Question

Distance from Palo Alto to Beijing. The surface of the earth is reasonably approximated as a sphere with radius R = 6367.5km. A location on the earth’s surface is traditionally given by its latitude θ and its longitude λ, which correspond to angular distance from the equator and prime meridian, respectively. The 3-D coordinates of the location are given by

Answer 1

The 3-D coordinates of the location of a point on the earth's surface is given by (R₀ₐ ×cosλ ×cosθ, R₀ₐ×sinλ×cosθ, R₀ₐ×sinλ) where λ =longitude and θ = latitude

and the distance between points on the surface such as two cities is given by P₁ = (x₁, y₁, z₁) and P₂ = (x₂, y₂, z₂) = |P₁ P₂| = √((x₂-x₁)² + (y₂-y₁)² + (z₂ - z₁)²)

Explanation:

If the center of the earth is marked O the location of a point is given in longitude and latitude as λ and θ respectively then where normally we have

-180 ° ≤ λ ≤ 180 ° and 90 ° ≤ θ ≤ 90°

However in cartesan coordinates, considering the radius of the earth and specifying the location of a point on the surface of the earth in three dimensions of the x, y and z coordinates with distance from the earth center = R where R = |Oa| = 6367.5km and the center of the Earth is O, then a point, a on the surface of the earth has the x, y and z Cartesian coordinates we have

a = (xₐ, yₐ, zₐ) = (R×cosλ×cosθ, R×sinλ×cosθ, R×sinλ)

Hence the 3-D coordinates of a location is given by

(R₀ₐ ×cosλ ×cosθ, R₀ₐ×sinλ×cosθ, R₀ₐ×sinλ) with R₀ₐ = the distance of the point a to the center of the earth

Distance between two points is given by P₁ = (x₁, y₁, z₁) and P₂ = (x₂, y₂, z₂) = |P₁ P₂| = √((x₂-x₁)² + (y₂-y₁)² + (z₂ - z₁)²)