Waypoint Encounter: the geometry involved in navigating an aircract or ship towards an en-route way point, including the implementation of a demonstration applet.
Barges follow the course of the rivers and canals on which they float. These naturally lead them to their destinations. Trains are guided by the tracks on which they travel. Cars and trucks follow roads. All they have to do to reach their destinations is to take the right turn at each junction along the way. But ships at sea and aeroplanes have no permanent way to guide them. They rely on natural geographical features, lighthouses and other artificial navigational aids to guide them on their journeys.
For a ship, aeroplane or off-road vehicle a journey is made up of a sequence of way points. A way point is simply a point on the Earth's surface. It may be the location of a hill, mountain, lake, town or some other geographical feature. It may be the location of a lighthouse or radio navigation station. Or it may be simply a latitude and longitude recorded in the memory of a G.P.S. [global positioning system] receiver or a futuristic astral navigator which fixes positions using signals from pulsars.
The featureless separation between way points plays no part in guiding the traveller to his destination.
This project is concerned essentially with the navigation of aircraft. They are capable of flying over any part of the Earth's surface at high speed. This allows them to follow paths much closer to mathematical ideals than can other kinds of vehicle. There is a strong tendency to think the ideal path for an aircraft to follow to be a succession of hyperbolic encounters with its way points - a path rather like that of a small positively charged particle weaving between a sequence of large stationary particles which are also positively charged:
However, aircraft are not ideal particles or planets moving in free space: they are vehicles battling through a rather dense complex dynamical atmosphere. For them, therefore, the ideal - the most economical - path is not a series of hyperbolae, but one of straight lines joined by small circular arcs whose radii are equal to the radius of the aircraft's minimum comfortable turning circle.
The aircraft's route is set by the radial on which the aircraft must leave each way point. Consequently the aircraft must fly over the way point in the direction of the next way point en-route. To achieve this, we construct a circle passing through the way point we are approaching such that the tangent to the circle at the way point is the direction of the next way point.
All angular computations are then performed relative to the centre of the constructed circle. The aircraft never passes near the centre of the circle. Nasty infinities therefore can never occur to cause trigonometrical confusion. If the aircraft drifts off course, a new circle is constructed, thus maintaining the integrity of the trigonometry.
As computer software, this could form the basis of an automatic flight control system of the future. Integrated with international flight plan processing, it could form the basis of a completely automated system of global air highways.
I have designed, programmed and exhaustively tested a software kernel to provide this navigational functionality. I have coded part of it in Java and installed it herewith as a demonstration applet to give some idea of my capabilities in this area. However, please be aware that a vast amount of applied expertise is needed (hopefully some of mine) to transform the core functions demonstrated in the applet into a safe automatic flight control system.
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© 1997 Robert John Morton