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Distances Along Great Circles & Small Circles - SS3 Mathematics Lesson Note

DISTANCES ALONG GREAT CIRCLES

Recall, great circles are the lines of longitude and the equator whilst small circles are the other parallels of latitude.

The shortest distance D between two points along a great circle on the earth surface is given by the length of the arc subtending an angle at the center of the earth. Thus, D=θ360×2πR, where R is the radius of the earth and θ is the angular difference between the two points.

Example 3 Find the distance between this pair of points on the earth surface: A(30N,  40W) and B(30S, 40W). Take R=6,400km

Solution

Angular difference = 30+30=60

Distance between the points=θ360×2πR

Distance between the points=60360×21×227×64001=6,704.76 km

DISTANCES ALONG SMALL CIRCLES

Recall as one moves from the equator along the parallels of latitude to the north and south poles, their radii decreases. The equator has the largest radius whilst the north and south poles have effective radii of zero.

The distance along small circles, d=α360×2πRcosθ, where R is the radius of the earth, α is the angular difference between the points and θ is the common latitude of both points.

Example 4 Find the distance between this pair of points on the earth surface: P(28S,  20E) and Q(28S, 60E). Take R=6,400km and π=3.142

Solution

Given P(28S,  20E) and Q(28S, 60E):

d=α360×2πRcosθ

α=6020=40

R=6,400km

π=3.142

θ=28

d=40360×2×3.142×6400×cos28=3946km

 

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