both
regular and supplementary contour lines. Elevation models introduced in the following
section are also included in the repository to ensure reproducibility of our results.
6. Examples
Figures 15–20 present supplementary contour lines created for six digital elevation
models. In all six figures, the map labelled with (a) shows supplementary contour lines
after vertex filtering by region width and the centrality threshold. Grey indicates areas
where vertices are filtered. The map labelled with (b) in Figures 15–20 shows the final
Figure 15. Supplementary contours for a flood plain (model 1): (a) after filtering vertices by region
width and the centrality threshold, (b) after filling gaps, removing short segments, and extending
segments. Grey represents areas where vertices are filtered by region width and the centrality
threshold. For clarity, contour lines use thicker line widths than normal maps. Regular contour
interval is 10 m.
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supplementary contour lines after filling gaps, removing short segments, and extending
segments. We used the default parameter values listed in Table 1 for all six examples.
Wopt was set to the half of the maximum region width for each model.
Models 1, 2, and 3 are taken from the TSNIIGAIK album of examples of topographic
relief representation (Goldman et al.1968).
Model 1 represents low mountainous relief (Figure 15). Wide regions are confined to
the floodplain occupying the lowest elevations. The central supplementary contour line
in south-south-west direction illustrates how short segments are extended and com-
bined into a long contour without gaps. The map on the left (Figure 15(a)) shows four
line segments of various lengths. After filling the gaps between segments, the map on
the right (Figure 15(b)) shows a single continuous line. A comparison of the left and the
right maps also illustrates how short line segments are removed, when there are no
nearby segments to connect to (see the four short segments to the left of the map
centre).
Model 2 represents a glacial relief with many elongated terrain features created by the
passing of a glacier (Figure 16). Multiple closed supplementary contours that do not enclose
any other contour line were identified, accentuating the round shape of small hilltops.
Model 2 contains many supplementary contour lines that were extended towards the
centre between neighbouring contour lines. For example, all final contours in the top fourth
of the two maps were extended, clearly illustrating the effectiveness of this technique.
Figure 16. Supplementary contours for a glacial relief (model 2). Regular contour interval is 2.5 m.
Also, refer to the caption of Figure 15.
16 T. SAMSONOV ET AL.

Model 3 represents a volcanic trap plateau with a representative wide horizontal
plateau at the highest elevation level (Figure 17). The supplementary contours on the
plateau successfully show the relatively minor variations in elevation and slope.
Model 4 is a section of the SRTM digital elevation model of the Caucasus Mountains
(Figure 18). It represents a foothill region with a main mountain range located in the south-
west corner of the map. Wide regions are confined to low and middle elevation levels. The
addition of supplementary contour lines facilitates the interpretation of this complex
topography. There are many examples of supplementary contours that were composed
from multiple shorter segments, which were separated by high centrality values.
Model 5 is a section of the SRTM digital elevation model of the Divača karst in
Slovenia with an elevation range between 325 and 500 m (Figure 19). Supplementary
contours are particularly useful for such complex and rugged karst topography, as they
clearly illustrate karst sinkholes, and also show subtle positive and negative topographic
variations. Short lines were manually placed perpendicular to contour lines to indicate
the direction of descending slope.
Model 6 is a fragment of the Protva river valley in Central European Russia (Figure 20).
The model has a horizontal resolution of 5 m, and was created from aerial surveys. The
elevation varies between 140 m and 180 m; the contour interval is 5 m. A pair of
supplementary contour lines are very useful for depicting the lowest areas of the river
floodplain, which regular contours cannot show. A few short candidate supplementary
contour lines indicating small peaks in the floodplain were not included in the final map,
because they are too short or too narrow.
7. Discussion
The supplementary contour lines Figures 15–20 created with the presented method are
of production-ready quality. We collected feedback and comments from two senior
expert cartographers specialising in topographic mapping at the Swiss national
Figure 17. Supplementary contours for a volcanic trap plateau (model 3). Regular contour interval is
10 m. Also, refer to the caption of Figure 15.
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 17

Figure 18. Supplementary contours for foothills in the Caucasus Mountains (model 4). Regular
contour interval is 100 m. Also, refer to the caption of Figure 15.
18 T. SAMSONOV ET AL.

mapping agency Swisstopo and the Institute of Cartography and Geoinformation of ETH
Zurich, two institutions that are internationally renown for their expertise in terrain
mapping (Tufte 1990, Field 2018). The experts were asked to comment on the selection
and the length of the automatically placed supplementary contour lines. The feedback
on the selection and the length of supplementary contour lines was very positive, and
the expert cartographers commented that additional manual editing could not improve
these supplementary contours substantially.
Default values for almost all workflow parameters (Table 1) can either be pre-set or
derived from the optimal region width parameter Wopt. This parameter is used to compute
default parameter values for region widths, gap filling, the removal of short line segments,
the extension of contours, and the filtering of empty supplementary contours for depres-
sions and peaks. These default parameter values do not guarantee ideal results for all
terrain types, but serve as a valid starting point for fine-tuning parameters, if needed.
Figure 19. Supplementary contours for sinkholes of the Divača karst in Slovenia (model 5). Regular
contour interval is 25 m. Also, refer to the caption of Figure 15.
Figure 20. Supplementary contours for the Protva river valley in Central European Russia (model 6).
Regular contour interval is 5 m. Also, refer to the caption of Figure 15.
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 19

The derivation of region width and centrality are computationally relatively expen-
sive operations. In particular, the circle-based computation of the region width raster
has an algorithmic complexity of On
2
ðÞ,wherenis the width or height of an elevation
model in cells. Our single-threaded Python implementation creates a region width
raster of 500 ,500 cells in about 20 s with an Intel Core i7 2.6 GHz laptop computer.
This computational overhead does not allow for real-time visualization of surfaces, but
is suitable for topographic map production. To improve performance, many processing
stages of the workflow could be multi-threaded and re-implemented using a high-
speed compiled language.
8. Conclusion
The desire to show as many details as possible with contour lines seems to entice some
map authors to select a small contour interval that results in very dense contour line
images. This is understandable, as creating contour lines from two-dimensional scalar fields
is simple and very fast with current software. However, a dense covering of contour lines
inevitably creates graphical conflicts with other map features, resulting in reduced read-
ability of text labels, lines, and other map symbols. In addition, densely adjoining contour
lines are often almost parallel across large areas. In these cases, the contour interval could
be increased, and the resulting less densely packed contours would still convey the same
information about the shape of the underlying surface. Another argument against overly
dense contour lines is their interference with the three-dimensional effect of shaded relief,
which is commonly used in combination with contour lines (Imhof 1982).
A well established and proven solution from manual cartography is the combination
of contour lines of moderate density with accurately placed supplementary contours.
With the presented method for the automated placement of supplementary contour
lines, we hope to contribute to improved contour line maps that are easy to read and
still show relevant details.
Acknowledgments
The authors thank the anonymous reviewers for their valuable comments, Sebastian Hennig for
his help with creating some of the figures, Jürg Gilgen of the Swiss Federal Office of Topography
and Stefan Räber of the Institute of Cartography and Geoinformation of ETH Zurich for their
comments and feedback on maps with automated supplementary contour lines, and Jane
Darbyshire, Oregon State University, for copy-editing this manuscript.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes on contributors
Timofey Samsonov is a leading researcher at Lomonosov Moscow State University (MSU), Faculty
of Geography, Moscow, Russia. He works in the field of automated cartography with a particular
interest in algorithms for cartographic generalization and visualization of spatial data.
20 T. SAMSONOV ET AL.

Sergey Koshel is a leading researcher at Lomonosov Moscow State University (MSU), Faculty of
Geography, Moscow, Russia. His research interests include digital terrain modeling and spatial
interpolation.
Dmitry Walther is currently a Bachelor student at Lomonosov MSU, Faculty of Geography,
Moscow, Russia.
Bernhard Jenny is an Associate Professor at Monash University, Melbourne, Australia. His research
focuses on immersive geovisualisation, map design, map projections, and terrain mapping in 2D
and 3D.
ORCID
Timofey Samsonov http://orcid.org/0000-0001-5994-0302
Sergey Koshel http://orcid.org/0000-0002-4540-2922
Bernhard Jenny http://orcid.org/0000-0001-6101-6100
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