The Polar Stereographic Map
This chapter explains what the polar stereographic map is, how it works, and how it can be used to our advantage, since it is the most widely used map in weather analysis and forecasting. A more extensive and detailed explanation can be found in other books. Skipping the equations in this chapter will not interfere with your understanding of the following chapters. Mathematical equations are quite often used as the proof that certain facts or conclusions are true. You can still accept the conclusions as being true, even if you don't know or understand the mathematics involved. I have added the mathematics for those who want the exact proofs. If you have studied the equivalent of high school geometry, the mathematics in this section should not be too difficult.The explanation I offer is perhaps easier to understand than any book on the polar stereographic map that I have been able to find. It is, nevertheless, precise and lays the foundation for meteorologists (and everybody else} as to some of the mathematics behind the charts in the latter part of the book. There is nothing new in my explanation of the map, but I have developed specific tools to aid in the use of the weather map that are indeed new to the field of meteorology.
Every map maker knows that there are all types of projections to show a spherical Earth on a flat piece of paper. But since no projection on paper can be completely right for all purposes, different ones are used for many different purposes. Some flat maps reproduce shapes, others are accurate for angles, directions, equal area comparisons, others for small areas, etc. With these known facts, a straight line or angle on one type of flat map doesn't have the same meaning as on another type. A straightline on one type of map can be compared with a straight line on another type of map by the means of transformation formulas.
Polar stereographic maps are used (among other reasons) because:
Construction of the Polar Stereographic Map
In the polar stereographic projection, a point of light is placed at the South Pole and a tangent plane is placed at the North Pole, which is the plane upon which the projection is made. To get a better understanding, see Figure 31. The point of light at the South Pole of the sphere is called the center of projection.
Figure 3-1. Light projected from the south pole onto a flat transparent screen, which is tangent to the north pole, shows lines of longitude (which are great circles) as straight lines; while the circles of latitude (which are small circles except for the equator) show as perfect circles.
In Figure 3-2, NESW represents the plane of a meridian, where N is the North Pole, S the South Pole, and EOW is a line joining two opposite points, E and W on the Equator, where O is the center of the sphere. If we divide arc NE into 9 equal parts, each part in the figure would represent 10° of latitude. Arc NE is of course 90°. All points on the sphere
Figure 3-2. The plane of any meridian (NESW)goes through the north and south poles, and shows the trace (NT) of the transparent plane shown in Figure 3-1.
with a distance of 10° from the North Pole will lie on a circle with a radius Na. The circle is on the tangent plane, with N as its center. Those points lying 20° from the North Pole will lie on a circle with a radius Nb, and so on.
The plane of projection could be placed at any latitude below the North Pole, but it must always be above the center of projection at the South Pole. The latitude at which the plane is drawn is called the standard latitude. A plane at the equator is known as a primitive plane or equatorial plane.
There are three things that determine the size of the map: the size of the globe used to construct the map, the standard latitude, and the scale.
The scale can be defined as a relation between a given distance on the ground and the corresponding distance on the map. The scale could also be defined as a ratio equal to the distance between two points on the map, divided by the distance between the same two points on the Earth. Remember, the scale represents distance only, not area. Consider the triangle in Figure 3-3, where side ab is 1/2 of side AB, but the area of triangle abc is not 1/2 of triangle ABC. The historical weather maps used in this book have a scale of 1/30,000,000 or 1 inch on the map is equal to 30,000,000 inches on the surface of the Earth. It is important to note here that the scale is exact only at the latitude where the plane was
Figure 3-3. Doubling the length of the sides of a triangle, more than doubles the area.
placed, when the projection was made. On the historical weather maps, the scale is accurate at 60° north latitude. The choice of 60° north was made to give smaller average absolute departures of scale over the most important working area of the map. However, to prove certain mathematical relationships on the map, it is easier to assume the plane to be at the North Pole or the Equator, since the results obtained will apply regardless of the standard latitude that may be used.
As can be seen from Figure 34, the size of the actual stereographic map is partially determined by the standard latitude used when making the projection. If the plane is large enough, the entire sphere could be shadowed upon it, except for the South Pole, which would be a line at infinity.
Figure 3-4. For a given sized globe, we see that a projection on a plane (shown by the line nm) at the north pole is larger than the projection on the plane (shown by the line op) at latitude 60 °
What is a Straight Line on the Polar Stereographic Map?
Every straight line on the polar stereographic map is actually part of a circle on the sphere. This is true because any circle that goes through the South Pole will project as a straight line. To understand this better, consider the following: when a straight pencil (used to represent a line) is placed between the source of light at the South Pole and the tangent plane, a shadow of a straight line will be projected on the plane; except when the pencil is parallel to the light rays, and then you'll see only a point projected. In geometry, a point and a straight line can lie on only one plane; in this case the light at the South Pole is considered to be the point, and the pencil is the straight line. This plane will always cut out a circle (which looks like a straight line on the map} on the sphere. The arc of any circle north of the Equator is the only part which will show, since the Equator is usually the outer boundary of most weather maps. All great circles that touch the South Pole will project as lines of longitude, just like AEC in Figure 35. Every other straight line on the map (or off
the map), regardless of length, orientation, or distance apart can be considered as physically representing the projection of an arc of a small circle (on the Earth's surface) that touches the South Pole-for example: HI and FG in Figure 35.
Figure 3-5. The polar stereographic map usually ends at the equator. The circle drawn with a thick line represents the equatorial circle of a polar stereographic map. Here we see how parts of some of the small circles and great circles would look if part of the southern hemisphere were to be shown on the map by extending the map plane past the equatorial circle.
When two straight lines intersect on the flat map, regardless of length, they can be considered as the intersection of the arcs of two different circles on the surface of the sphere. The plane angle of intersection measured on the map will be an exact measure of the angle of intersection between these two circles. This is also true for lines that do not even cross in the boundaries of the map, as can be seen in Figure 3-6.
Figure 3-6. It is sometimes necessary to extend the line off the map past the Equatorial circle to measure the angle between two straight lines (provided they are not parallel). The two lines, AB and CD, meet at O to give the angle AOC.
Straight lines on the polar stereographic map can be interpreted in three different ways:
Stereographic maps are used not only in weather, but in many other fields. They are used for mapping crystal faces in crystallometry, and the mapping of geological structures in the field of geology. They have used straight lines on their maps for over a century. This is the first time that attention has been focused on the characteristics of a straight line on a polar stereographic map in meteorology. The significance of straight lines will be seen in the analysis we will make of weather patterns later on in the text.
The Nature of Pattern Distortions Created by the Polar Stereographic Map
What is the actual distortion of measurements on the polar stereographic map? To find the amount of distortion at any latitude, we must first find the actual radius of that latitude from the Earth's north-south axis, and then compare this with the radius as measured on the map. To derive the formula for the actual radius, see Figure 3-7, where nb is the radius of the latitude, and NB is the projected radius. Let Ø (phi) be the
Figure 3-7. S is the point of projection, N is the north pole, line XY is in the plane of projection, and the latitude Ø(phi) is 50°N. The projected radius from the north-south axis, NB, is longer than the actual radius, nb. This difference in length increases as you go to lower latitudes, and decreases as you go to higher latitudes.
latitude for any point on the Earth. Then, (theta) is the angle of the co-latitude. Co-latitude is defined as being equal to 90° minus the latitude Ø (phi). Let XY represent the plane which is tangent to the North Pole at N, b is any point on the meridian NESW at latitude ¢(phi), where the colatitude is (theta}, or angle bON. By simple geometry, the angle bON equals two times angle bSN. NO equals R-the radius of the Earth.
Referring to Figure 3-2, the radius at the points a, b, c, . . . T. can be found similarly. The projection of the latitude circles on the polar stereographic map is made by drawing circles with radii Na, Nb, Nc, . . . Nt. a, b, c, . . . T. are 10°, 20°, 30°, . . . 90°, distant from the North Pole so as to correspond to latitudes of 80°, 70°, 60°, . . . 0°, while the angles NSa, NSb, NSc, . . . NST, are 5°, 10°, 15°, . . . 45°, respectively.
We get Table I by substituting different values of latitude in Equation 3-1. The length of the radius for different latitudes of the projection in terms of the radius of the Earth is shown in Column E of Table I below, the length of the actual radius on the Earth for the different parallels is shown in Column D.
Now let's look at some of the distortions created by the map. On the Earth, the circumference of a circle of latitude at Ø (phi) is 2 PI RcosØ . On the projection, the circumference of this same circle becomes 4 PI R tan (45 - 1/2Ø) . The error in scale of the circles can thus be obtained by their ratio: