Classwork Exercise and Series (Mathematics-SS3): Spherical Geometry

Concept and Location of Longitude and Latitude

Lines of latitude run East & West or horizontal but measure distance North & South of the Equator—vertically. The equator is labeled as zero degrees latitude. The greatest amount of latitude is 90 degrees at the North or South poles. We can then label our equator as 0° and our North and South poles as 90°. These lines of latitude are parallel to the equator and are even referred to as “parallels” or “parallels of latitude”.

Moving to lines of longitude, they run perpendicular to lines of latitude. That is, longitude lines run North and South but measure East and West of zero degrees longitude which is a line called the Greenwich Meridian or Prime Meridian. This arbitrary north/south line was marked by the British in the 17th century and runs through a town just outside of London called Greenwich.

Starting at the Prime Meridian, we measure the earth east or west with these north/south-running lines called “meridians.” We can measure halfway around the world till these meridians meet at 180 degrees. This meridian line at 180° east or west is called the International Date Line. So unlike latitude, where the greatest or maximum latitude is 90° at either the north or south poles, the greatest amount of longitude is 180°—halfway around the world from the prime meridian. One other important way these longitude lines differ from parallel latitude lines is that lines of longitude are not parallel, and in fact converge at both the North and South poles.

Latitude is measured by the angle at the centre of the earth between the equator and a radius of the earth.

Look at the lines running left to right on the map. These are the horizontal lines of latitude. All lines of latitude run parallel to the equator. The equator has a degree rating of zero. Lines running above the equator are north degrees, while lines running below the equator are south degrees.

Inspect the lines running top to bottom, or vertically. These are lines of longitude. Longitude lines run parallel to the prime meridian. The prime meridian, like that of the equator, has a degree rating of zero. Lines to the west of the prime meridian are west degrees, while lines to the east are east degrees

Angular Difference

The angle subtended at the centre of the great or small circle by the minor arc formed by joining two places on the great or small circle respectively is known as the angular difference between the two places.

Note At this level, these angular differences involve places usually along the same latitude or same longitude.

On the same Longitude

Find the angular difference between the following pairs of places on the earth’s surface.

(i) E (30⁰N, 45⁰E) and F (25⁰S, 45⁰E)

Angular difference between E and F = 30 + 25 = 55⁰

On the same Latitude

Find the angular difference between the following pairs of places on the surface of the earth.

(i) A (40⁰N, 35⁰W) and B (40⁰N, 50⁰E)

Solution

35⁰ + 50⁰ =85⁰

i.e. the angular difference between A and B = 85⁰

Distance along Great Circles

Recall that great circles are circles are lines of longitude and the equator, while the small circles are the other parallels of latitudes. So distances along great circles are distances along the longitude and the equator.

However, there are other great circles formed by planes which pass through the centre of the earth but are not necessarily perpendicular to the equator.

The distance (i.e. shortest distance) D between two points along a great circle on the earth’s surface is given by the length of the arc subtending an angle at the centre of the earth. Thus D = q/360 x 2πr is the distance where R is the radius of the earth and q is the angular difference between the two points.

Radii and Lengths of Latitudes

Latitude is an angular measurement that denotes a position on the surface of the planet with respect to the equator, or the major axis that runs east and west and divides the planet into the Northern and Southern hemispheres. When the Earth’s radius points to any position above or below the equator, and is revolved about the major north to south axis of the planet, the radius traces out a smaller circle that runs parallel to the equator. This smaller circle is the circle whose radius may be used to determine latitude.

Recall that the radii of parallel of latitude decrease as one moves away from the equator towards the poles. The radius of the equator is the largest, and the radii at North and South poles zero.

Draw a diagram of the planet clearly marking the equator with a horizontal line that circles about the planet in the plane of the midpoint between the North and South Poles. Sketch the radius of the circle as a straight line that extends from the perimeter of the circle to the mid-point of the entire sphere at a zero degree angle. Label the radius as “R”.

Draw a circle of latitude in the same way that you sketched the equator in Step 1, except with the plane translated upward or downward from the equator such that it traces out a plane that runs parallel to the plane marked by the equator. The circle of latitude should clearly show a smaller perimeter than that of the circle denoted by the equator.

Draw the radius of the circle of latitude. This line will be located in the plane traced out by this circle, such that the radius extends from the perimeter to the north-south planetary axis at a zero degree angle. Label this radius as “r”.

Sketch the radius of the Earth again as a straight line that connects the mid-point of the planetary sphere to the perimeter of the circle of latitude that you drew in Step 2. Label the angle between this radius (R) and the circle of latitude radius (r) as q. This quantity is the latitude.

Write the relationship among R, r and q. From basic trigonometry, it can be seen from our diagram that r = R(cos(q)) when considering the triangle formed between r, R and the segment of the north-south axis that connects these two segments. We can solve this equation for q using basic algebra, ending up with q = arccos(r/R), where “arccos” is the inverse cosine function. Plug the given value for the latitude radius and the radius of the Earth (6,400 kilometers) into the inverse cosine equation to calculate q.

d = a/360 x 2πr

d = a/360 x 2πRcosq.

We use this formula to find the distance along a line of latitude between two points. In a special case when the latitude is the equator

D = d = a/360 x 2πR

The Nautical Mile

The distance of the arc of a great circle which subtends an angle of 1 minute (1’) at the centre of the earth is called a nautical mile.

So if the radius R of the earth is 6400 km, then

     1 nautical mile      = 1’/360⁰ x 2 x π x R

                                     = (1/360 x 60) x 2 x 3.142 x 6400

                                     ≡ 1.862 km

 1 nautical mile      ≡ 1.86 km

     Recall that 1 mile ≡ 1.6 km

 1 nautical mile      = 1.86/1.6 mile

                                     = 1.163 miles

 1 Nautical mile      ≡ 1.16 miles

This shows that 1 nautical mine is longer than 1 ordinary mile. A speed of 1 nautical mile per hour is called 1 knot. This is widely used in air and sea navigation.

So we can say that a ship has a speed of 100 knots. That means a speed of 100 nautical miles per hour which will be equivalent to (1.862 x 100km/hr) 186.2 km/hr.

World Time

The surface of the time is divided into 24 time zones. Each zone represents 15⁰ of longitude or one hour of time. The passage of time follows the path of the sun in a westerly direction so that countries to the east of London and the Greenwich meridian are ahead of Greenwich Mean Time GMT and countries to the west are behind. All times given are based on 1 p.m. in Lagos but there may be inconsistencies of one or two hours in some cities caused by seasonal use of daylight saving.

Questions

1. A ship sails from point A(40⁰N, 28E) to point B(40⁰N, 25⁰W) along Lat 40⁰N and then to point C(15⁰S, 25⁰W) along Long 25⁰W. Take radius of earth R = 6400km and π 3.142. Calculate

1. The distance from A to B and then to C in nautical miles

A. 5003 nautical miles B. 5745 nautical miles C. 5674 nautical miles D. 4573 nautical miles

2. The average speed of the ship if the whole journey takes 20 hrs.

A. 287 Knots B. 257 Knots C. 327 knots D. 357 Knots

3. Find the angular difference between the following pairs of places on the surface of the earth

C(300⁰ S, 20⁰ E) and D(30⁰ S, 60⁰ E)

A. 50⁰ B. 60⁰ C. 40⁰ D. 45⁰

4. Find the distance between the following pairs of points on the earth’s surface. Take radius of earth R = 6400km C(40⁰N, 15⁰E) and D(10⁰N, 15⁰E)

A. 3574.45 km B. 3352.38 km C. 4253.40 km D. 4532.38 km

5. Find the angular difference between the following pairs of places on the earth’s surface

A. 0⁰ B. 27 C. 29⁰ D. 30⁰

Answers

1.  B  2.  A  3.  C  4.  B  5. C