Before too long, we will be relieved of the burden of long-distance driving. Given the desired destination and access to a mapping system, electronic algorithms will select the best route and control the autonomous vehicle, constantly monitoring and adjusting its direction and speed of travel. The origins of the methods used for autonomous navigation lie in the early 1960s, when the space race triggered by the Russian launch of Sputnik I was raging [TM214 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Kalman Filters: from the Moon to the Motorway’
we can usually find 
, whereas we may not be able to find 
.
“
, the ratio of the circumference of a circle to its diameter. We review several historical methods and describe a recently-discovered and completely original and ingenious method.
, the length of the major axis. The gardener puts down two stakes and loops a piece of rope around them. Using a stick, he pulls the loop taut, marking the points around a curve. This is illustrated here.
Ireland has four provinces, the principal city in each being the provincial capital: Belfast, Cork, Dublin and Galway. The map here shows the location of these cities. Now imagine a company that needs to visit and to deliver goods frequently to all four cities. Where might they locate their HQ to minimize transport costs and travel times?
is the position of the HQ and 
are the positions of the cities.
![Solution of Steiner 5-point problem with soap film [from Courant and Robbins].](https://thatsmaths.files.wordpress.com/2015/01/steiner-5point.jpg)
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