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The earthly globe is spherical; more precisely, it is an oblate spheroid, like a ball slightly flattened at the poles. More precisely still, it is a triaxial ellipsoid that closely approximates a “geoid”, a surface of constant gravitational potential [TM199 or search for “thatsmaths” at irishtimes.com].
Transverse Mercator projection with central meridian at Greenwich.
Continue reading ‘Ireland’s Mapping Grid in Harmony with GPS’
On a glorious sunny June day we reached the summit of Céidín, south of the Glen of Imall, to find a triangulation station or trig pillar. These concrete pillars are found on many prominent peaks throughout Ireland, and were erected to aid in surveying the country [see TM142, or search for “thatsmaths” at irishtimes.com].
Trig pillar on summit of Croaghan Moira, Wicklow [Image from https://mountainviews.ie/%5D.
Continue reading ‘Trigonometric Comfort Blankets on Hilltops’
If you fly 14,500 km due westward from New York you will come to Beijing. The two cities are on the fortieth parallel of latitude. However, by flying a great circle route over the Arctic, you can reach Beijing in 11,000 km, saving 3,500 km and much time and aviation fuel. [TM124 or search for “thatsmaths” at irishtimes.com].
Great circle route from New York to Beijing (gnomonic projection).
On a gnomonic projection (as above) each point on the Earth’s surface is projected from the centre of the Earth onto a plane tangent to the globe. On this map, great circles appear as straight lines.
Continue reading ‘From Sailing on a Rhumb to Flying on a Geodesic’
In a nutshell: In web maps, geographical coordinates are projected as if the Earth were a perfect sphere. The results are great for general use but not for high-precision applications. 
Continue reading ‘Maps on the Web’
Bragging about mathematical ineptitude is not cool. There is nothing admirable about ignorance and incompetence. Moreover, everyone thinks mathematically all the time, even if they are not aware of it. Can we all do maths? Yes, we can! [See this week’s That’s Maths column (TM064) or search for “thatsmaths” at irishtimes.com].
When you use a map of the underground network, you are doing topology.
Steiner’s minimal tree problem is this: Find the shortest possible network interconnecting a set of points in the Euclidean plane. If the points are linked directly to each other by straight line segments, we obtain the minimal spanning tree. But Steiner’s problem allows for additional points – now called Steiner points – to be added to the network, yielding Steiner’s minimal tree. This generally results in a reduction of the overall length of the network.
![Solution of Steiner 5-point problem with soap film [from Courant and Robbins].](https://thatsmaths.files.wordpress.com/2015/01/steiner-5point.jpg)
A solution of Steiner 5-point problem with soap film [from Courant and Robbins].
Special Curves
A large number of curves, called special curves, have been studied by mathematicians. A curve is the path traced out by a point moving in space. To keep things simple, we assume that the point is confined to two-dimensional Euclidean space 
![{\mathbf{\gamma} : [a,b]\longrightarrow \mathbb{R}^2}](https://s0.wp.com/latex.php?latex=%7B%5Cmathbf%7B%5Cgamma%7D+%3A+%5Ba%2Cb%5D%5Clongrightarrow+%5Cmathbb%7BR%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002)
In the 1820s Carl Friedrich Gauss carried out a surveying experiment to measure the sum of the three angles of a large triangle. Euclidean geometry tells us that this sum is always 180º or two right angles. But Gauss himself had discovered other geometries, which he called non-Euclidean. In these, the three angles of a triangle may add up to more than two right angles, or to less.
10 Deutschmark currency note (front)
Continue reading ‘Gauss’s Great Triangle and the Shape of Space’
The article in this week’s That’s Maths column in the Irish Times ( TM035 ) is about the remarkable Christmas Eve journey of Santa Claus.
Reports of the length of Ireland’s coastline vary widely. The World Factbook of the Central Intelligence Agency gives a length of 1448 km. The Ordnance Survey of Ireland has a value of 3,171 km (http://www.osi.ie). The World Resources Institute, using data from the United States Defense Mapping Agency, gives 6,347km (see Wikipedia article [3]).
ThatsMaths 