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Pythargoras Theorem

Issue No 22, 3 April 2023

By: Anthony O. Ives

Pythargoras theorem enables you to determine the length of the remaining side of a right angle triangle if you already know the length of the other two sides. Pythargoras theorem is an important theorem of trigonometry [1]. Like trigonometry it is important for a wide range of applications including engineering, vector analysis, navigation, architecture, etc. The distance formula for cartesian co-ordinates is also derived from pythargoras theorem. The distance is used in a wide range of applications including navigation and is very important for modern satelite navigation systems such as GPS Navstar [2].

The equation for pythargoras theorem is given below:

\[R = \sqrt{x^2 + y^2}\]

Where R is the length of hypotenuse side of the triangle, x is the length of adjacent side of the triangle and y is the length of the opposite side of the triangle. The hypotenuse, adjacent and opposite sides of a triangle as well x, y and R are graphically defined in the diagram below:

Trigonometry Diagram

The pythargoras equation can be manipulated so that say 'y' can be determined from 'R' and 'x' or 'x' can be determined from 'R' and 'y'. The two equations for 'x' and 'y' are given below:

\[x = \sqrt{R^2 - y^2}\]

\[y = \sqrt{R^2 - x^2}\]

Pythargoras theorem can be used to derive an equation that is used to determine the distance between two points [3] in space as in the diagram below:

Distance between Two Points in a 2D Space

The equation for determining the distance between to two points in a 2D space are given below:

\[s_{2D} = \sqrt{ \left( x_1 - x_2 \right)^2 + \left( y_1 - y_2 \right)^2}\]

Where s_2D is the distance between the two points in a 2D space, x₁ is the x coordinate of the first point, x₂ is the x coordinate of the second point, y₁ is the y coordinate of the first point and y₂ is the y coordinate of the second point.

The equation for distance can be extended to a 3D space and would look like the following for a 3D space:

\[s_{3D} = \sqrt{ \left( x_1 - x_2 \right)^2 + \left( y_1 - y_2 \right)^2 + \left( z_1 - z_2 \right)^2}\]

Where s_3D is the distance between the two points in a 3D space, z₁ is the z coordinate of the first point, z₂ is the z coordinate of the second point.

Global Navigation Satelite Systems (GNSS) use a ground receiver which recieves the distance between the receiver and a number of satelites. The position of each of the satelites is known and transmitted to the reciever using this information it can be seen that you could calculate your position using the distance equation. Essentially you want to find out your own positon which is your x, y, z coordinates that is three unknowns. You know the distance from your position to each satelite giving you three distances as well as the x, y, z coordinates for each satelite. Using this information you can setup three equations with three unknowns and with a bit of complex simulateous equation manipulation you can determine your x, y, z coordinates and hence your position. This is simplfied version of the procedure that Global Navigation Satelite Systems (GNSS) use however actual Global Navigation Satelite Systems (GNSS) use more a complex calculation method and require a minimum of 4 satelites to determine your position [2] where as you may have noticed the simplfied procedure requires only three satelites. The fourth satelite is required to calcuate a time which is mainly to do with accuracy just using three satelites would in reality produce large errors in the position calculation.

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References:

[1] http://www.eiteog.com/EiteogBLOG/No12EiteogBlogTrig.html

[2] Instrumental Flying Handbook, FAA-H-8083-15B, 2012, United States Department of Transportation, Federal Aviation Administration, https://www.faa.gov/sites/faa.gov/files/regulations_policies/handbooks_manuals/aviation/FAA-H-8083-15B.pdf

[3] Engineering Mathematics, K.A. Stroud, Fourth Edition, 1995, Macmillan

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