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POLAR ZENITHAL MAP Part 1 PROJECTIONS Five basic by Keith Selkirk, School of Education, University of Nottingham projections

Map projections used to be studied as part of the geography syllabus, but have disappeared from it in recent years. They provide some excellent practical work in trigonometry and , and deserve to be more widely studied for this alone. In addition, at a time when world-wide travel is becoming increasingly common, it is un- fortunate that few people are aware of the effects of map projections upon the resulting maps, particularly in the polar regions. In the first of these articles we shall study five basic types of and in the second we shall look at some of the which can be developed from them.

Polar Zenithal Projections The zenithal and cylindrical projections are limiting cases of the conical projection. This is illustrated in Figure 1. The A football cannot be wrapped in a piece of paper without either semi-vertical of a cone is the angle between its axis and a stretching or creasing the paper. This is what we mean when straight line on its through its vertex. Figure l(b) and we say that the surface of a is not developable into a (c) shows cones of semi-vertical 600 and 300 respectively . As a result, any plane map of a sphere must contain dis- resting on a sphere. If the semi-vertical angle is increased to 900, tortion; the way in which this distortion takes place is deter- the cone becomes a disc as in Figure l(a) and this is the zenithal mined by the . There are three fundamental case. If the semi-vertical angle decreases, the vertex moves types of projection, the zenithal, the conical, and the cylindrical. towards the top of the diagram and in the limit, when it tends In the first of these, the map is regarded as part of a plane, in to , the cone tends to a cylinder as in Figure 1(d). the second as part of a cone and in the third as part of a cylinder. Because they start with a plane, the zenithal projections are In the latter two cases the cone and the cylinder are unrolled easier to visualise, and we shall restrict ourselves to them, even and laid flat, these two surfaces, unlike the sphere's surface though they are less common in practice than the other types. being developable into planes. The point where the plane touches the sphere is called the

Fig. 1 The conical projection and its limiting cases.

300

600o

(b) Conical (c) Conical (a) Zenithal (semi-vertical angle 600) (semi-vertical angle 300) (d) Cylindrical

2 Mathematics in School, March 1982 zenith, and this point may be anywhere on the sphere. There show that NSQ= 1/2 (900 - 0), whence is no fundamental difference in the projection wherever the NQ= 2R tan /2 (900 -0). zenith is, but because of the way in which we define position on the surface of the earth by means of and , The important advantage of the stereographic projection is the case where the zenith is at a pole is the easiest to study from that it is shape-preserving or orthomorphic. Although the scale a mathematical point of view. These are the polar zenithal is not constant, the shape at any point is correct because the projections we shall examine below. distortion is the same in all directions. We shall show why this is the case in the second article. The In the gnomonic projection the map may be considered as a The Orthographic touching the earth at the north pole. A light at the centre If the light source is now removed a long way off so that a of the earth then casts a shadow of the earth's surface on the parallel beam of light is produced, then Figure 4 is the result. plane map. Thus the word projection may be used literally, and In this figure K is the foot of the perpendicular from P onto this is illustrated in the lower half of Figure 2. This shows a the equatorial plane. In this diagram cross-section through the earth and the map, the centre of the NQ=R cos 0. earth being labelled O and the north pole N. The point P is a typical point in the northern hemisphere with latitude 0 The gives a view of the earth as seen (measured in degrees). The point P maps or projects onto the from outer space. The projection would be a suitable one to point Q of the map. The upper part of the figure shows the use when mapping the visible surface of the for earth- map itself with the pole N and the of latitude for P. bound astronomers. If R is the radius of the earth, then NQ=R cot 0 The Equidistant Projection On the map, therefore, the radius of the 0 will The above three projections are called projections be R cot 0. (In practice this will be reduced by the scale of the since they are views of the northern hemisphere when seen map, but we can ignore this for these articles, and imagine the from particular points, namely the centre of the earth, the map drawn at full scale.) south pole and an infinitely distant point in a southerly direc- The gnomonic projection has some grave disadvantages as tion respectively. (In fact we are in these cases viewing the map we shall see, but it has one outstanding advantage. The shortest from behind.) The last two projections do not have this distance between any two points on the surface of the earth is property and are therefore known as conventional projections. an arc of a circle on the surface whose centre is at the centre The equidistant projection is designed in such a way that of the earth. This is called a great circle to distinguish it from distances measured from the zenith are correct. This means smaller which may also be drawn on the earth's surface that the concentric circles forming the circles of latitude on the such as circles of latitude. Since the projection is from the map have their radii equally spaced. This makes it a very centre of the earth which is the centre of all great circles, any natural map to use, since this is the most obvious way to draw given great circle will have a shadow which is a plane, and the basic grid of the map. In Figure 5 it means that which must intersect the plane of the map in a straight line. NQ=arc NP Thus great circles project into straight lines on the map, and any straight line on the map is the projection of a great circle. _ (900 - 0) 2tnR 3600 The gnomonic projection is therefore of great use in since it makes it easier to the shortest route on the surface This property makes it suitable for use in cases where distances of the earth. We shall return to this point in the second article. from the zenith need to be compared, remembering that the zenith can be placed anywhere we wish. The Stereographic Projection If the light (or centre of projection) is placed at the south pole, The Equal Area Projection S, Figure 2 changes to Figure 3 and the stereographic projec- In the equal area projection, equal areas on the earth cor- tion is obtained. To determine NQ in this case, we must first respond, as might be expected, to equal areas on the map.

Fig. 2 Polar zenithal Fig. 3 Polar zenithal Fig. 4 Polar zenithal Fig. 5 Polar zenithal Fig. 6 Polar zenithal gnomonic projection. stereographic projection. orthographic projection. equidistant projection. equal area projection.

N 0 N ao N o N i N 0Q

A N N N NcN N '0 0 Q a/ x x ---- p x p 0 il Xt p

X 0 0 0 K 0

0

Mathematics in School, March 1982 3 Figure 6 illustrates this as far as is possible on a plane diagram. (The same result for the whole sphere shows that the surface The crux of the matter is that for all points P, the area of the area of the sphere is 4itR2 and is illustrated in the crest of the spherical "cap" of the earth bounded by the circle of latitude Mathematical Association.) Now on which P lies must equal the area of the circle of radius NQ on the map, where Q is the point onto which P projects. The XN= R(1 - sin 0) cap is indicated by the shaded area of Figure 6; clearly because Hence of the curvature of the earth, the radius XP of its bounding circle is less than NQ. 7tNQ2= 2r R2 (1 - sin 0) The area of the cap is impossible to find without the use of and the calculus, or some method such as the Greeks used which effectively assumes the methods of the integral calculus. NQ=R,2 (1--sin 0) The result is, however, comparatively well known and rather This type of projection is of great importance and should be surprising. It states that the area of the cap is equal to the area used whenever symbols are used to indicate the density of a of the curved surface of the cylinder of radius R and height feature on the earth's surface. If this is not done, a very mis- XN. Thus the area of the cap is leading impression of the relative densities in different parts of 27tR .XN the map may be given.

Getting the answer wrong by Lesley R. Booth, SESM Project, Chelsea College

Teachers are quite used to the idea that children sometimes get and-pencil tests of understanding in different topic areas, such mathematics questions wrong. Sometimes the wrong answer is as ratio, generalised arithmetic, measurement, fractions, graphs due to a "careless" mistake which is easily corrected; often and so on. Examples of the kinds of wrong answer which were however, the teacher feels that the error is symptomatic of some found are shown in Figure 1. misunderstanding on the part of the child. If only we could By carefully examining these wrong answers, and the responses find out why the child had made that error, we might be able which children had given when asked about their answers in to help the child restructure his or her concept of the prob- interviews conducted as part of the CSMS test development lem, and so avoid the error. Unfortunately, teachers usually programme, it was possible to suggest reasons why these par- don't have time to diagnose every child's mistakes in this way. ticular errors may have occurred. Nor, of course, do research workers, but they may have a In essence it was suggested that many of these errors might chance of studying "common errors" - particular wrong be due to children's use of intuitive "child-methods" which are answers which are made by large numbers of children. perfectly adequate for handling "easy" problems, but which This is the task which the Strategies and Errors in Secondary do not generalise to harder questions, where success really Mathematics (SESM) Project1 has set itself. Funded by the requires the use of the "proper" mathematical methods taught Social Science Research Council and based at Chelsea College, in the classroom6,7. In addition, of course, it was thought this project aims to investigate particular mathematical errors highly likely that there would be specific kinds of misconception identified by the earlier Concepts in Secondary Mathematics associated with the different areas of mathematics - algebra8, and Science (CSMS) Project2'34,"5. The errors chosen for study number""., fractions", and so on. Consequently, the SESM were those which were made by a large proportion (in some Project set out to examine these ideas by interviewing indi- cases 50% or more) of the children tested by CSMS on paper- vidual children on relevant problems, in order to discover precisely which kinds of misunderstanding contribute most to the particular errors under study. Percentage CSMS item "Error" giving While the results of this (still continuing) examination seem (abridged) answer answer to support the notion that children often do have their own (13 year olds) methods which break down when the questions get harder, the investigation into children's work on beginning algebra, or

2+3 5 30 "generalised arithmetic", has also revealed another source of 7 4 11 error. It seems that in this topic at least, some of the mistakes which children make are due not to their misinterpretation of the question or their ideas of what letters mean, nor to the Add 4 onto 3n 3n4, 7n 45 methods which they use to solve the problem, but rather to their misconceptions concerning the way in which the answer

Percentage CSMS "Error" giving Perimeter s s 2u556, 2ul 6 46 item answer answer 6 (13 year olds)

c C and D are 48 1. What can you write for 5e2, e 10, 10e, 42 LD I 1 same length the area of this e + 10 rectangle: 912 13 44 21 I e 21 I Fig. 1 Fig. 2

4 Mathematics in School, March 1982