Mathematics in Traffic Control

Air Traffic Control

• Air traffic controlling involves navigating aircrafts for safe take offs & landing.

• This involves the use of a lot of mathematical tools like Vectors, Trigonometry etc.

Introduction to ATC

Vectors

• Vectors are quantities with defined magnitude & direction.

• Vectors are greatly used in air traffic control.

• For example: we can find the angle of drift between ground & air vectors using the relation of direction of the 2 vectors.

Aerial Field in 3D

Trigonometry

Trigonometry• Trigonometry becomes a great aid to

calculate distances.• Using the knowledge of trigonometry we can

find things like the distance of the signals.• It helps planes to send radio signals to the

ground station.

Spherical Geometry • It is used in evaluating the coordinates of the sphere, thus integral in navigation.

Calculus

• Calculus is widely used in finding various things like velocity, solutions of problems involving variations in distances.

• Suppose if we want the rate at which 2 planes move to avoid collision we can use the concept of instantaneous change in position.

Radar

• RADAR is a very useful in finding airplanes in airspace, It is used to detect obstacles like storms, It is used in landings, takeoffs during unfavorable conditions.

• The equation of the power of the radar is given by:

Road Safety

• Road safety is one of the major concerns of the traffic polices all over the world, especially in India.

• It is observable that the structures, methods of construction etc. are the major reasons the such accidents.

• Mathematicians have developed certain algorithms for road safety, of which one is discussed.

THE MATHEMATICAL MODEL OF CITY TRAFFIC

• For analysis of local route planning algorithms, a mathematical model of city traffic is used.

• The model is:

• It uses the 4D coordinate system to define the area & hence can find the set of coordinates of the form:

• 𝑠 = ( , , , ).𝑥 𝑦 𝜃 𝑣• Where, for object: : × → .𝐽 𝑆 𝑆 ℝ• It can help find things like distance & can also find the permissible

speed of the object.

Doppler Effect

• Doppler effect is a very common & efficient way to find the speeds, paths of satellites, position of the planes.

Δλc=λ˳ν

Banked Turns

• The bank angle is the angle at which the vehicle is inclined about its longitudinal axis with respect to its path.

• The formulae for velocity of a car at a banked turn of angle θ is given by