• The Concept of Frame of Reference
• Position as a Function of Axes
• Distance
• Calculation on Distance

Frame of reference is a concept used for determining the position of a point or object in space. Any frame of reference comprises of at least two axes: the horizontal, vertical. The concept is used in Physics, Mathematics, Geography, etc.

In Physics and Mathematics, the frame of reference is called the Cartesian plane which comprises basically two axes: x and y axes. As shown below:

The horizontal (x) axes increases to the right and the vertical (y) axes increases upwards. This cartesian frame is also applied in computer graphics, for example: CorelDRAW, to show the printable area in the workspace/drawing area.

In Geography, the frame of reference is the 0^o latitude (equator) and the 0^o longitude (meridian). Every position on the earth surface is represented as a combination of latidude and longitude from the equator and meridian respectively.

Let's consider the cartesian below:

The point A is the intersection of where x = 3 and y = 4, thus, we represent the point A as (3,4).
The point B is the intersection of where x = 6 and y = 10, thus, we represent the point B as (6, 10).
The point C is the intersection of where x = 18 and y = 6, thus, we respresent the point C as (18, 6).

Let's consider the map:

The point A is the intersection of latitude = 5^oN and longitude = 7^oE, thus, we represent the point A as 5^oN, 7^oE.
The point B is the intersection of latitude = 4.5^oN = 4^o30'N and longitude = 7^oE, thus we represent the point B as 4^o30'N, 7^oE.
The point C is the intersection of latitdue = 4.5^oN = 4^o30'N and longitude = 7.5^oE = 7^o30'E, thus, we represent point C as 4^o30'N, 7^o30'E.

Distance is the space between two points. This could be on a flat surface or a spherical surface.

For flat surfaces, the formula is:

For spherical surfaces, the formula is:

Q1. Find the distance between points A (3, 10) and B (5, 9).

$x_1=3$; $x_2=5$
$y_1=10$; $y_2=9$
$d=\sqrt{{(\Delta x)}^2+{(\Delta y)}^2}=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$
$d=\sqrt{{(5-3)}^2+{(9-10)}^2}=\sqrt{2^2+{\left(-1\right)}^2}$
$d=\sqrt{4+1}=\sqrt{5}$

Q2. Find the distance between Towns A (3^oN, 10^oE) and B (5^oN, 9^oE) on a sphere.

$a={\sin }^2\left(\frac{9-10}{2}\right)+\cos \left(10\right)\cdot \cos \left(9\right)\cdot {\sin }^2\left(\frac{5-3}{2}\right)$
$a={\sin }^2\left(0.5\right)+\cos \left(10\right)\cdot \cos \left(9\right)\cdot {\sin }^2\left(1\right)$
$a=0.00007615+0.00029627$
$a=0.00037242$
$d=2\cdot R\cdot {sin}^{-1}\left(\sqrt{a}\right)$
$d=2\cdot R\cdot {sin}^{-1}\left(\sqrt{0.0003724}\right)$
$d=12742\cdot 0.01929$
$d=245.89\text{ km}$

• The position of an object is determined with respect to the frame of reference.
• The position of an object on a flat surface is represented based on the x and y axes as (x, y).
• The position of an object on spherical surace is represented based on the latitudes and longitudes e.g. (N, E), (S,E).
• Distance is the space between two positions or points.
• To calculate distances on a flat surface, we use:
  $$d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$$
• To calculate distances on a spherical surface (e.g. the earth), we use:
  $$d=2R⋅sin^{-1}(\sqrt{sin^2(\frac{Φ_2-Φ_1}{2})+cos(Φ_1)⋅cos(Φ_2)⋅sin^2(\frac{λ_2 - λ_1}{2})})$$