Contour Mapping

7/31/2019 Contour Mapping

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CONTOUR MAPPING

This article describes the contour map , a common type of drawing incivil engineering. It is an ideal means of representing a three-dimensional surface using a single two-dimensional view. Contourmaps not only convey a qualitative impression of the features of thesurface but also enable, using a single view, complete quantitativeinformation to be extracted from the drawing. Although they are oftenused for topographic maps, contour maps have other applications thatwill be briefly mentioned in the article.

1. what it isWe will define what a contour map actually is by considering a simpleexample of representing a three-dimensional landform using only twodimensions.

A three-dimensional surface is given. For example, we can consider a

the landform shown in the figure to the right. The perspective viewshown is obviously a two-dimensional representation of a three-dimensional object. This conveys a reasonable qualitative impressionof the overall characteristics of the landform, but it does have thefollowing shortcomings:

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Perspective view of a three-dimensional landform


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1. Depending on the point from which the view is taken, we may missvaluable information. For example, when the landform is viewed toobtain the upper of the two views shown to the right, we have noinformation on what lies beyond the red line. We need to rotate toa different viewpoint to obtain this information.

2. There is no reliable way of extracting quantitative informationfrom this drawing. For example, we may wish to know the elevationof a specific point on this landform or the difference in elevationbetween two given points. Not only is it difficult to locate a givenpoint exactly in a horizontal plane, it is likewise difficult to gainmore than a general impression of its elevation. It is thus notpossible to use this drawing to answer questions such as What isthe elevation of a point 400 m to the north and 300 m to the eastof Point A.

We can solve the first problem to some extent by using multiple viewsof the same landform. This is shown in the figure to the right, wherewe have rotated the original viewpoint by approximately 90 degrees toobtain a second view. With the help of the second view, we can seewhat lies beyond the ridge. By selecting a sufficient number of views,we can generally provide a correct qualitative impression of the entirelandform. The problem with this approach is that it is not compact, inthe sense that one view is generally not enough. Furthermore, to be of value as a basis for extracting quantitative information, this approachalso requires a means of relating one view to another. This is not astraightforward task.

contour mapping civ 235s civil engineering gra

Perspective views of a three-dimensional landform from two different viewpoints


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We can solve the second problem to some extent by identifyingspecific points and writing in the elevation of these points on thedrawing. This has been done for two points in the drawing to the right.Elevations are given in metres above sea level. This increases thequantitative content of the drawing. If a sufficiently large number of such elevations were given, it would be possible to estimate, at leastapproximately, elevations of other points by interpolation. There is still

no reliable quantitative basis, however, for locating points in thehorizontal plane. It is practically impossible, for example, to know thedistance and bearing of the point with elevation 1019 relative to thepoint with elevation 1178. Reliable answers to questions relating to theelevation of a point of known coordinates relative to a given point thusremain difficult to obtain.

Neither providing multiple perspective views nor providing elevationvalues for given points thus allows us to represent the threedimensional surface using a single two-dimensional drawing thatenables quantitative information to be extracted (i.e., the z coordinateof a point given it s x and y coordinates).


Perspective view of a three-dimensional landform (elevations are given in metres above s

El. 1178

El. 10


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To accomplish this objective, we need a more suitable twodimensional representation of the three dimensional surface. Todevelop this representation, we return to the previous perspectiveviews. We first imagine that the landform has been sliced by ahorizontal plane of constant elevation. In this case, say its elevation is1000 m. Where this plane cuts the landform defines one or morecurves in a horizontal plane. We can erase the plane itself but leave

the line created by the intersection of the plane and the landform. Thiscurve joints points of equal elevation. We define any curve joiningpoints of equal elevation a contour line , or simply a contour .


Landform sliced by a plane at Elevation 1000.

Contour formed by intersection of plane and landform. All points along contour are at Ele

El. 1000

El. 1


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We can repeat this construction for planes at other elevations. Forexample, we can do so for planes with elevation 1100 m and 900 m.

We can provide greater detail by showing more contour lines. In thelower view to the right, we have shown contours at increments of 20 m.We say that the contour interval in this drawing is 20 m. This contourinterval appears to cover the landform reasonably well and capturethe changes in topography.

Although this drawing conveys significantly more information thanthe original perspective view given on the first page of this article, it isin itself is not particularly useful, since it has most of the shortcomingsof the drawings developed initially. But it can be transformed into apowerful drawing by representing this information on a horizontalplane.


Landform with three contours (top), landform with many contours (bottom)

El. 1100

El. 1000

El. 900

El. 900

El. 1000

El. 1100


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We do this by viewing the landform, with the contour lines, lookingdirectly down from a point above it. By choosing this viewpoint, wegain a dimensionally true representation of the horizontal plane,which permits us to use true x-y coordinates to locate points. Thevertical dimension disappears visually but is now represented in amore abstract way through the contour lines, which now appear to bedrawn on a horizontal plane.

A given point can now be located and measured in x-y coordinatesfrom any other point on the map. Its elevation can be read directlyfrom the contour that intersects the point or, for points located inbetween contours, by interpolation. For example, a point 500 m to theeast and 500 m to the north of the origin in the lower left portion of the drawing (shown with the red dot) is found to have an elevation of approximately 1008 m.


Landform with contours viewed from above. Contour interval is 20 m.

E l . 1

1 0 0

E l . 1 0 0 0

E l . 9 0 0

E l . 1 0 0

0

E l . 9 0 0 1000 m

10


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It is important that contour maps always incorporate a constantcontour interval. By doing this, we can get a visual sense of the three-dimensional properties of the surface, even when the shading of theoriginal landform has been removed, by considering the patternsformed by the contours.

The closest path from one contour line to an adjacent contour givesthe steepest path. This follows directly from the definition of slope,which is equal to rise over run. For a constant rise (fixed contourinterval), the greatest slope corresponds to the smallest value of run(closest distance between contours). It therefore follows that closelyspaced contours denote regions that are steep and widely spacedcontours are regions that are relatively flat.

Closed curves denote either hills or depressions. We distinguishbetween the two by considering whether the contours are increasingor decreasing. In the case of the map shown to the right, the changeof the contours indicates that the triangular figure enclosed by thegreen rectangle would be the top of a hill.

A series of adjacent contours that all point in the same directionoften indicates the path of a river, since this corresponds to thelandform created by the flow of water through the earth. The blue linedrawn onto the map to the right indicates one possible river.

We can summarize the essence of contour maps as follows. Contourmaps allow us to represent three-dimensional surfaces using a singletwo-dimensional drawing. They maintain the ability to locate pointsaccurately in two horizontal dimensions. Everything in these twodimensions is drawn to scale. We lose a direct means of visualizingthe third dimension, but are able to represent it accurately throughlines joining points of equal elevation, called contour lines. These areequivalent to the curve formed by intersecting the surface of the givenlandform with planes of constant elevation. By working with a


Landform with contours viewed from above. Contour interval is 20 m.

E l . 1

1 0 0

E l . 1 0 0 0

E l . 9 0 0

E l . 1 0 0

0

E l . 9 0 0


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constant contour interval, we gain the ability to visualize the three-dimensional characteristics of the surface.

The data used to generate the landform considered in this exampleoriginate from the National Map of Switzerland. The correspondingsection of this map is shown in the figure to the right. The contourinterval in this case is 10 m.


Excerpt from National Map of Switzerland used for creation of landform. The grid square


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2. suitable applicationsContour drawings are not the best way to represent all three-dimensional objects. They are best suited to representing objects thathave a single surface , a significant and well defined reference plane ,and a significant third coordinate perpendicular to the reference plane .

It follows from these conditions that contour drawings are well suited

to the representation of landforms with a single two-dimensionaldrawing. They have a single surface (the surface of ground), a welldefined reference plane (the horizontal plane) and a significant thirdcoordinate perpendicular to the reference plane (the z coordinaterepresents elevation, which is of primary significance).

Contour drawings are not suitable for representing other types of three dimensional objects when at least one of these conditions is notsatisfied. The bridge pictured in the figure to the right, for example,does not have a single surface, but rather several including a nearvertical surface, a far vertical surface, and an upper surface. A singlecontour diagram cannot adequately represent all of these surfaces.This type of object is best represented in other ways, such as with

multiple views based on standard viewing planes. In these drawingsonly two dimensions are depicted in each view. No quantitative (andoften no qualitative) information regarding the third coordinate canbe extracted from a given view. For this reason, more than one view isrequired to describe the object completely.

3. how to do itThere are several ways to produce contour drawings from a set of x, y,and z coordinates describing a given three-dimensional surface. Thissection describes one way that is relatively straightforward.

Given:


Bridge represented by multiple two-dimensional views


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1. A regular square grid of points, with x, y, and z coordinates of thesurface defined for each point of the grid. This grid is shown in theimage to the right. It has nine points. The scale in the horizontalplane is given graphically. Elevations in metres are given for eachgrid point.

Required:

Produce a contour drawing representing the surface defined by thegiven x, y, and z coordinates.

How to proceed:

1. Set the contour interval. To do this, it is necessary first to scan the zcoordinates to extract the minimum and maximum values. Withinthis range, define an interval that is regular and that captures therelevant features of the surface with good fidelity. In this case,regular means taken from the series 1 m, 2 m, 5 m, 10 m, 20 m, 50m, 100 m, etc. When working with a set of several diagrams, it isusually preferable to use a constant contour interval over theentire set of diagrams to enable comparisons across the set of drawings. In such cases, the choice of interval should be made in

consideration of the properties of the entire set of data. For thisexample, the minimum and maximum elevations are 12 and 45 mrespectively. For simplicity, we will use a contour interval of 10 mfor this example. So relevant contours will be at the 20, 30, and 40m elevations.

2. Draw the grid to a suitable scale. As always, use regular scales (i.e.,from the series 1:1, 1:2, 1:5, 1:10, 1:20, 1:50, 1:100, etc...). In this case,this has already been done for us.

3. For easy reference, write in the z values next to the correspondingpoints of the grid. This has already been provided.

4. For each segment joining adjacent points of the grid, identifypoints of intersection of contour lines with the segment, based on


Given information for contour map


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the assumption that the change in z within a given segment of the grid is linear . Proceed according to the following example:(a) Given: The top left horizontal segment AB of the grid has the

following z values: z(A)=19.0 m, z(B)=22.0 m.

(b) So one contour will intersect this segment. It is the 20 mcontour, since 19 < 20 < 22.

(c) Locate the point of intersection of the contour within thissegment by linear interpolation. One straightforward way toaccomplish this is to use a scale in a way similar to the methodused to subdivide a line into several equal segments:

(i) On the vertical gridline passing through the left end point,align the scale to a value corresponding to the elevation atthat location. In this case, the scale is set to 90.

(ii) On the vertical gridline passing through the right end point,align the scale to a value corresponding to the elevation atthat location. In this case, the scale is set to 120. The scaleremains at 90 along the left gridline. The series 90, 100, 110,120 defined by the scale is similar to the series 19, 20, 21, 22defined by the given elevations. So the intersection of the 20m contour with the given line segment will correspond to100 on the scale.

(iii) Draw a line perpendicular to the given segmentcorresponding to the point of intersection identified withthe scale. This locates the intersection of the contour withthe given segment.

(iv) Write the value of the contour next to the intersection point.

(v) Note: the accuracy of this procedure increases as the anglebetween the scale and Segment AB gets smaller. So it isusually helpful to try fitting several scales to the given

segment to minimize this angle.


Locating intersection of contour and gridline

A

B20


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(vi) The outcome of this phase of the process is shown in thefigure to the right.


Intersections of contours and gridlines completely identified


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(d) When all of the points of intersection of contours and segmentshave been thus identified, draw the contour lines. Proceed on asquare by square basis. For a given square bounded by fouradjacent grid points, the following two cases must beconsidered:

(i) A given contour intersects exactly two bounding segmentsof the square. This is the case, for example, for the 20 mcontour in the upper left-hand square. In this case, simplydraw a line joining the points of intersection. This line isthe path of the contour within the square. The image to theright shows the 20 m and the 30 m contours drawn for theupper left hand square in the grid.

(ii) A given contour intersects all four bounding segments of the square. In this case, it is not clear how to draw thecontours. The figure to the right shows that a singlearrangement of intersecting points can correspond toseveral arrangements of contours. In this case, only onearrangement (the middle one) corresponds to the giventhree-dimensional figure shown.


Irregular situation: contour intersects grid square on four sides. There are several possiblecontours, only one of which correctly represents the three dimensional figure shown

Drawing the contours


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(e) The completed contours are shown in the figure to the right.

(f) Once the complete contour diagram has been drawn, trace thecontours onto a new sheet of paper. This leaves only thecontours and does not show the working grid and other marksthat were made to produce them. It is generally necessary tolabel specific contours and spot elevations. As with all planviews (i.e. top views), a north arrow is required.


Completed contours (before tracing)


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4. variationsIn some cases, it i s preferable to draw smooth curves for the contours.This will often provide a more realistic rendition of the features of agiven landscape.

All of the contour maps we will draw in this course will be done bystraight line segments linking points of equal elevation along gridlines,

as described in the previous section.

It is common to create contour maps from survey data obtained in thefield. In such cases, it is sometimes difficult to get elevation values fora square grid of points in the plane. It is also possible to create acontour map following the principles outlined in the previous sectionfor an irregular collection of points. In such a case, it is necessary to

establish a triangular network of lines joining the available points inthe plane. This is shown in the leftmost diagram. Along these lines,contour values are interpolated, as shown in the middle diagram.Finally, for a given triangle, straight line segments are drawn linkingpoints on the boundary of identical contour value. This is shown inthe rightmost diagram, where contour lines have been highlighted ingreen.


Contour map showing contours as smooth curves

Contour mapping when points are not arranged on a square grid


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5. examples from practiceThis section describes several types of contour drawing in commonuse in engineering.

5.1. Standard Topographic Maps Topographic maps describe, with a high level of detail and accuracy,

the topography (i.e., the shape) of a given geographical area. Thestandard way of representing the three dimensional features of landforms is contour lines.

The first example is from Canadas National Topographic System of maps. The most detailed scale available is 1:50 000.

The second example is from the National Map of Switzerland. Thismap is drawn to a scale of 1:50 000.

The third example is also from the National Map of Switzerland, thistime from their 1:25 000 series.

The Swiss maps are produced with much greater detail and withadditional visual cues to help the user gain a qualitative impression of

the three-dimensional landforms from the contours. This isaccomplished by: (1) a relatively small contour interval (in this case 10metres), (2) subtle shading that corresponds to the shadows that wouldbe cast on the landforms when the sun shines from the northwestquadrant of the map, and (3) pictorial symbols such as the cliff symbol,which is used when the slope of the land is so steep as to causeexcessive bunching of the contour lines.


Canadian National Topographic System 1:50 000 (to scale). Each square represents 1 km. Contour interval is 100 feet (about 30 m)

National Map of Switzerland (Landeskarte der Schweiz) 1:50 000 (to scale). Each square represents 1km. Contour interval is 20 m.

National Map of Switzerland (Landeskarte der Schweiz) 1:25 000 (to scale). Each squarerepresents 1 km. Contour interval is 10 m.


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5.2. Project-specific topographic plans and diagrams When the level of detail given on standard topographic maps isinsufficient, it is possible to produce topographic plans for a given sitebased on specific survey data. These plans are used, for example, forthe layout of bridges. In this diagram, for example, the contour intervalis 5 m, which is considerably less than the contour interval used onstandard topographic maps.

This type of map will generally be prepared by a specialist land surveyfirm.

Although contour diagrams are most often used to represent naturalfeatures such as topography, they are sometimes used to representfeatures of the facility to be built. Contour maps are sometimes made,for example, of bridge decks to validate that drainage will workproperly.

5.3. Contour graphs It is also common to use contours to represent abstract surfaces, i.e.,mathematical functions of two variables z=f(x,y). In such cases, x and

y need to be spatial coordinates in a well defined and meaningfulplane. Function z then defines a three-dimensional surface, similar toa landform. The same principles used to draw contour maps of physical landforms can be used to draw contour graphs of suchfunctions.

The figure to the right shows one such application. This diagram iscalled an inf luence surface for a slab free along the bottom, fixed inshear and bending along the top, and extending to infinity in theother two directions. It is based on the function z = Ma(x,y), where A isa given fixed point and Ma is the bending moment at Point A due to aunit load applied at Point (x,y). It is the two-dimensional analog of theone-dimensional influence line .


Project-specific topographic map used for location of a bridge

Influence surface for a cantilever slab. Fixed end is at the top. Contours give bending moper metre of length for a unit load applied at that location (Pucher, Einflussfelder elastiscPlatten)


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This type of diagram can be used to calculate bending moments inbridge deck slabs due to loads applied by the wheels of a heavy truck.Its use is illus trated in the figure to the right.


Use of influence surfaces in calculating bending moments in cantilever bridge deck slabs


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6. applications: cutting sections from contourmaps

Contour maps are useful in and of themselves, but they are also usedas a basis for producing relevant two-dimensional drawings. These canbe visualized as the curve formed by the intersection of the giventhree dimensional surface and a vertical plane. The curve thus formedis often referred to as a section . The process of drawing this curve froma given contour map is referred to as cutting a section .A common application of sections cut from contour maps is theproduction of the elevation of a valley to be crossed by a bridge. Theprocedure for cutting such a section is relatively straightforward.

Given:

1. Contour map of the area under consideration

2. A straight line drawn on the contour map locating the verticalplane that defines the section (Line A-A).

Required:

A two-dimensional drawing (x-z view) showing the shape of thelandform along the given line.How to proceed:

1. Draw a line parallel to Line A-A on a blank portion of the page. Thiswill be the horizontal datum of the section to be cut.

2. Based on these maximum and minimum elevations intersected byLine A-A, draw a vertical scale of elevations to the left of, andperpendicular to, the horizontal datum. Provide suitable labels tothis axis. Draw horizontal gridlines based on the labels given onthe axis. The drawing to the right represents the progress thus far.The area just created is called the section diagram.

3. For each contour that intersects Line A-A, do the following:


Setting up to cut section from contour map along Line A-A


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(a) Identify the points of intersection of the contour and Line A-A.We will call one such point Point P1.

(b) Draw lines perpendicular to Line A-A from Points Pi to thesection diagram. The outcome of this step is shown in thediagram to the right.


Draw lines perpendicular to Line A-A from intersections with contours to the section diag


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(c) Draw horizontal lines in the section diagram corresponding tothe contour elevations.

(d) Identify the points of intersection Q1 for specific perpendicularsoriginating from the contours and the corresponding elevationin the section diagram.

4. Join the points Qi to form a continuous curve. The resulting curveis the section cut along Line A-A.

The figure to the right shows the finished section cut along Line A-A.The location of one point of the profile is highlighted. It correspondsto the 1000 level contour.


Finished section cut from the contours

Contour Mapping

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